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\section{Introduction}
For the past two decades, additivity conjectures have been extensively
studied in quantum information theory e.g.
\cite{Bennett_EtAl, Pomeransky, AmosovEtAl, Osawa, Shor,
hayden_winter_2008}.
In this paper, we concentrate on the issue of additivity of classical
Holevo capacity of a quantum channel $\Phi$, denoted henceforth by
$C(\Phi)$.
The quantity $C(\Phi)$
is the number of classical bits of information per channel use that
can reliably be transmitted in the limit of infinitely many independent
uses of $\Phi$. Capacities of classical memoryless
channels are known to be additive, that is, the capacity of two
channels $\Phi$ and $\Psi$, used independently, is the sum of the
individual capacities. In other words,
$C(\Phi \otimes \Psi) = C(\Phi) + C(\Psi)$.
This additivity property leads to
a single letter characterization of the capacity of classical channels
viz. the capacity is nothing but the mutual information between the
input and channel output maximised over all possible input distributions
for one channel use
\cite{vajda_shannon_weaver_1950}.
For a long time, in analogy with the classical setting, it was generally
believed that the classical Holevo capacity of a quantum channel is
additive. In fact, this belief was proven to be true for several classes
of quantum channels e.g.
\cite{KingUnital, FujiwaraEtAl, KingDepolarising, ShorEntang,
KingEtAl}. Thus, it came as a major surprise to
the community
when Hastings, in a major breakthrough, showed that there are indeed
quantum channels with superadditive classical Holevo capacity
\cite{hastings_2009} i.e. there are quantum channels
$\Phi$, $\Psi$ such that $C(\Phi \otimes \Psi) > C(\Phi) + C(\Psi)$.
Hastings' proof proceeds by showing that a
Haar random unitary leads to such channels with high probability,
in the sense that the unitary, when viewed suitably, is the
Stinespring dilation of a quantum channel with superadditive classical
Holevo capacity.
The drawback of using Haar random unitaries is that they are inefficient
to implement. In fact, it takes at least
$\Omega(n^2 \log (1/\epsilon))$
random bits in order to pick an $n \times n$ Haar random unitary to
within a precision of $\epsilon$ in the $\ell_2$-distance
\cite{Vershynin}. Hence, it is of considerable interest to find an explicit
efficiently implementable unitary that gives rise to a quantum channel
with superadditive classical Holevo capacity.
In this paper, we take the first
step in this direction. We show that with high probability
a uniformly random $n \times n$ unitary from an
approximate $n^{2/3}$-design
leads to a quantum channel with
superadditive classical Holevo capacity. Though no efficient algorithms for
implementing approximate $n^{2/3}$-designs are known, nevertheless,
it is known that a uniformly random unitary from an exact
$n^{2/3}$-design can be sampled using
only $O(n^{2/3} \log n)$ random bits \cite[Theorem 3.3]{Kuperberg}.
Also, efficient constructions of approximate $(\log n)^{O(1)}$-designs
are known \cite{sen:zigzag, brandao2012local}.
Thus, our work can be viewed as
a partial derandomisation of Hastings' result, and a step towards the
quest of finding an explicit quantum channel with superadditive classical
Holevo capacity.
Hastings' proof was considerably simplified by Aubrun, Szarek and
Werner~\cite{aubrun_szarek_werner_2010_main} who showed that existence
of channels with subadditive minimum output von Neumann entropy follows
from a sharp Dvoretzky-like theorem which states that, under the
Haar measure, random subspaces of large dimension make a Lipschitz function
take almost constant value.
Dvoretzky's original theorem \cite{Dvoretzky}
stated that any centrally
symmetric convex body can be embedded with low distortion into a
section of a high dimensional unit $\ell_2$-sphere.
Milman \cite{Milman} extended Dvoretzky's theorem by proving that,
with high probability, Haar random subspaces of an appropriate dimension
make a Lipschitz function take almost constant value. Dvoretzky's theorem
becomes the special case of Milman's theorem where the Lipschitz
function happens to be norm induced by the centrally symmetric convex
body i.e. the norm under which the convex body becomes the unit ball.
Milman's work started a whole body of research sharpening the various
parameters of the extended Dvoretzky theorem
e.g. \cite{Schechtman, Gordon} etc. However, all these works
use Haar random subspaces. A Haar random subspace of $\mathbb{C}^n$ of
dimension $d$
can be obtained by applying a Haar random unitary to a fixed subspace
of dimension $d$ e.g. the subspace spanned by the first $d$ standard
basis vectors of $\mathbb{C}^n$. Our work is the first one to replace the
Haar random unitary in any Dvoretzky-type theorem by a uniformly
random unitary chosen from an
approximate $t$-design for a suitable value of $t$. In other words,
our main technical result is an Aubrun-Szarek-Werner style result for
approximate $t$-designs instead of Haar random unitaries.
As a corollary, we obtain the subadditivity of minimum output von
Neumann entropy for unitaries chosen from an approximate $n^{2/3}$-design.
As another corollary, we obtain the subadditivity
of minimum output R\'{e}nyi $p$-entropy for all $p > 1$ for
quantum channels arising from unitaries
chosen from an approximate unitary
$(n^{1.7} \log n)$-design.
Such a unitary can in fact be chosen from an exact
$(n^{1.7} \log n)$-design using only
$n^{1.7} (\log n)^2$ random
bits \cite{Kuperberg}, which is
much less than $\Omega(n^2)$ random bits required to choose a
Haar random unitary. Subadditivity
of minimum output R\'{e}nyi $p$-entropy for all $p > 1$
was originally proved for Haar random unitaries
by Hayden and Winter \cite{hayden_winter_2008}.
To prove our main technical result, we use a concentration of
measure result by
Low~\cite{low_2009} for approximate unitary $t$-designs,
combined with a stratified analysis of the variational
behaviour of Lipschitz functions on the unit sphere in high
dimension.
We need such a fine grained stratified analysis for the
following reason. Aubrun, Szarek and
Werner~\cite{aubrun_szarek_werner_2010_main} worked with the function
$f(M) := \lVert MM^\dag - (I/k) \rVert_2$, where the argument $M$
is a $k^3$-tuple rearranged to form a $k \times k^2$ matrix. They
found subspaces of dimension $k^2$ where $f$ took almost constant value.
For this, they had to do a two step analysis. The global Lipschitz
constant of $f$ was $2$ which, under naive Dvoretzky type arguments,
would only guarantee the existence of subpaces of dimension
$\frac{k^2}{\log k}$ where $f$ is almost constant. This does not
suffice to find a counter example to minimum output von Neumann
entropy. In order to shave off the $\log k$ term in the denominator,
they had to use several sophisticated arguments. One of them was the
observation that there is a high probablity subset $T$ of
$\mathbb{S}_{\mathbb{C}^{k^3}}$
on which the Lipschitz constant of $f$ was $k^{-1/2}$. They exploited
this by their two step analysis, where they separately analysed the
behaviour of $f$ on $T$ and on $T^c$, and managed to shave off
the $\log k$ term. For us, since we are working with designs, we
need the function to be a polynomial. Hence, instead of $f$, we have
to work with $f^2$. This seemingly trivial change introduces severe
technical difficulties. The main reason behind them is that the Lipschitz
constant of $f^2$ is about twice the Lipschitz constant for $f$
but the variation that we are looking to bound is around square of the
earlier variation! This contradiction lies at the heart of the technical
difficulty. In order to overcome this, we have to partition
$\mathbb{S}_{\mathbb{C}^{k^3}}$ into a number of sets
$\Omega_1, \Omega_2, \ldots, \Omega_{\log k}$,
called `layers',
with local Lipschitz constants for $f^2$ running as
$k^{-3/2}, 2^3 k^{-3/2}, 3^3 k^{-3/2}, \ldots, (\log k)^3 k^{-3/2}$.
We have to bound
the variation of $f^2$ individually on $\Omega_i$ as well as put
them together to bound the variation on large subspheres of
$\mathbb{S}_{\mathbb{C}^{k^3}}$. This leads to a challenging stratified
analysis, which
forms the main technical advance of this paper.
Another tool developed in this work which should find use in other
situations also, is a systematic way to approximate a monotonic
differentiable function and its derivative using moderate degree
polynomials. This tool is crucially used to prove strict subadditivity
of R\'{e}nyi $p$-entropy for any $p > 1$ for channels whose unitary
Stinespring dilation is chosen from an approximate design instead of
a Haar random unitary.
The power of our
stratified analysis shows up in the consequence that the dimension of
the subspace on which the Lipschitz function is almost constant depends
only on the smallest local Lipschitz constant, provided some mild
niceness conditions are satisfied. This gives larger dimensional subspaces
than a naive analysis which would depend on the global Lipschitz
constant. In fact, the stratified analysis
allows us to prove a sharper Dvoretzky-type theorem even for the
Haar measure. As a result, we can recover Aubrun, Szarek and Werner's
result for the function $f$ directly and elegantly
instead of applying their Dvoretzky-type result twice which is rather
messy. Another powerful consequence of our stratified analysis is that
with probability
exponentially close to one random, over Haar measure or $t$-design
measure, large subspaces
make the Lipschitz function almost constant. In contrast,
Aubrun, Szarek and Werner could only guarantee constant probability close
to one for the Haar measure, and they did not consider $t$-designs.
They also stated without providing details that the existence probability
could be made exponentially close to one using a deep Levy-type lemma for
unitary matrices. In contrast our stratified analysis uses only
the elementary Levy lemma for the unit sphere, yet it manages to prove
existence with probability exponentially close to one.
The rest of the paper is organised as follows. Section~\ref{sec:prelim}
contains notations, symbols
definitions and preliminary tools required for the
paper.
Section~\ref{sec:main} states and proves the main technical theorems viz.
the stratified analyses for Haar measure and approximate $t$-designs.
Section \ref{sec:vonNeumannentropy} describes the
application to subadditivity of minimum output von Neumann entropy.
Section~\ref{sec:Renyipentropy} describes the
application to subadditivity of minimum output R\'{e}nyi $p$-entropy
for $p > 1$. Section~\ref{sec:conclusion} concludes the paper and states
some open problems for future work.
\section{Preliminaries}
\label{sec:prelim}
All Hilbert spaces used in this paper are finite dimensional.
The $n$ dimensional space over complex numbers, $\mathbb{C}^n$, is endowed
with the standard inner product aka the dot product:
$\langle x, y \rangle := \sum_{i=1}^n x_i^* y_i$.
The unit radius sphere in $\mathbb{C}^n$ is denoted by $\mathbb{S}_{\mathbb{C}^n}$.
The symbol $\mathcal{M}_{k,d}$
denotes the Hilbert space of $k \times d$ linear operators over the
complex field under the Hilbert-Schmidt inner product
$\langle M, N \rangle := \mathrm{Tr}\, [M^\dag N]$,
and $\mathcal{M}_d := \mathcal{M}_{d,d}$.
Let $\mathcal{U}(n)$ denote the set of $n \times n$ unitary matrices with
complex entries.
For a composite Hilbert space
$\mathbb{C}^k \otimes \mathbb{C}^d$,
the notation $\mathrm{Tr}\,_{\mathbb{C}^d}[\cdot]$ denotes the operation of
taking partial trace i.e.
tracing out the mentioned subsystem $\mathbb{C}^d$. We use $\mathrm{Tr}\,[\cdot]$
to denote the trace of the underlying operator.
Fix standard bases for Hilbert spaces $A \cong \mathbb{C}^k$, $B \cong \mathbb{C}^d$.
Let $\ket{e_i}^A$, $\ket{e_i}^B$ denote standard basis vectors of
$A$, $B$ respectively. Any vector $x \in A \otimes B$ can be
written as $x = \sum_{i,j} \alpha_{ij} \ket{e_i}^A \otimes \ket{e_j}^B$.
We use $\mathrm{op}_{d \rightarrow k}(x)$ to denote the operator
$\sum_{i,j} \alpha_{ij} \ket{e_i}^A \otimes \bra{e_j}^B$
in $\mathcal{M}_{k, d}$. Conversely, given an operator
$M = \sum_{ij} m_{ij} \ket{e_i}^A \otimes \bra{e_j}^B$ in
$\mathcal{M}_{k,d}$, we let
$\mathrm{vec}(M) := \sum_{ij} m_{ij} \ket{e_i}^A \otimes \ket{e_j}^B$ denote
the vector in $\mathbb{C}^k \otimes \mathbb{C}^d$.
For Hermitian positive semidefinite operators $M$, we define
$M^\alpha$ for any $\alpha > 0$ to be the unique Hermitian
operator obtained by keeping the eigenbasis same and
taking the $\alpha$th power of the eigenvalues. We can define
$\log M$ similarly.
For $p > 1$, the notation
$\lVert M \rVert_p$ denotes the Schatten $p$-norm
of the matrix $M$, which is
nothing but the $\ell_p$-norm of the vector of its singular values.
Alternatively, $\lVert M \rVert_p = (\mathrm{Tr}\, [(M^\dagger M )^{p/2}])^{1/p}$.
Then $p=2$ gives the Hilbert Schmidt
norm aka the Frobenius norm which is nothing but
$\lVert M \rVert_2 = \lVert \mathrm{vec}(M) \rVert_2$.
Also, $p = \infty$ gives the operator norm aka spectral norm
which is nothing but
$
\lVert M \rVert_\infty =
\max_{v: \lVert v \rVert_2 = 1} \lVert M v \rVert_2.
$
Unless stated otherwise, the symbol $\rho$ denotes a quantum
state aka density matrix which is nothing but a Hermitian, positive
semidefinite matrix with unit trace. A rank one density matrix is called
a pure state. By the spectral theorem, any density matrix is a convex
combination of pure states.
The notation $\mathcal{D}(\mathbb{C}^d)$ denotes the convex
set of all $d \times d$ density matrices. We use $\ket{\cdot}$ to denote
a unit vector. By a slight abuse of notation, we shall often use a unit
vector $\ket{\psi}$ to denote a pure state $\ket{\psi}\bra{\psi}$.
A linear mapping $\Phi: \mathcal{M}_m \to \mathcal{M}_d $ is called
a superoperator. A superoperator is trace preserving if
$\mathrm{Tr}\, \Phi(M) = \mathrm{Tr}\, M$ for all $M \in \mathcal{M}_m$. It is said to be
positive if $\Phi(M)$ is positive semidefinite for all positive
semidefinite $M$. Furthermore, $\Phi$ is said to be completely positive
if $\Phi \otimes \mathbb{I}$ is a positive superoperator for identity
superoperators $\mathbb{I}$ of all dimensions.
Completely positive and trace preserving (CPTP) superoperators
are referred to as quantum channels.
Unless stated otherwise, $\Phi$, $\Psi$ are used to denote
quantum channels.
A compact convex set $\mathcal{S}$ in $\mathbb{C}^n$ is called a convex body.
The radius $r(\mathcal{S})$ of a convex body $\mathcal{S}$ is defined as
\[
r(\mathcal{S}) := \min_{x \in \mathcal{S}} \max_{y \in \mathcal{S}} \lVert x - y \rVert_2.
\]
Any point $x \in \mathcal{S}$ achieving the minimum above is said to be a
centre of $\mathcal{S}$.
The convex body $\mathcal{S}$ is said
to be centrally symmetric iff for every $x \in \mathbb{C}^n$,
$x \in \mathcal{S} \leftrightarrow -x \in \mathcal{S}$. The zero vector is a centre
of a centrally symmetric convex body. A centrally symmetric convex body
lying in $\mathbb{C}^n$
can be thought of as the unit sphere of a suitable notion of norm
in $\mathbb{C}^n$. Conversely for any norm in $\mathbb{C}^n$, the unit sphere under
the norm forms a centrally symmetric convex body.
\subsection{Entropies and norms}
\begin{definition}
The von Neumann entropy of a quantum state
$\rho$ is defined as
\[
S(\rho) := -\mathrm{Tr}\, [\rho \log \rho].
\]
For all $p > 1$, the R\'{e}nyi $p$-entropy of a quantum state
$\rho$ is defined as
\[
S_p(\rho) := \frac{1}{1-p} \log \mathrm{Tr}\, \rho^p =
-\frac{p}{p-1} \log \lVert \rho \rVert_p.
\]
\end{definition}
It turns out that
$
S(\rho) = \lim_{p \downarrow 1} S_p(\rho) =: S_1(\rho).
$
\noindent
Also, it can be shown that for $p \geq 1$, $S_p(\cdot)$
is concave in its argument.
\begin{definition}
For $p \geq 1$, the minimum output R\'{e}nyi $p$-entropy of
a quantum channel $\Phi$ is defined as :
\[
S_p^{\mathrm{min}}(\Phi) :=
\min_{\rho \in \mathcal{D}(\mathbb{C}^m)} S_p(\Phi(\rho))
\]
\end{definition}
By an easy concavity argument it can be seen that above
minimum is achieved on a pure state.
Equivalently, to obtain $S_p^{\mathrm{min}}(\Phi)$ for $p > 1$
we must maximise
$\lVert \Phi(\rho) \rVert_p$ for all input states $\rho$. This quantity is
also known as the
$1 \rightarrow p$ superoperator norm of superoperator
$\Phi: \mathcal{M}_m \rightarrow \mathcal{M}_d$:
\[
\lVert \Phi \rVert_{1 \rightarrow p}
:= \max_{M \in \mathcal{M}_m: \lVert M \rVert_1 = 1}
\lVert \Phi(M) \rVert_p.
\]
By an easy convexity argument it can be seen that the above
maximum is achieved on a pure state i.e.
\[
\lVert \Phi \rVert_{1 \rightarrow p} =
\max_{x \in \mathbb{C}^m: \lVert x \rVert_2 = 1} \lVert \ket{x}\bra{x} \rVert_{p}.
\]
Thus, the additivity conjecture for minimal output p-R\'{e}nyi $p$-entropy,
$p > 1$,
for quantum channels $\Phi$ and $\Psi$ is equivalent to
multiplicativity of $1 \rightarrow p$-norms of quantum channels viz.
$
\lVert \Phi \otimes \Psi \rVert_{1 \rightarrow p}
\stackrel{?}{=} \lVert \Phi \rVert_{1 \rightarrow p} \cdot
\lVert \Psi \rVert_{1 \rightarrow p}.
$
This equivalence will be used in Section~\ref{sec:Renyipentropy} to
give a counter example to additivity
conjecture for all $p > 1$ where the Stinespring dilation of the
quantum channel will be described
from a unitary chosen uniformly at random from an approximate $t$-design.
The equivalent result for Haar random unitaries was originally
proved by Hayden and Winter \cite{hayden_winter_2008}.
We heavily use the one-one correspondence between quantum channels and
subspaces of
composite Hilbert spaces, originally proved by Aubrun, Szarek and Werner
\cite{aubrun_szarek_werner_2010}, in this paper.
Let $\mathcal{W}$ be a subspace of
$\mathbb{C}^k \otimes \mathbb{C}^d$ of dimension $m$. Identify $\mathcal{W}$ with
$\mathbb{C}^m$ through an isometry
$V : \mathbb{C}^m \to \mathbb{C}^k \otimes \mathbb{C}^d$ whose range is
$\mathcal{W}$. Then, the corresponding quantum channel
$\Phi_\mathcal{W} : \mathcal{M}_m \to \mathcal{M}_k$ is defined by
$
\Phi_\mathcal{W}(\rho) := \mathrm{Tr}\,_{\mathbb{C}^d}(V \rho V^\dagger).
$
Using this equivalence and the fact that for $p > 1$ the
$1 \rightarrow p$-superoperator norm is achieved on pure input
states, we can write \cite{aubrun_szarek_werner_2010}
\begin{equation}
\label{eq:redefpnorm}
\lVert \Phi_\mathcal{W} \rVert_{1 \rightarrow p} =
\max_{x \in \mathcal{W}: \lVert x \rVert_2 = 1}
\lVert \mathrm{Tr}\,_{\mathbb{C}^d} \ket{x}\bra{x} \rVert_p =
\max_{x \in \mathcal{W}: \lVert x \rVert_2 = 1}
\lVert \mathrm{op}_{d \rightarrow k}(x) \rVert_{2p}^2.
\end{equation}
In an important paper, Shor \cite{Shor} proved that several
additivity conjectures
for quantum channels were in fact equivalent to the additivity
of minimum output von Neumann entropy of a quantum channel.
More specifically, Shor showed that if there is a quantum channel $\Phi$
whose minimum output von Neumann entropy is subadditive, then there
are quantum channels $\Psi_1$, $\Psi_2$ exhibiting superadditive
classical Holevo capacity viz.
$C(\Psi_1 \otimes \Psi_2) > C(\Psi_1) + C(\Psi_2)$. This equivalence
was used as a starting point by Hastings \cite{hastings_2009} in his
proof that there are
channels with superadditive classical Holevo capacity. Aubrun, Szarek and
Werner \cite{aubrun_szarek_werner_2010_main}, as well as this paper
also have the same starting point. For this, we need the following
fact.
\begin{fact}[\mbox{\cite[Lemma 2]{aubrun_szarek_werner_2010_main}}]
\label{fact:minoutputentropy}
Let a quantum
channel $\Phi_\mathcal{W}: \mathcal{M}_m \rightarrow \mathcal{M}_k$ be described by a subspace
$\mathcal{W} \leq \mathbb{C}^k \otimes \mathbb{C}^d$ of dimension $m$. Then,
\begin{eqnarray*}
S_{\mathrm{min}}(\Phi_\mathcal{W})
& = &
\log k -
k \cdot
\max_{\rho \in \mathcal{D}(\mathbb{C}^m)} \lVert \Phi(\rho) - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}
\rVert_2^2 \\
& = &
\log k -
k \cdot
\max_{x \in \mathcal{W}: \lVert x \rVert_2 = 1}
\lVert
(\mathrm{op}_{d \rightarrow k}(x)) (\mathrm{op}_{d \rightarrow k}(x))^\dagger
- \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}
\rVert_2^2.
\end{eqnarray*}
\end{fact}
We will need the following result proved by Hayden and
Winter \cite{hayden_winter_2008}
that upper bounds $S_p^{\mathrm{min}}(\Phi \otimes \bar{\Phi})$ where
$\bar{\Phi}$ denotes the CPTP superoperator obtained by taking complex
conjugate of the CPTP superoperator $\Phi$.
\begin{fact}
\label{fact:maxeigenvalue}
Let $V : \mathbb{C}^m \rightarrow \mathbb{C}^k \otimes \mathbb{C}^d$ be an
isometry describing the quantum channel
$\Phi: \rho \mapsto \mathrm{Tr}\,_{\mathbb{C}^d} [V \rho V^\dagger]$. Let
$\ket{\phi}$ denote the maximally entangled state in
$\mathbb{C}^m \otimes \mathbb{C}^m$. Suppose $m \leq d$.
Then $(\Phi \otimes \bar{\Phi})(\ket{\phi} \bra{\phi})$ has a
singular value
not less than $\frac{m}{kd}$.
Hence for all $p > 1$,
\[
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p}
\geq
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow \infty} \geq
\frac{m}{kd}.
\]
Moreover,
\[
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) \leq
2 \log k - \frac{m}{kd} \log k +
O \left(\frac{m}{kd} \log \frac{d}{m} + \frac{1}{k}\right).
\]
\end{fact}
\subsection{Polynomial approximation of monotonic functions}
We will need the following facts about step functions and their analytic
and polynomial approximations when we prove our result on strict
subadditivity of minimum output R\'{e}nyi $p$-entropy for
channels chosen from approximate $t$-designs.
\begin{definition}
\label{def:stepfunction}
The {\em (Heaviside) step function} is a function
$\mathbb{R} \rightarrow [0, 1]$ defined as follows:
\[
s(x) :=
\begin{array}{l l}
0 & \mbox{for $x < 0$} \\
\frac{1}{2} & \mbox{for $x = 0$} \\
1 & \mbox{for $x > 0$}.
\end{array}
\]
\end{definition}
\begin{definition}
\label{def:errorfunction}
The {\em error function} is a function $\mathbb{R} \rightarrow (-1, 1)$ defined
as follows:
\[
\mathrm{erf}(x) :=
\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt.
\]
\end{definition}
\noindent
The error function is a monotonically increasing function.
For positive $x$, $\mathrm{erf}(x)$ is nothing but the probability that the
normal distribution with mean $0$ and variance $1/2$ gives a point in
the interval $[-x, x]$. From the error function,
we get the so-called {\em sigmoid function}
$\Phi(x) := \frac{1}{2} + \frac{1}{2} \mathrm{erf}(x)$ which is nothing but
the cumulative distribution function of the above normal distribution.
The sigmoid function is a monotonically increasing function approximating
the step function in the following sense. Let $0 < \epsilon < 1$.
\begin{equation}
\label{eq:PhiVsS}
\Phi(x) ~
\begin{array}{l l l}
= & s(x) = \frac{1}{2} & \mbox{for $x=0$}, \\
> & s(x) = 0 & \mbox{for $x<0$}, \\
< & \frac{1}{2} & \mbox{for $x<0$}, \\
< & s(x) = 1 & \mbox{for $x>0$}, \\
> & \frac{1}{2} & \mbox{for $x>0$}, \\
> & s(x) - \epsilon = 1 - \epsilon &
\mbox{for
$x>\sqrt{\ln \epsilon^{-1}}$
}, \\
< & s(x) + \epsilon = \epsilon &
\mbox{for
$x<-\sqrt{\ln \epsilon^{-1}}$
},
\end{array}
~~~
\Phi'(x) ~
\begin{array}{l l l}
= & \frac{1}{\sqrt{\pi}} & \mbox{for $x=0$}, \\
< & \frac{1}{\sqrt{\pi}} & \mbox{for $x \neq 0$}, \\
> & 0 & \mbox{for all $x$}, \\
< & \epsilon &
\mbox{for
$|x| > \sqrt{\ln \epsilon^{-1}}$
}.
\end{array}
\end{equation}
The last two statements for $\Phi(x)$ above hold for
small $\epsilon$ and follow from
the bound
$
1 - \Phi(x) \leq
\frac{1}{2 x \sqrt{\pi}} e^{-x^2}.
$
The error function has the following rapidly converging Maclaurin
series:
\[
\mathrm{erf}(x) =
\frac{2}{\sqrt{\pi}}
\sum_{i=0}^\infty (-1)^i \frac{x^{2i+1}}{i! (2i+1)}.
\]
It is obtained by integrating termwise the Maclaurin series
$
e^{-x^2} =
\sum_{i=0}^\infty (-1)^i \frac{x^{2i}}{i!}.
$
Since both the above series are alternating series of positive and
negative terms,
truncating the Maclaurin expansion of $\Phi(x)$ at $i = n$ for
odd $n > x^2$ gives us
a polynomial $p_n(x)$ of degree $2n+1$ such that
\begin{equation}
\label{eq:PhiVsP}
p_n(x) ~
\begin{array}{l l l}
= & \Phi(x) = \frac{1}{2} & \mbox{for $x=0$}, \\
> & \Phi(x) & \mbox{for $-\sqrt{n} \leq x < 0$}, \\
< & \Phi(x) & \mbox{for $0 < x \leq \sqrt{n}$}, \\
> & \Phi(x) - \epsilon
& \mbox{for
$
0 \leq x \leq \frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2}
$
},\\
< & \Phi(x) + \epsilon
& \mbox{for
$
-\frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2} \leq x \leq 0
$
}.
\end{array}
\end{equation}
Moreover, the derivative $p'_n(x)$ is a polynomial of degree $2n$
satisfying
\begin{equation}
\label{eq:PhiprimeVsPprime}
p'_n(x) ~
\begin{array}{l l l}
= & \Phi'(x) = \frac{1}{\sqrt{\pi}} & \mbox{for $x=0$}, \\
\leq & \Phi'(x)
& \mbox{for
$
-\sqrt{n} \leq x \leq \sqrt{n}
$
}, \\
> & \Phi'(x) - \epsilon
& \mbox{for
$
-\frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2}
\leq x \leq \frac{\epsilon^{\frac{1}{2n}} \sqrt{n}}{2}
$
}.
\end{array}
\end{equation}
For the last two claims in Equation~\ref{eq:PhiVsP} and the last claim in
Equation~\ref{eq:PhiprimeVsPprime},
we used Stirling's approximation
$
n^n e^{-n} < n!
$
which holds for all positive integers $n$.
We will also need to upper bound the sum of absolute values of the
coefficients of $p_n(x)$, denoted by $\alpha(p_n(x))$. For this we
observe that
$
\alpha(p_n(x)) =
|p_n(\sqrt{-1})| \leq
\frac{1}{2} + \frac{e}{\sqrt{\pi}}.
$
We can now conclude that for $m > 0$, $0 \leq q \leq A$,
\begin{equation}
\label{eq:alpha}
\begin{array}{rcl}
\alpha(p_n(m(x-q)))
& = &
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\alpha(
\sum_{i=0}^n (-1)^i
\frac{(m(x-q))^{2i+1}}{i! (2i + 1)}
)
\;\leq\;
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\alpha(
\sum_{i=0}^n
\frac{(m(x+q))^{2i+1}}{i! (2i + 1)}
) \\
& = &
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\sum_{i=0}^n
\frac{(m(1+q))^{2i+1}}{i! (2i + 1)}
\;\leq\;
\frac{1}{2} + \frac{1}{\sqrt{\pi}}
\sum_{i=0}^\infty
\frac{(m(1+A))^{2i+1}}{i!} \\
& = &
\frac{1}{2} + \frac{m(1+A) e^{(m(1+A))^2}}{\sqrt{\pi}}
\;\leq\;
e^{2 (m(1+A))^2}.
\end{array}
\end{equation}
Let $f: [0, A] \rightarrow \mathbb{R}$ be a continuous non-decreasing function.
The {\em global Lipschitz constant} of $f$ is defined by
\[
L :=
\sup_{x,y \in [0, A], x < y}
\frac{f(y) - f(x)}{y - x}.
\]
If $L$ is finite, then we say that $f$ is $L$-Lipschitz.
Let $\epsilon > 0$.
For an element $x \in [0, A]$, the {\em $\epsilon$-smoothed local
Lipschitz constant of $f$ at $x$} is defined by
\[
L^\epsilon_x :=
\sup_{x,y \in f^{-1}((f(x) - \epsilon, f(x) + \epsilon)), x < y}
\frac{f(y) - f(x)}{y - x}.
\]
It is obvious that $L_x^\epsilon \leq L$. If $f$ is differentiable,
then $f'(x) \leq L^\epsilon_x$.
We now give a general proposition showing how to approximate
a continuous non-decreasing Lipschitz function by a polynomial
of moderate degree.
\begin{proposition}
\label{prop:poly}
Let $f: [0,A] \rightarrow [0,1]$ be a continuous non-decreasing onto
function with global Lipschitz constant $L$.
Fix $0 < \epsilon < 1$.
Let $L_x^\epsilon$ denote the $\epsilon$-smoothed local Lipschitz
constant of $f$ at $x$.
Let $n$ be the minimum positive odd integer satisfying
$
m A \leq \frac{\epsilon^{\frac{1}{n}} \sqrt{n}}{2},
$
where
$
m := \frac{2 L}{\epsilon} \sqrt{\ln \epsilon^{-2}}.
$
Define
$
m_x := \frac{2 L_x^\epsilon}{\epsilon} \sqrt{\ln \epsilon^{-2}}.
$
Then there is a polynomial $p(x)$ of degree at most $2n + 1$
such that
\[
p(x) - 2 \epsilon \leq f(x) \leq p(x) + 3 \epsilon, ~~~
-m \epsilon^2 < p'(x) < \epsilon m_x + m \epsilon^2, ~~~
\forall x \in [0,A].
\]
Moreover the sum of absolute values of the coefficients of $p(x)$,
denoted by $\alpha(p(x))$, is at most
$e^{2 ((A+1)m)^2}$.
\end{proposition}
\begin{proof}
Subdivide the range $[0,1]$ into $t := \lceil 1/\epsilon \rceil$ many
closed subintervals each of length $\epsilon$ except possibly the last one
whose length $\epsilon'$ may be less than $\epsilon$.
Denote their inverse images
under $f$ by $I_1, I_2, \ldots, I_{t}$.
For $1 \leq i < t$, let $p_i$ be the single
point intersection of closed subintervals $I_i$ and $I_{i+1}$; define
$p_0 := 0$, $p_t := A$.
The subinterval $I_i$, $1 \leq i < t$ is of length at least
$
\frac{\epsilon}{2 L_{p_i}^{\epsilon/2}} +
\frac{\epsilon}{2 L_{p_{i-1}}^{\epsilon/2}},
$
$I_t$ is of length at least
$
\frac{\epsilon'}{2 L_{p_t}^{\epsilon/2}} +
\frac{\epsilon'}{2 L_{p_{t-1}}^{\epsilon/2}}.
$
Observe that $\max_i L_{p_i}^{\epsilon/2} \leq L$.
Define the function
\[
g_1(x) :=
\epsilon \sum_{i=1}^{t - 1} s(x - p_i).
\]
Then $g_1(x) \leq f(x) \leq g_1(x) + \epsilon$ for all $x \in [0,A]$.
Define
$
m_i := \frac{2 L_{p_i}^{\epsilon/2}}{\epsilon} \sqrt{\ln \epsilon^{-2}},
$
$1 \leq i \leq t$. Then $m \geq \max_i m_i$.
Approximate the step function $s(x - p_i)$ by the sigmoid function
$\Phi(m_i (x - p_i))$. By Equation~\ref{eq:PhiVsS},
\[
\Phi(m_i (x-p_i)) ~
\begin{array}{l l l}
= & s(x - p_i) = \frac{1}{2} & \mbox{for $x=p_i$}, \\
> & s(x - p_i) = 0 & \mbox{for $x<p_i$}, \\
< & \frac{1}{2} & \mbox{for $x<p_i$}, \\
< & s(x - p_i) = 1 & \mbox{for $x>p_i$}, \\
> & \frac{1}{2} & \mbox{for $x>p_i$}, \\
> & s(x - p_i) - \epsilon^2 = 1 - \epsilon^2
& \mbox{for $x> p_i +\frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}$}, \\
< & s(x - p_i) + \epsilon^2 = \epsilon^2
& \mbox{for $x< p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}$}.
\end{array}
\]
Define the function
\[
g_2(x) :=
\epsilon \sum_{i=1}^{t - 1} \Phi(m_i (x - p_i)).
\]
It is now easy to see that
$
g_2(x) - \epsilon \leq g_1(x) \leq g_2(x) + \epsilon
$
for all $x \in [0, A]$. Thus,
\[
g_2(x) - \epsilon \leq f(x) \leq g_2(x) + 2 \epsilon ~~~
\forall x \in [0,A].
\]
Also,
\[
0 < g'_2(x) < \epsilon m_i + m \epsilon^2, ~~~
\mbox{if\ }
x \in
[
p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}},
p_i + \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}
]
\mbox{\ for some $i$},
\]
and
$
0 < g'_2(x) < m \epsilon^2
$
otherwise.
We now approximate the sigmoid function $\Phi(m_i (x-p_i))$ by
the polynomial $p_n(m_i (x - p_i))$ for
$m_i A \leq m A < \frac{\epsilon^{\frac{1}{n}} \sqrt{n}}{2}$, $n$ odd.
From Equations~\ref{eq:PhiVsP}, \ref{eq:PhiprimeVsPprime} we get
\[
p_n(m_i (x-p_i)) ~
\begin{array}{l l l}
= & \Phi(m_i (x-p_i)) = \frac{1}{2} & \mbox{for $x=p_i$}, \\
> & \Phi(m_i (x - p_i)) & \mbox{for $0 \leq x<p_i$}, \\
< & \Phi(m_i (x - p_i)) & \mbox{for $p_i < x \leq A$}, \\
> & \Phi(m_i (x - p_i)) - \epsilon^2 & \mbox{for $p_i \leq x \leq A$}, \\
< & \Phi(m_i (x - p_i)) + \epsilon^2 & \mbox{for $0 \leq x \leq p_i$},
\end{array}
\]
\[
p'_n(m_i (x-p_i)) ~
\begin{array}{l l l}
= & \Phi'(m_i (x-p_i)) = \frac{m_i}{\sqrt{\pi}} & \mbox{for $x=p_i$}, \\
\leq & \Phi'(m_i (x - p_i)) & \mbox{for $0 \leq x \leq A$}, \\
> & \Phi'(m_i (x - p_i)) - \epsilon^2 & \mbox{for $0 \leq x \leq A$}.
\end{array}
\]
Define the degree $2n+1$ polynomial
\[
p(x) :=
\epsilon \sum_{i=1}^{t - 1} p_n(m_i (x - p_i)).
\]
It is now easy to see that
\[
p(x) - \epsilon^2 \leq g_2(x) \leq p(x) + \epsilon^2,
~~~
p'(x) \leq g'_2(x) \leq p'(x) + m \epsilon^2,
\]
for all $x \in [0, A]$. Thus,
\[
p(x) - 2 \epsilon \leq f(x) \leq p(x) + 3 \epsilon ~~~
\forall x \in [0,A],
\]
and
\[
-m \epsilon^2 < p'(x) < \epsilon m_i + m \epsilon^2, ~~~
\mbox{if\ }
x \in
[
p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}},
p_i + \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}
]
\mbox{\ for some $i$},
\]
and
$
-m \epsilon^2 < p'(x) < m \epsilon^2
$
otherwise. Now observe that if
$
x \in
[
p_i - \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}},
p_i + \frac{\epsilon}{2 L_{p_i}^{\epsilon/2}}
],
$
$m_x \geq m_i$. Hence we can always say that
\[
-m \epsilon^2 < p'(x) < \epsilon m_x + m \epsilon^2 ~~~
\forall x \in [0, A].
\]
Finally by Equation~\ref{eq:alpha},
\[
\alpha(p(x)) \leq
\epsilon \sum_{i=1}^{t-1} \alpha(p_n(m_i (x-p_i))) \leq
\epsilon \sum_{i=1}^{t-1} e^{2((A+1)m_i)^2} \leq
e^{2((A+1)m)^2}.
\]
This completes the proof of the proposition.
\end{proof}
\paragraph{Remarks:}
\ \\
\noindent
1. \
Any continuous non-decreasing Lipschitz
function on a closed bounded interval can be converted into a function
of the above type by translating the domain and the range and scaling
the range.
\noindent
2. \
A similar proposition can be proved for approximating a monotonically
non-increasing Lipschitz function by a polynomial.
\subsection{Concentration results for Lipschitz functions}
We now state some basic definitions and facts from geometric functional
analysis that will be used in the proof of our main result.
\begin{definition}
A function $f:X \rightarrow \mathbb{C} $ defined over a metric
space $X$ is said to
be $L$-Lipschitz if
$\forall x, y \in X$ it satisfies the following inequality:
\[
\lvert f(x)-f(y) \rvert \leq L \cdot d(x,y).
\]
\end{definition}
\begin{definition}
Let $X$ be a compact metric space. An $\epsilon$-net $\mathcal{N}$ of
$X$ is a finite set of points such that for any point $x \in X$, there is
a point $x' \in \mathcal{N}$ such that $d(x,x') \leq \epsilon$.
\end{definition}
\noindent
Note that compactness guarantees that finite sized $\epsilon$-nets exist
for all $\epsilon > 0$.
We will need the following definition and fact from
\cite{aubrun_szarek_werner_2010_main}.
\begin{definition}
A function $f:X \rightarrow \mathbb{C} $ defined over a normed linear
space $X$ is said to be circled if
$f(e^{i\theta} x) = f(x)$ for all
$\theta \in \mathbb{R}$ and $x \in X$.
\end{definition}
\begin{fact}
\label{fact:extension}
Let $f: X \rightarrow \mathbb{R}$ be a function defined on a metric space $X$.
Suppose there exists a subset $Y \subseteq X$ such that $f$ restricted
to $Y$ is $L$-Lipschitz. Then there is a function
$\hat{f}: X \rightarrow \mathbb{R}$
that is $L$-Lipschitz on all of $X$ satisfying $\hat{f}(y) = f(y)$ for all
$y \in Y$. If $X$ is a normed linear space over real or complex numbers
and $f$ is circled
then the extension $\hat{f}$ is also circled.
\end{fact}
\begin{proof}
{\bf (Sketch)}
Define
$
\hat{f}(x) := \inf_{y \in Y} [f(y) + L d(x, y)].
$
\end{proof}
In this paper, we endow $\mathbb{C}^n$ with the $\ell_2$-metric and
$\mathbb{U}(n)$ with the Schatten $\ell_2$-metric aka Frobenius metric. The
following fact gives a reasonably tight upper bound on the
size of an $\epsilon$-net of $\mathbb{S}_{\mathbb{C}^n}$.
\begin{fact}[\mbox{\cite[Corollary~4.2.13]{Vershynin}}]
\label{fact:net}
Let $\epsilon > 0$. There exists an $\epsilon$-net of $\mathbb{S}_{\mathbb{C}^n}$ of
size less than
$
(\frac{3}{\epsilon})^{2n}.
$
\end{fact}
A fundamental result about concentration of Lipschitz functions defined
on the unit sphere or the unitary group, known as Levy's lemma, lies at
the heart of all
proofs of Dvoretzky-type theorems via the probabilistic method.
We now state the version of Levy's lemma that will be used in this
paper.
\begin{fact}[Levy's lemma, \mbox{\cite[Corollary~4.4.28]{AGZ}}]
\label{fact:levy}
Consider the Haar probability measure on $\mathbb{S}_{\mathbb{C}^n}$.
Let $f: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{C}$ be an $L$-Lipshitz function.
Let $\mu := \mathbb{E}_x[f(x)]$ and $\lambda > 0$. Then
\[
\Pr_x(\lvert f(x) - \mu \rvert \geq \lambda) \leq
2 \exp(-\frac{n \lambda^2}{4 L^2}).
\]
\end{fact}
An elementary proof of the above fact, without explicitly calculated
constants, can be found in \cite[Theorem~5.1.4]{Vershynin}.
For our work, we need a measure concentration inequality like Levy's
lemma for difference of function values on two distinct arbitrary
points which is sensitive to the distance between those points.
Such an inequality is stated in the following fact.
\begin{fact}[\mbox{\cite[Lemma~9]{aubrun_szarek_werner_2010_main}}]
\label{fact:LevyLipschitz}
Let $f: \mathbb{S}_{\mathbb{C}^n} \to \mathbb{C}$ be a circled $L$-Lipschitz
function.
Consider the Haar probability measure on $\mathbb{U}(n)$.
Then for any
$x, y \in \mathbb{S}_{\mathbb{C}^n}$, $x \neq y$ and for
any $\lambda > 0$,
\[
\Pr_U[\lvert f(Ux) - f(Uy) \rvert > \lambda]
\leq 2 \exp(-\frac{\lambda^2 n}{8 L^2 \lVert x-y \rVert_2^2}).
\]
\end{fact}
The derandomisation in our paper is carried out by replacing the
Stinespring dilation unitary of a quantum channel,
which is chosen from the Haar measure in
\cite{aubrun_szarek_werner_2010_main},
with a unitary chosen uniformly at random
from a finite cardinality approximate unitary $t$-design for a
suitable value of $t$. The next few statements lead us to the definition
of an approximate unitary $t$-design.
\begin{definition}[\mbox{\cite[Definition~2.2]{low_2009}}]
A monomial in the entries of a matrix $U$ is of degree $(r,s)$ if
it contains $r$ conjugated elements and $s$ unconjugated elements. The
evaluation of monomial $M$ at the entries of a matrix $U$ is denoted
by $M(U)$. We call a monomial balanced if $r = s$,
and say that it has degree $t$ if it is of degree $(t,t)$.
A polynomial is said to be balanced of degree
$t$ if it is a sum of balanced monomials of degree at most $t$.
\end{definition}
\begin{definition}[\mbox{\cite[Definition~2.3]{low_2009}}]
A probability distribution $\nu$ supported on a finite set of $d \times d$
unitary matrices is said to be an exact unitary $t$-design if
for all balanced monomials $M$ of degree at most $t$,
$
\mathbb{E}_{U \sim \nu}[M(U)] = \mathbb{E}_{U \sim \mathrm{Haar}}[M(U)].
$
\end{definition}
\begin{definition}[\mbox{\cite[Definition~2.6]{low_2009}}]
A probability distribution $\nu$ supported on a finite set of $d \times d$
unitary matrices is said to be
an $\epsilon$-approximate unitary $t$-design if for all
balanced monomials $M$ of degree at most $t$
\[
\lvert
\mathbb{E}_{U \sim \nu}(M(U)) -
\mathbb{E}_{U \sim \mathrm{Haar}}(M(U))
\rvert \leq
\frac{\epsilon}{d^t}.
\]
\end{definition}
We will need the following fact.
\begin{fact}[\mbox{\cite[Lemma~3.4]{low_2009}}]
\label{fact:TvsHaar}
Let $Y: \mathbb{U}(n) \rightarrow \mathbb{C}$ be a balanced polynomial of
degree $a$ in the entries of the unitary matrix $U$ that is provided
as input. Let
$\alpha(Y)$ denote the sum of absolute values of the coefficients of $Y$.
Let $r$, $t$ be positive integers satisfying $2ar < t$.
Let $\nu$ be an $\epsilon$-approximate unitary $t$-design. Then
\[
\mathbb{E}_{U \sim \nu}[{\lvert Y_U\rvert}^{2r}] \leq
\mathbb{E}_{U \sim \mathrm{Haar}}[{\lvert Y_U \rvert}^{2r}]+
\frac{\epsilon \alpha(Y)^{2r}}{n^t}.
\]
\end{fact}
\section{Sharp Dvoretzky-like theorems via stratified analysis}
\label{sec:main}
In this section, we prove our main technical results viz. sharp
Dvoretzky-like theorems for Haar measure as well as approximate
$t$-designs using stratified analysis.
We start by proving the following two lemmas which are `baby stratified'
analogues of Fact~\ref{fact:LevyLipschitz} for Haar measure and
approximate unitary $t$-designs.
\begin{lemma}
\label{lem:expectationHaar}
Let $Y: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ be a circled function with
global Lipschitz constant $L_1$. Suppose that
there exists a subset $\Omega \subseteq \mathbb{S}_{\mathbb{C}^n}$ such that
$Y$ restricted to $\Omega$
has a smaller Lipschitz constant $L_2$.
Let $x, y \in \mathbb{S}_{\mathbb{C}^n}$.
Let $Y_x := Y(Ux)$, $Y_y := Y(Uy)$ be two correlated random variables,
under the choice of a Haar random unitary $U$. Let $\lambda > 0$.
Then
\[
\Pr_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert > \lambda] \leq
2 \exp(-\frac{n \lambda^{2}}{8 L_2^2 \lVert x - y\rVert_2^2}) +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c].
\]
\end{lemma}
\begin{proof}
By Fact~\ref{fact:extension}, there is a circled function $Y'$ that
agrees with $Y$ on $\Omega$ and is $L_2$-Lipschitz on all
of $\mathbb{S}_{\mathbb{C}^n}$.
Define correlated random variables $Y'_x$,
$Y'_y$ in the natural manner. Then
using Fact~\ref{fact:LevyLipschitz}, we get
\begin{eqnarray*}
\lefteqn{\Pr_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert > \lambda]} \\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert > \lambda|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[
(Ux, Uy) \not \in \Omega \times \Omega
] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert > \lambda|(Ux, Uy) \not \in \Omega \times \Omega
] \\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y'_x - Y'_{y}\rvert > \lambda|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[
(Ux, Uy) \not \in \Omega \times \Omega
] \cdot
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert > \lambda|(Ux, Uy) \not \in \Omega \times \Omega
] \\
& \leq &
\Pr_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y}\rvert > \lambda] +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \\
& \leq &
2 \exp(-\frac{n \lambda^{2}}{8 L_2^2 \lVert x - y\rVert_2^2}) +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c].
\end{eqnarray*}
This finishes the proof of the lemma.
\end{proof}
\begin{lemma}
\label{lem:expectationtdesign}
Let $Y: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ be a
balanced polynomial of degree $a$ in entries of the vector $x \in \mathbb{C}^n$
that is
provided as input. Let
$\alpha(Y)$ denote the sum of absolute values of the coefficients of $Y$.
Suppose $Y$ has global Lipschitz constant $L_1$.
Suppose that
there exists a subset $\Omega \subseteq \mathbb{S}_{\mathbb{C}^n}$ such that
$Y$ restricted to $\Omega$
has a smaller Lipschitz constant $L_2$.
Let $x, y \in \mathbb{S}_{\mathbb{C}^n}$.
Let $Y_x := Y(Ux)$, $Y_y := Y(Uy)$ be two correlated random variables,
under the choice of a unitary $U$ chosen uniformly at random from an
$\epsilon$-approximate unitary $t$-design $\nu$.
Let $r$ be a positive integer satisfying $2ar \leq t$.
Let
$
0 < \epsilon <
\frac{
n^{t-r} (4 r L_2^2 \lVert x - y \rVert_2^2)^r
}{\alpha(Y)^{2r}}.
$
Then
\[
\mathbb{E}_{U \sim \nu}[\lvert Y_x - Y_{y}\rvert^{2r}] \leq
3 \left(\frac{4 r L_2^2 \lVert x - y\rVert_2^2}{n}\right)^{r} +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \cdot
(L_1^2\lVert x- y \rVert_2^2)^{r}.
\]
\end{lemma}
\begin{proof}
Since
$Y_x -Y_y$ is a balanced polynomial in the entries of the unitary
matrix $U$, from Fact~\ref{fact:TvsHaar} we have
\[
\mathbb{E}_{U \sim \nu}[\lvert Y_x - Y_{y}\rvert^{2r}]
\overset{\mathrm{a}}{\leq}
\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}] +
\frac{\epsilon \alpha(Y)^{2r}}{n^t}.
\]
By choosing
$\epsilon$ small enough to satisfy the constraint above, we get
$
\frac{\epsilon \alpha(Y)^{2r}}{n^t}
\overset{\mathrm{b}}{\leq}
\left(\frac{4 r L_2^2 \lVert x - y \rVert_2^2}{n}\right)^r.
$
Combining (a) and (b) gives
\[
\mathbb{E}_{U \sim \nu}[\lvert Y_x - Y_{y}\rvert^{2r}]
\overset{\mathrm{c}}{\leq}
\mathbb{E}_{U \sim \mathrm{Haar}}({\lvert Y_x - Y_{y}\rvert}^{2r}) +
\left(\frac{4 r L_2^2 \lVert x - y \rVert_2^2}{n}\right)^r.
\]
Now we find
$\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}]$.
Since $Y$ is a balanced polynomial, it is circled.
By Fact~\ref{fact:extension}, there is a circled function $Y'$ such that
$Y'$ agrees with $Y$ on $\Omega$ and $Y'$ is $L_2$-Lipschitz on all
of $\mathbb{S}_{\mathbb{C}^n}$.
Define correlated random variables $Y'_x$,
$Y'_y$ in the natural manner. Then
\begin{eqnarray*}
\lefteqn{\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}]} \\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert^{2r}|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \not \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert^{2r}|(Ux, Uy) \not \in \Omega \times \Omega]\\
& = &
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y'_x - Y'_{y}\rvert^{2r}|(Ux, Uy) \in \Omega \times \Omega
] \\
& &
{} +
\Pr_{U \sim \mathrm{Haar}}[(Ux, Uy) \not \in \Omega \times \Omega] \cdot
\mathbb{E}_{U \sim \mathrm{Haar}}[
\lvert Y_x - Y_{y}\rvert^{2r}|(Ux, Uy) \not \in \Omega \times \Omega]\\
& \overset{\mathrm{d}}{\leq} &
\mathbb{E}_{U \sim Haar}[\lvert Y'_x - Y'_{y}\rvert^{2r}]+
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \cdot
(L_1^2\lVert x- y \rVert_2^2)^{r}.
\end{eqnarray*}
Now we find
$\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y}\rvert^{2r}]$
using Fact~\ref{fact:LevyLipschitz} and
Low's method \cite[Lemma~3.3]{low_2009}.
\begin{eqnarray*}
\lefteqn{\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y}\rvert^{2r}]} \\
& = &
\int_0^\infty
\Pr_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y} \rvert^{2r}>\lambda] \,
d\lambda \
\;= \;
\int_0^\infty
\Pr_{U \sim \mathrm{Haar}}[\lvert Y'_x - Y'_{y} \rvert>\lambda^{1/(2r)}] \,
d\lambda \\
& \leq &
2 \int_0^\infty
\exp(-\frac{n \lambda^{1/r}}{8 L_2^2 \lVert x - y\rVert_2^2}) \,
d\lambda
\; \overset{\mathrm{e}}{\leq} \;
2 \left(\frac{4 r L_2^2 \lVert x - y\rVert_2^2}{n}\right)^{r}.
\end{eqnarray*}
Combining inequalities (d) and (e), we have
\[
\mathbb{E}_{U \sim \mathrm{Haar}}[\lvert Y_x - Y_{y}\rvert^{2r}] \leq
2 \left(\frac{4 r L_2^2 \lVert x - y\rVert_2^2}{n}\right)^{r} +
2 \Pr_{z \sim \mathrm{Haar}}[z \in \Omega^c] \cdot
(L_1^2\lVert x- y \rVert_2^2)^{r}.
\]
Further combining with (c) gives us the desired conclusion of the lemma.
\end{proof}
We also need a so-called {\em chaining inequality} for probability
similar to Dudley's
inequality in geometric functional analysis
\cite{aubrun_szarek_werner_2010_main,pisier_1989}. The original
Dudley's inequality bounds the expectation of the supremum, over pairs
of correlated random variables, of the difference between them
in terms of an integral, over $\eta$, of
a certain function of the size of an $\eta$-net of $\mathbb{S}_{\mathbb{C}^n}$.
Our chaining lemma differs from it in two important respects.
First, instead of the expectation it bounds a tail probability of
the supremum, over pairs
of correlated random variables, of the difference between them.
Second, it replaces the integral by a finite summation
over $\eta$-nets of $\mathbb{S}_{\mathbb{C}^n}$ with geometrically decreasing
$\eta$. Despite the fancy name, our chaining lemma is a simple consequence
of the union bound of probabilities. Nevertheless, it is crucial to
proving our main result as it allows us to efficiently invoke
powerful measure concentration results in order to bound the variation of a
Lipschitz function on subspaces of $\mathbb{C}^n$.
\begin{lemma}[Chaining]
\label{lem:chaining}
Let $\{X_s\}_{s \in \mathcal{S}}$ be a family of correlated complex valued
random variables indexed
by elements of a compact metric space $\mathcal{S}$.
Let $\lambda, L_1 > 0$. The family is said to be $L_1$-Lipschitz if
for all $s, t \in \mathcal{S}$, $|X_s - X_t| \leq L_1 d(s,t)$ for
all points of the sample space.
Define $i_0$ to be the unique
integer such that the radius of $\mathcal{S}$ lies in the interval
$(2^{-i_0-1}, 2^{-i_0}]$. Define
$i_1 := \max\{i_0, \lceil \log \frac{2 L_1}{\lambda}\rceil\}$.
Let $p : \mathbb{Z} \rightarrow \mathbb{R}_+$ be a non-decreasing
function. Suppose the infinite series
$\sum_{i > i_0} \frac{\sqrt{|i| p(i)}}{2^i}$ is convergent with
value $C$.
Then,
\[
\Pr[\sup_{s,t \in \mathcal{S}} \lvert X_s-X_t\rvert > \lambda] \leq
\sum_{i = i_0+1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
d(u, u^\prime) < 2^{-i+2}
} \Pr [\lvert X_u - X_{u^\prime}\rvert >
\frac{\lambda \sqrt{|i| p(i)}}{4C \cdot 2^i}
],
\]
for a sequence of $2^{-i}$-nets $\mathcal{N}_i$, $i_0 \leq i \leq i_1$,
$|\mathcal{N}_{i_0}| = 1$, of $\mathcal{S}$.
\end{lemma}
\begin{proof}
For every $i \in \mathbb{Z}$, let $\mathcal{N}_i$ be a $2^{-i}$-net of $\mathcal{S}$.
Let $i_0$ be such that radius of $\mathcal{S}$
lies in $(2^{-(i_0+1)},2^{-i_0}]$. The net $\mathcal{N}_{i_0}$
consists of a single element, say $s_0$. For every $s \in \mathcal{S}$ and
$i \in \mathbb{Z}$, let $\pi_i(s)$ be an element of $\mathcal{N}_i$ satisfying
$d(s,\pi_i(s)) \leq 2^{-i}$. We have the following chaining equation
for every $s \in \mathcal{S}$:
\[
X_s =
X_{s_0} +
\left(\sum_{i = i_0}^{i_i} (X_{\pi_{i+1}(s)}-X_{\pi_{i}(s)})\right) +
(X_s - X_{\pi_{i_1 + 1}(s)}).
\]
Lipschitz property of the family implies that
\begin{eqnarray*}
\sup_{s,t \in \mathcal{S}} \lvert X_s-X_t\rvert
& \leq &
2 \sum_{i = i_0}^{i_1}
\sup_{s \in \mathcal{S}} \lvert X_{\pi_{i+1}(s)}-X_{\pi_{i}(s)} \rvert +
L_1 2^{-i_1} \\
& \leq &
2 \sum_{i = i_0}^{i_1}
\sup_{(u,u^\prime) \in \mathcal{N}_{i} \times \mathcal{N}_{i+1} : d(u,u')<2^{-i+1}}
\lvert X_u - X_{u'} \rvert +
L_1 2^{-i_1} \\
& \leq &
2 \sum_{i = i_0+1}^{i_1+1}
\sup_{(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i} : d(u,u')<2^{-i+2}}
\lvert X_u - X_{u'} \rvert +
\frac{\lambda}{2}.
\end{eqnarray*}
Now if
$\sup_{s,t \in \mathcal{S}} \lvert X_s-X_t\rvert > \lambda$, there must
exist an $i$, $i_0 + 1 \leq i \leq i_1 + 1$ such that
\[
\sup_{(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i} : d(u,u')<2^{-i+2}}
\lvert X_u - X_{u'} \rvert >
\frac{\lambda \sqrt{|i| p(i)}}{4C \cdot 2^i}.
\]
Applying the union bound on probability leads us to the conclusion
of the lemma.
\end{proof}
We now prove our sharp Dvoretzky-like theorem for subspaces chosen
from the Haar measure using stratified analysis.
\begin{theorem}
\label{thm:mainHaar}
Let $p : \mathbb{N} \rightarrow \mathbb{R}_+$ be a non-decreasing
function. Suppose the infinite series
$\sum_{i > 0} \frac{\sqrt{i p(i)}}{2^i}$ is convergent with
value $C$.
Let $f: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ have
global Lipschitz constant $L_1$.
Let $L_2, c_1, c_2, c_3, \lambda > 0$.
Define $m := \lceil \frac{c_1 n \lambda^2}{L_2^2} \rceil$.
Suppose there is an increasing sequence of subsets
$\Omega_1 \subseteq \Omega_2 \subseteq \cdots$ of $\mathbb{S}_{\mathbb{C}^n}$
such that with probability at least $1 - c_2 e^{-c_3 m i}$,
a Haar random subspace of dimension
$m$ lies in $\Omega_i$ and $f$ restricted to $\Omega_i$ has Lipschitz
constant $L_2 \sqrt{p(i)}$.
Then there exists a constant $c$ depending on $c_3$, $C$,
$0 < c < 1$, such that for
$
m^\prime := c m
$
with probability at least $1- (c_2 + 1) 2^{-m'}$, a
subspace $W$ of dimension
$m^\prime$ chosen with respect to Haar measure
satisfies the property that
$\lvert f(w)- \mu \rvert < \lambda$
for all points $w \in W \cap \mathbb{S}_{\mathbb{C}^n}$.
\end{theorem}
\begin{proof}
In this proof $\mathbb{S}_{\mathbb{C}^{n}}$ denotes the unit $\ell_2$-length sphere in
$\mathbb{C}^n$ together with the origin point $0$. The radius of
$\mathbb{S}_{\mathbb{C}^{n}}$ is one which makes $i_0 = 0$ in
Lemma~\ref{lem:chaining}.
Consider a canonical embedding of $\mathbb{S}_{\mathbb{C}^{m'}}$ into
$\mathbb{S}_{\mathbb{C}^{m}}$ and further into $\mathbb{S}_{\mathbb{C}^n}$. Define
\[
B_i := \{U \in \mathbb{U}(n): \forall z \in \mathbb{S}_{\mathbb{C}^{m}}, Uz \in \Omega_i\}.
\]
For $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, define the random variable
$Y_s := f(Us) - \mu$,
where the randomness arises solely from the choice of $U \in \mathbb{U}(n)$.
Then $\Pr_{U \sim \mathrm{Haar}}[B_i] \geq 1 - c_2 e^{-c_3 m i}$.
Let $i_1 := \lceil \log \frac{2 L_1}{\lambda} \rceil$.
Let $\mathcal{N}_i$, $i = 0, 1, \ldots, i_1$ be a sequence of
$2^{-i}$-nets in $\mathbb{S}_{\mathbb{C}^{m'}}$ of minimum cardinality,
where $\mathcal{N}_0 := \{0\}$
and $Y_0 := 0$. We can take $|\mathcal{N}_i| \overset{a}{\leq} 2^{2 (i+2) m'}$ by
Fact~\ref{fact:net}. By Lemma~\ref{lem:chaining}
\[
\Pr_{U \sim \mathrm{Haar}}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m'}}}
\lvert Y_s -Y_t \rvert > \lambda
] \leq
2 \sum_{i=1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert_2 < 2^{-i+2}
}
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
].
\]
Applying Lemma~\ref{lem:expectationHaar} to the set $B_{i}$ gives, for
$u$, $u'$ satisfying $\lVert u- u^\prime \rVert_2 < 2^{-i+2}$,
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \mathrm{Haar}}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
]
} \\
& \leq &
2 \exp\left(-\frac{n \lambda^2 i p(i)}{2^7 C^2 2^{2i} L_2^2 p(i)
\lVert u -u^\prime \rVert_2^2} \right) +
2 \Pr_{z \sim \mathrm{Haar}} [z \in \Omega_{i}^c] \\
& \leq &
2 \exp\left(-\frac{n i \lambda^2 }{2^9 C^2 L_2^2} \right) +
2 \Pr_{z \sim \mathrm{Haar}} [z \in \Omega_{i}^c] \\
\\
& \leq &
2 \exp\left(-\frac{im}{2^9 C^2} \right) +
2 c_2 \exp(-c_3 m i)
\; \leq \;
2 (c_2 + 1) \exp(-c_4 m i),
\end{eqnarray*}
for a constant $c_4$ depending only on $C$ and $c_3$.
This gives us
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \mathrm{Haar}}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m'}}}
\lvert Y_s -Y_t \rvert > \lambda
]
} \\
& \leq &
4 (c_2 + 1) \sum_{i=1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert_2 < 2^{-i+2}
}
e^{-c_4 m i}
\;\leq\;
4 (c_2 + 1) \sum_{i=1}^{i_1+1}
|\mathcal{N}_{i-1}| \cdot |\mathcal{N}_{i}| \cdot e^{-c_4 m i} \\
& \leq &
4 (c_2 + 1) \sum_{i=1}^{i_1+1}
2^{4 m' (i+2)}
e^{-c_4 m i}
\;\leq\;
(c_2 + 1) 2^{-m'},
\end{eqnarray*}
where the third inequality follows from (a) and the fourth inequality
follows from the definition $m' := c m$ for an appropriate choice
of $c$ depending only on $c_4$. In other words, $c$ depends only on
$C$ and $c_3$.
Taking $t = 0$, we see that
with probability at least $1 - (c_2 + 1) 2^{-m'}$ over
the choice of a Haar random unitary, we have
that for all $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, $|Y_s| \leq \lambda$.
This completes the proof of the theorem.
\end{proof}
\paragraph{Remark:}
The sets $\Omega_i$ and the Lipschitz constants $L_2 \sqrt{p(i)}$
for $1 \leq i \leq \lceil \log \frac{2 L_1}{\lambda} \rceil + 1$
formalise the idea of stratified analysis mentioned intuitively in the
introduction. As $i$ increases the relevant Lipschitz constant increases.
So we need a finer net i.e. a $2^{-i}$-net for the $i$th layer
$\Omega_i$ in order to control the variation of $f$ for subspaces
lying inside $\Omega_i$.
With exponentially high probability, we thus get a
Haar random subspace of dimension $m^\prime$, slightly smaller than $m$,
where $f$ is almost constant.
Note that the definition of $m$ involves only the smallest local
Lipschitz constant $L_2$. Thus the dimension of the space $m'$ that
we obtain is larger than what would be obtained by a naive
analysis which would be constrained by the
global Lipschitz constant $L_1$. Moreover, a naive analysis would not
give exponentially high probability, just an arbitrary constant close
one. These two properties underscore the power of our stratified
analysis. However, applying the stratified analysis to a concrete
function is not always straightforward. We need to define the
layers $\Omega_1, \Omega_2, \ldots, $ properly and show separately that
Haar random subspaces of dimension $m$ lie in $\Omega_i$ with
probability $1 - c_2 e^{-c_3 m i}$. But for several interesting functions
this can be done without much difficulty.
This will become clearer in
Section~\ref{sec:vonNeumannentropy} where we will show how to
recover Aubrun, Szarek and Werner's result for the Haar measure directly
from
Theorem~\ref{thm:mainHaar}, without having to apply a Dvoretzky-style
theorem twice in a messy fashion as in the original paper
\cite{aubrun_szarek_werner_2010_main}. Moreover, we get success probability
exponentially close to one unlike Aubrun, Szarek and Werner who could
get only a constant close to one. Furthermore, our
methods extend to approximate $t$-designs and allows us to prove
exponentially close to one probability even for that setting.
We now prove our sharp Dvoretzky-like theorem for subspaces chosen
from approximate $t$-designs using stratified analysis.
\begin{theorem}
\label{thm:maintdesign}
Let $p : \mathbb{N} \rightarrow \mathbb{R}_+$ be a non-decreasing
function. Suppose the infinite series
$\sum_{i > 0} \frac{\sqrt{i p(i)}}{2^i}$ is convergent with
value $C$.
Let $f: \mathbb{S}_{\mathbb{C}^n} \rightarrow \mathbb{R}$ be a balanced degree $`a'$
polynomial with global Lipschitz constant $L_1$.
Let $0 \leq L_2 \leq 1$, $c_1, c_2, c_3, \lambda > 0$.
Define $m := \lceil \frac{c_1 n \lambda^2}{L_2^2} \rceil$.
Suppose there is an increasing sequence of subsets
$\Omega_1 \subseteq \Omega_2 \subseteq \cdots$ of $\mathbb{S}_{\mathbb{C}^n}$
such that with probability at least $1 - c_2 e^{-c_3 m i}$,
a Haar random subspace of dimension
$m$ lies in $\Omega_i$ and $f$ restricted to $\Omega_i$ has Lipschitz
constant $L_2 \sqrt{p(i)}$.
Suppose
\[
0 < \epsilon <
\left(\frac{\lambda}{4 L_1}\right)^{2m} \cdot
\frac{n^{(2a-1)m} (L_2^2 p(1))^{m}}{\max\{\alpha(f)^{2m},1\}}.
\]
Then there exists a constant $c$ depending on $c_1$, $c_3$, $C$, $p(1)$,
$0 < c < 1$ such that for
\[
m^\prime :=
c m
\frac{\log \log \frac{C^2 L_1^2}{\lambda^2 p(1)}}
{\lceil \log \frac{C^2 L_1^2}{\lambda^2 p(1)} \rceil},
\]
with probability at least $1- (c_2+1) 2^{-m'}$, a
subspace $W$ of dimension
$m^\prime$ chosen under an $\epsilon$-approximate $(2am)$-design $\nu$
satisfies the property that
$\lvert f(w)- \mu \rvert < \lambda$
for all points $w \in W \cap \mathbb{S}_{\mathbb{C}^n}$.
\end{theorem}
\begin{proof}
In this proof $\mathbb{S}_{\mathbb{C}^{n}}$ denotes the unit $\ell_2$-length sphere in
$\mathbb{C}^n$ together with the origin point $0$. The radius of
$\mathbb{S}_{\mathbb{C}^{n}}$ is one which makes $i_0 = 0$ in
Lemma~\ref{lem:expectationtdesign}.
Consider a canonical embedding of $\mathbb{S}_{\mathbb{C}^{m'}}$ into
$\mathbb{S}_{\mathbb{C}^{m}}$ and further into $\mathbb{S}_{\mathbb{C}^n}$. Define
\[
B_i := \{U \in \mathbb{U}(n): \forall z \in \mathbb{S}_{\mathbb{C}^{m}}, Uz \in \Omega_i\}.
\]
For $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, define the random variable
$Y_s := f(Us) - \mu$,
where the randomness arises solely from the choice of $U \in \mathbb{U}(n)$.
Then $\Pr_{U \sim \mathrm{Haar}}[B_i] \geq 1 - c_2 e^{-c_3 m i}$.
Let $i_1 := \lceil \log \frac{2 L_1}{\lambda} \rceil$.
Let $\mathcal{N}_i$, $i = 0, 1, \ldots, i_1$ be a sequence of
$2^{-i}$-nets in $\mathbb{S}_{\mathbb{C}^{m'}}$ of minimum cardinality,
where $\mathcal{N}_0 := \{0\}$
and $Y_0 := 0$. We can take $|\mathcal{N}_i| \overset{a}{\leq} 2^{2 (i+2) m'}$ by
Fact~\ref{fact:net}. By Lemma~\ref{lem:chaining}
\begin{equation}
\label{eq:chainingtdesign}
\Pr_{U \sim \nu}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m'}}}
\lvert Y_s -Y_t \rvert > \lambda
] \leq
2 \sum_{i=1}^{i_1+1}
\sum_{
(u,u^\prime) \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert_2 < 2^{-i+2}
}
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
].
\end{equation}
Let $r$ be a positive integer such that $r (i_1+1) < m$.
Applying Lemma~\ref{lem:expectationtdesign} to the set $B_{i}$ gives, for
$u$, $u'$ satisfying $\lVert u- u^\prime \rVert_2 < 2^{-i+2}$,
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda \sqrt{i p(i)}}{4C \cdot 2^i}
]
} \\
& = &
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert^{2 r i} >
\left(\frac{\lambda^2 i p(i)}{2^4 C^2 2^{2i}}\right)^{r i}
]
\;\leq\;
\left(\frac{2^{2i+4} C^2}{\lambda^2 i p(i)} \right)^{r i}
\mathbb{E}_{U \sim \nu}[\lvert Y_u - Y_{u^\prime}\rvert^{2 r i}] \\
& \leq &
3 \left(\frac{2^{2i+4} C^2}{\lambda^2 i p(i)} \right)^{r i}
\left(
\left(
\frac{4 r i L_2^2 p(i) \lVert u - u^\prime \rVert_2^2}{n}
\right)^{r i}
+ c_2 e^{-c_3 m i} \cdot (L_1^2 \lVert u- u^\prime \rVert_2^2)^{r i}
\right) \\
& \leq &
3 \left(
\frac{2^{2i+6} C^2 r L_2^2 \lVert u- u^\prime \rVert_2^2}
{n \lambda^2}
\right)^{r i} +
3 c_2 e^{-c_3 m i}
\left(
\frac{2^{2i+4} C^2 L_1^2 \lVert u- u^\prime \rVert_2^2}{\lambda^2 i p(i)}
\right)^{r i} \\
& \leq &
\underbrace{
3 \left(
\frac{2^{10} C^2 r L_2^2}
{n \lambda^2}
\right)^{r i}
}_{=: \mathrm{I}} +
\underbrace{
3 c_2 e^{-c_3 m i}
\left(
\frac{2^{8} C^2 L_1^2}{\lambda^2 p(1)}
\right)^{r i}
}_{=: \mathrm{II}}.
\end{eqnarray*}
We now analyse the two terms in the above expression.
Take
\[
r :=
\frac{c_4 n \lambda^2}{2^{10} C^2 L_2^2} \cdot
\frac{1}{\lceil \log \frac{2^8 C^2 L_1^2}{\lambda^2 p(1)} \rceil}
\]
for a constant $c_4$, $0 < c_4 < 1$, $c_4$ depending only on
$C$, $c_1$, $c_3$, $p(1)$ chosen to be small enough so that
$r (i_1+1) < m$ and
$\frac{c_4 n \lambda^2}{2^{10} C^2 L_2^2} \leq \frac{c_3 m}{2}$.
Substitute $r$ back in I and II to get
\[
\mathrm{I}
\leq
3 \cdot 2^{-r i \log \log \frac{2^8 C^2 L_1^2}{\lambda^2 p(1)}},
~~~
\mathrm{II}
\leq
3 c_2 e^{-c_3 m i} 2^{\frac{c_3 m i}{2}}
<
3 c_2 e^{-c_3 m i / 2}.
\]
We choose
\[
m'' :=
r \log \log \frac{2^8 C^2 L_1^2}{\lambda^2 p(1)} <
\frac{c_3 m}{2}.
\]
This gives us
\[
\mathrm{I}
\leq
3 \cdot 2^{-m'' i},
~~~
\mathrm{II}
\leq
3 c_2 e^{-m'' i}.
\]
Thus, we have shown that
\[
\Pr_{U \sim \nu}[
\lvert Y_u - Y_{u^\prime}\rvert >
\frac{\lambda\sqrt{p(i)}}{4 C \cdot 2^i}
] \leq 3 (c_2 + 1) 2^{-m'' i}.
\]
Substituting above in Equation~\ref{eq:chainingtdesign}, we get
\begin{eqnarray*}
\lefteqn{
\Pr_{U \sim \nu}[
\sup_{s,t \in \mathbb{S}_{\mathbb{C}^{m^\prime}}}\lvert Y_s -Y_t \rvert > \lambda
]
} \\
& \leq &
2 \sum_{i=1}^{i_1+1}
\sum_{u,u^\prime \in \mathcal{N}_{i-1} \times \mathcal{N}_{i}:
\lVert u- u^\prime \rVert < 2^{-i+2}
} 3 (c_2 + 1) 2^{-m'' i} \\
& \leq &
6 (c_2+1) \sum_{i=1}^{i_1+1}
|\mathcal{N}_{i-1}| \cdot |\mathcal{N}_{i}| \cdot
2^{-m'' i}
\;\leq\;
6 (c_2+1) \sum_{i=1}^{i_1+1}
2^{4 m'(i+2)} 2^{-m'' i}
\;\leq\;
(c_2 + 1) 2^{-m'},
\end{eqnarray*}
if $m'$ is chosen as indicated above for a small enough constant
$c$, $0 < c < 1$, $c$ depending only on $c_4$, $c_1$, $C$ i.e.
$c$ depending only on $C$, $c_1$, $c_3$, $p(1)$.
Taking $t = 0$, we see that
with probability at least $1 - (c_2 + 1) 2^{-m^\prime}$ over
the choice of a uniformly random unitary from the approximate
$(2am)$-design, we have
that for all $s \in \mathbb{S}_{\mathbb{C}^{m'}}$, $|Y_s| \leq \lambda$.
This completes the proof of the theorem.
\end{proof}
\section{Strict subadditivity of minimum output von Neumann entropy for
approximate $t$-designs}
\label{sec:vonNeumannentropy}
We first apply Theorem~\ref{thm:mainHaar}
in order to directly recover Aubrun, Szarek and Werner's result
\cite{aubrun_szarek_werner_2010_main}
that channels with Haar random unitary Stinespring dilations
exhibit strict subadditivity of minimum output von Neumann entropy. In
fact, we
go beyond their result in the sense that we obtain exponentially
high probability close to one as opposed to constant
probability. After this warmup, we apply Theorem~\ref{thm:maintdesign}
in order to show that channels with approximate $n^{2/3}$-design
unitary Stinespring dilations exhibit strict
subadditivity of minimum output von Neumann entropy with exponentially high
probability close to one.
Let $k$ be a positive integer.
Consider the sphere $\mathbb{S}_{\mathbb{C}^{k^3}}$. Define the $k \times k^2$
matrix $M$ to be the rearrangment of a $k^3$-tuple from
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Note that the $\ell_2$-norm on $\mathbb{C}^{k^3}$ is
the same as the Frobenius norm on $\mathbb{C}^{k \times k^2}$.
In Step~I, we define the
function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$ as
$f(M) := \lVert M \rVert_\infty$. The function $f$ has global Lipschitz
constant $L_1 = 1$ since
\[
|f(M) - f(N)| \leq
\lVert M - N \rVert_\infty \leq
\lVert M - N \rVert_2.
\]
For large enough $k$ the mean $\mu$ of $f$, under the Haar measure, is less
than $2 k^{-1/2}$ \cite[Corollary~7]{aubrun_szarek_werner_2010_main}.
We use the notation of Theorem~\ref{thm:mainHaar}.
Define $L_2 := 1$, $p(i) := 1$ for all $i \in \mathbb{N}$. Then $C < 2$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be all of
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Let $j$, $4 \leq j \leq k$ be a positive integer.
Let $\lambda_j := \sqrt{\frac{j}{k}}$. Define $c_1 := 1$,
$m = k^2$, $c_2 := 0$, $c_3 := 1$. Trivially,
a Haar random subspace of dimension $m j$ lies in $\Omega_i$ with
probability at least $1 - c_2 e^{-c_3 m j i}$. Theorem~\ref{thm:mainHaar}
tells us that there is a universal constant $\hat{c}_1$ such that
for $m' := \hat{c}_1 k^2$, with probability at least
$1 - 2^{-m' j}$, a Haar random subspace $W$ of dimension $m' j$
satisfies
\[
\lVert M \rVert_\infty <
\frac{2}{\sqrt{k}} + \sqrt{\frac{j}{k}} <
2 \sqrt{\frac{j}{k}}
\]
for all $M \in W$.
In Step~II, we define the
function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$ as
$f(M) := \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2$.
The function $f$ has global Lipschitz
constant $L_1 = 2$ since
\begin{eqnarray*}
|f(M) - f(N)|
& \leq &
\lVert M M^\dagger - N N^\dagger \rVert_2
\; \leq \;
\lVert M M^\dagger - M N^\dagger \rVert_2 +
\lVert M N^\dagger - N N^\dagger \rVert_2 \\
& \leq &
\lVert M \rVert_\infty \lVert M^\dagger - N^\dagger \rVert_2 +
\lVert N^\dagger \rVert_\infty \lVert M - N \rVert_2 \\
& = &
(\lVert M \rVert_\infty + \lVert N \rVert_\infty)
\lVert M - N \rVert_2
\;\leq\;
2 \lVert M - N \rVert_2.
\end{eqnarray*}
The mean $\mu$ of $f$, under the Haar measure, is less
than $c_0 k^{-1}$ for a universal constant $c_0$
\cite[Corollary~7]{aubrun_szarek_werner_2010_main}.
We use the notation of Theorem~\ref{thm:mainHaar}.
Let $j$, $c_0 < j \leq k$ be a positive integer.
Define $L_2 := 4 \sqrt{\frac{j}{k}}$, $p(i) := i+3$ for all $i \in \mathbb{N}$.
Then $C \leq 4$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be
the subsets
\[
\Omega_i :=
\left\{
M \in \mathbb{S}_{\mathbb{C}^{k^3}}:
\lVert M \rVert_\infty \leq 2 \sqrt{\frac{j(i+3)}{k}}
\right\}.
\]
It is easy to see that $f$ restricted to $\Omega_i$ has local
Lipschitz constant at most $L_2 \sqrt{p(i)}$.
Let $\lambda := \frac{j}{k}$. Define $c_1 := 16 \hat{c}_1$,
$m = \hat{c}_1 j k^2$, $c_2 := 1$, $c_3 := \ln 2$. By the previous
paragraph,
a Haar random subspace of dimension $m (i+3)$ lies in $\Omega_i$ with
probability at least
$1 - c_2 e^{-c_3 m (i+3)} \geq 1 - c_2 e^{-c_3 m i}$.
Theorem~\ref{thm:mainHaar}
tells us that there is a universal constant $\hat{c}_2$ such that
for $m' := \hat{c}_2 k^2$, with probability at least
$1 - 2^{-m' j}$, a Haar random subspace $W$ of dimension $m' j$
satisfies
\[
f(M) = \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2 <
\frac{c_0}{k} + \frac{j}{k} <
\frac{2j}{k}
\]
for all $M \in W$. Setting $j = 1$ allows us to recover Aubrun, Szarek
and Werner's technical result \cite{aubrun_szarek_werner_2010_main} with
probability exponentially close to one viz.
with probability at least
$1 - 2^{-m'}$, a Haar random subspace $W$ of dimension $m'$
satisfies
$
\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2 <
\frac{2}{k}
$
for all $M \in W$. We will now see how this implies the existence
of a channel with strictly subadditive minimum output von Neumann
entropy.
\begin{fact}
\label{fact:subadditivity}
Let $k$ be a positive integer.
Let $W$ be a Haar random subspace of dimension $m := \hat{c}_2 k^2$
chosen from the Hilbert space $\mathbb{C}^{k^3}$, where $\hat{c}_2$ is a
universal constant. Let $\Phi$ be the channel with output dimension
$k$ corresponding to the
subspace $W$. Then with probability at least $1 - 2^{-m}$ over the
choice of $W$,
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - \frac{4}{k},
~~~
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) \leq
2 \log k - \frac{\hat{c}_2 \log k}{k} +
O \left(\frac{1}{k} \right).
\]
In other words,
$
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) <
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi})
$
for large enough $k$.
\end{fact}
\begin{proof}
The input dimension of the channel $\Phi$ is $\dim W = m$. The
Stinespring dilation
of the channel $\Phi$ is the $k^3 \times k^3$ unitary matrix
that defines the subspace $W$. The subspace $W$ is obtained by
taking the span of first $m$ columns of a Haar random unitary matrix.
Let $M$ be a unit $\ell_2$-norm vector in $\mathbb{C}^{k^3}$ rearranged as
a $k \times k^2$ matrix.
From Fact~\ref{fact:minoutputentropy}, we get
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - k \max_{M \in W} \lVert M M^\dag - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2^2 \geq
\log k - \frac{4}{k}.
\]
And from Fact~\ref{fact:maxeigenvalue}, with $d = k^2$, we get
\begin{eqnarray*}
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi})
& \leq &
2 \log k -
\frac{m}{kd} \log k +
O \left( \frac{m}{kd} \log \frac{d}{m} + \frac{1}{k} \right) \\
& = &
2 \log k - \frac{\hat{c}_2 \log k}{k} +
O \left(\frac{1}{k} \right) \\
& < &
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi}),
\end{eqnarray*}
for large enough $k$.
\end{proof}
Thus we have shown
that for large enough $n$, Haar random $n \times n$ unitaries give rise
to channels
exhibiting strict subadditivity of minimum output von Neumann
entropy implying that classical Holevo capacity of quantum channels can
be superadditive.
In Step~III, we define the
function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$ as
$f(M) := \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2^2$ i.e.
this $f$ is the square of the $f$ defined in Step~II above.
Now, $f$ is a balanced polynomial of degree $a = 2$ and
$1 < \alpha(f) < k^{6}$ as can be seen by considering
$f(J)$ where $J$ is the $k \times k^2$ all ones matrix.
The function $f$ has global Lipschitz
constant $L_1 = 4$ since
\begin{eqnarray*}
|f(M) - f(N)|
& \leq &
|\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2 -
\lVert N N^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2| \cdot
|\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2 +
\lVert N N^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2| \\
& \leq &
(\lVert M \rVert_\infty + \lVert N \rVert_\infty)
(\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2 +
\lVert N N^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2)
\lVert M - N \rVert_2 \\
& \leq &
4 \lVert M - N \rVert_2.
\end{eqnarray*}
The mean $\mu$ of $f$ under the Haar measure is less
than $c_0^2 k^{-2}$ for the same universal constant $c_0$
\cite[Corollary~7]{aubrun_szarek_werner_2010_main}.
We use the notation of Theorem~\ref{thm:maintdesign}.
Define $L_2 := 16 k^{-3/2} $, $p(i) := i^3$ for all $i \in \mathbb{N}$.
Then $C \leq 5$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be
the subsets
\[
\Omega_i :=
\left\{
M \in \mathbb{S}_{\mathbb{C}^{k^3}}:
\lVert M \rVert_\infty \leq 2 \sqrt{\frac{i}{k}},
\lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2 < \frac{2i}{k}
\right\}.
\]
It is easy to see that $f$ restricted to $\Omega_i$ has local
Lipschitz constant at most $L_2 \sqrt{p(i)}$.
Let $\lambda := k^{-2}$. Define $c_1 := 2^8 \hat{c}_2$,
$m = \hat{c}_2 k^2 < \hat{c}_1 k^2$, $c_2 := 2$,
$c_3 := \ln 2$. By the previous two
paragraphs,
a Haar random subspace of dimension $m i$ lies in $\Omega_i$ with
probability at least
$1 - c_2 e^{-c_3 m i}$.
In particular,
a Haar random subspace of dimension $m$ lies in $\Omega_i$ with
probability at least
$1 - c_2 e^{-c_3 m i}$.
Let
\[
0 \leq \epsilon <
\left(\frac{1}{16 k^2}\right)^{2m} \frac{k^{9m} k^{-3m}}{k^{12m}} =
(4 k)^{-10 \hat{c}_2 k^2}.
\]
Theorem~\ref{thm:maintdesign}
tells us that there is a universal constant $\hat{c}_3$ such that
for
\[
m' := \hat{c}_3 k^2 \frac{\log \log k}{\log k},
\]
with probability at least
$1 - 3 \cdot 2^{-m'}$, a subspace $W$ of dimension $m'$ chosen from an
$\epsilon$-approximate $(4 \hat{c}_2 k^2)$-design $\nu$ satisfies
\[
f(M) = \lVert M M^\dagger - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k}\rVert_2^2 <
\frac{c_0^2}{k^2} + \frac{1}{k^2} =
\frac{c_0^2 + 1}{k^2}
\]
for all $M \in W$.
We shall now see how this result gives us a channel with strict
subadditivity of minimum output von Neumann entropy.
\begin{theorem}
\label{thm:subadditivity}
Let $k$ be a positive integer.
Let $W$ be a subspace of dimension
$
m' := \hat{c}_3 k^2 \frac{\log \log k}{\log k}
$
chosen with uniform probability from a
$k^{-8 \hat{c}_2 k^2}$-approximate unitary $(4 \hat{c}_2 k^2)$-design
from the Hilbert space $\mathbb{C}^{k^3}$, where $\hat{c}_2$, $\hat{c}_3$ are
universal constants. Let $\Phi$ be the channel with output dimension
$k$ corresponding to the
subspace $W$. Then with probability at least $1 - 3 \cdot 2^{-m'}$ over the
choice of $W$,
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - \frac{c_0}{k},
~~~
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) \leq
2 \log k - \frac{\hat{c}_3 \log \log k}{k} +
O \left(\frac{(\log \log k)^2}{k \log k} + \frac{1}{k} \right),
\]
for a universal constant $c_0$.
In other words,
$
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi}) <
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi})
$
for large enough $k$.
\end{theorem}
\begin{proof}
The input dimension of the channel $\Phi$ is $\dim W = m'$. The
Stinespring dilation
of the channel $\Phi$ is the $k^3 \times k^3$ unitary matrix
that defines the subspace $W$. The subspace $W$ is obtained by
taking the span of first $m'$ columns of the unitary matrix. This unitary
matrix is chosen uniformly at random from a
$k^{-8 \hat{c}_2 k^2}$-approximate
unitary $(4 \hat{c}_2 k^2)$-design.
Let $M$ be a unit $\ell_2$-norm vector in $\mathbb{C}^{k^3}$ rearranged as
a $k \times k^2$ matrix.
From Fact~\ref{fact:minoutputentropy}, we get
\[
S_{\mathrm{min}}(\Phi)\geq
\log k - k \max_{M \in W} \lVert M M^\dag - \frac{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}{k} \rVert_2^2 \geq
\log k - \frac{c_0^2 + 1}{k}.
\]
And from Fact~\ref{fact:maxeigenvalue}, with $d = k^2$, we get
\begin{eqnarray*}
S_{\mathrm{min}}(\Phi \otimes \bar{\Phi})
& \leq &
2 \log k -
\frac{m^\prime}{kd} \log k +
O \left( \frac{m'}{kd} \log \frac{d}{m'} + \frac{1}{k} \right) \\
& = &
2 \log k - \frac{\hat{c}_3 \log \log k}{k} +
O \left( \frac{(\log \log k)^2}{k \log k} + \frac{1}{k} \right) \\
& < &
S_{\mathrm{min}}(\Phi) + S_{\mathrm{min}}(\bar{\Phi}),
\end{eqnarray*}
for large enough $k$.
\end{proof}
Thus we have shown
that for large enough $n$, approximate unitary $n^{2/3}$-designs give rise
to channels
exhibiting strict subadditivity of minimum output von Neumann
entropy, implying that classical Holevo capacity of quantum channels can
be superadditive.
\paragraph{Remark:}
Observe that the counter example we get for additivity conjecture for
classical Holevo capacity of quantum channels, when the channel is chosen
from an approximate unitary $t$-design has weaker parameters
than a channel chosen from Haar random unitaries.
Nevertheless, as explained in the introduction our work
is the first partial derandomisation of a construction of quantum
channels violating additivity of classical Holevo capacity.
\section{Strict subadditivity of minimum output R\'{e}nyi $p$-entropy for
approximate $t$-designs}
\label{sec:Renyipentropy}
In this section, we apply Proposition~\ref{prop:poly} and
Theorem~\ref{thm:maintdesign}
in order to show that channels with approximate
$(n^{1.7} \log n)$-design
unitary Stinespring dilations exhibit strict
subadditivity of minimum output R\'{e}nyi $p$-entropy for
$p > 1$ with exponentially high
probability close to one.
Let $k$ be a positive integer.
Consider the sphere $\mathbb{S}_{\mathbb{C}^{k^3}}$. Define the $k \times k^{2}$
matrix $M$ to be the rearrangment of a $k^3$-tuple from
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Note that the $\ell_2$-norm on $\mathbb{C}^{k^3}$ is
the same as the Frobenius norm on $\mathbb{C}^{k \times k^{2}}$.
Let $1 < p \leq 1.1$.
In Step~I, we define the function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$
as $f(M) := \lVert M \rVert_{2p}$. The function $f$ has global Lipschitz
constant $L_1 = 1$ since
\[
|f(M) - f(N)| \leq
\lVert M - N \rVert_{2p} \leq
\lVert M - N \rVert_{2}.
\]
For large enough $k$ the mean $\mu$ of $f$, under the Haar measure,
is less than $2 k^{\frac{1}{2p} - \frac{1}{2}}$
\cite[Section~VIII]{aubrun_szarek_werner_2010},
\cite[Corollary~7]{aubrun_szarek_werner_2010_main}. We use the notation
of Theorem~\ref{thm:mainHaar}.
Define $L_2 := 1$, $p(i) := 1$ for all $i \in \mathbb{N}$. Then $C < 2$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be all of
$\mathbb{S}_{\mathbb{C}^{k^3}}$. Let $j$, $4 \leq j \leq k$ be a positive integer.
Let $\lambda_j := j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}$.
Define $c_1 := 1$,
$m = k^{2 + \frac{1}{p}}$, $c_2 := 0$, $c_3 := 1$. Trivially,
a Haar random subspace of dimension $m j$ lies in $\Omega_i$ with
probability at least $1 - c_2 e^{-c_3 m j i}$. Theorem~\ref{thm:mainHaar}
tells us that there is a universal constant $\hat{c}_1$ such that
for $m' := \hat{c}_1 k^{2 + \frac{1}{p}}$, with probability at least
$1 - 2^{-m' j}$, a Haar random subspace $W$ of dimension $m' j$
satisfies
\[
\lVert M \rVert_{\infty} \leq
\lVert M \rVert_{2p} <
2 k^{\frac{1}{2p} - \frac{1}{2}} +
j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}} <
2 j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}
\]
for all $M \in W$. In particular, with probability at least
$1 - 2^{-\hat{c}_1 j k^{\frac{4}{3p} + \frac{5}{3}} (\log k)^{-1}}$,
a Haar random
subspace $W$ of dimension
$\hat{c}_1 j k^{\frac{4}{3p} + \frac{5}{3}} (\log k)^{-1}$ satisfies
\[
\lVert M \rVert_{\infty} \leq
\lVert M \rVert_{2p} <
2 j^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}
\]
for all $M \in W$.
Let $j$, $4 \leq j \leq k$ be a positive integer.
Define the function $f: [0, 1] \rightarrow [0, 1]$ as
$f(x) := x^p$.
Set $\epsilon := k^{-p}$ in Proposition~\ref{prop:poly}. Let
$n$ be the minimum positive odd integer satisfying
$
2 p k^p \sqrt{\ln k^{2p}} \leq
\frac{k^{-\frac{p}{n}} \sqrt{n}}{2};
$
$n < 2^7 p^3 k^{2p} \log k$.
Proposition~\ref{prop:poly} implies
that there is a polynomial $p(x)$ of degree at most
$2n + 1 < 2^{9} p^3 k^{2p} \log k$ such that
\begin{equation}
\label{eq:derivative}
\begin{array}{l l}
p(x) - 2 k^{-p} \leq x^p \leq p(x) + 3 k^{-p}, &
\forall x \in [0, 1], \\
|p'(x)| <
4 p (j+1)^{p-1} \sqrt{\ln k^{2p}}
k^{\frac{5}{3} - \frac{2}{3p} - p}, &
\forall x \in [0, j k^{\frac{2}{3p} - 1}], \\
|p'(x)| <
4 p (5j)^{p-1} \sqrt{\ln k^{2p}}
k^{2 - p - \frac{1}{p}}, &
\forall x \in (j k^{\frac{2}{3p} - 1}, 5 j k^{\frac{1}{p} - 1}], \\
|p'(x)| <
4 p \sqrt{\ln k^{2p}}, &
\forall x \in (5 j k^{\frac{1}{p} - 1}, 1].
\end{array}
\end{equation}
Also, Proposition~\ref{prop:poly} guarantees that
$\alpha(p(x)) < e^{2^7 p^3 k^{2p} \log k}$.
In Step~II, we define the function $f: \mathbb{S}_{\mathbb{C}^{k^3}} \rightarrow \mathbb{R}$
as $f(M) := \mathrm{Tr}\, [p(M M^\dag)]$, where $p$ is the polynomial defined in
Equation~\ref{eq:derivative}.
Now, $f$ is a balanced polynomial of degree
$a = 2n + 1 < 2^{9} p^3 k^{2p} \log k$ and
\[
\alpha(f) = \mathrm{Tr}\, [p(J J^\dag)] = k^3 \alpha(p(x)) <
e^{2^8 p^3 k^{2p} \log k},
\]
where $J$ is the $k \times k^2$ all ones matrix.
For a $k \times k$ matrix $X$, define
$\mathrm{Sing}(X)$ to be the $k \times k$ diagonal matrix consisting of the
singular values of $X$ arranged in decreasing order.
The function $f$ has global Lipschitz
constant $L_1 = 2^{4} p^{3/2} \sqrt{\log k}$ since
\begin{eqnarray*}
|f(M) - f(N)|
& = &
|\mathrm{Tr}\, [p(\mathrm{Sing}(M)^2)] - \mathrm{Tr}\, [p(\mathrm{Sing}(N)^2)]|
\; = \;
|\mathrm{Tr}\, [p(\mathrm{Sing}(M)^2)-p(\mathrm{Sing}(N)^2)]| \\
& \leq &
8 p^{3/2} \sqrt{\log k} \cdot
\lVert \mathrm{Sing}(M)^2 - \mathrm{Sing}(N)^2 \rVert_1 \\
& \leq &
8 p^{3/2} \sqrt{\log k} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \cdot
\lVert \mathrm{Sing}(M) + \mathrm{Sing}(N) \rVert_2 \\`
& \leq &
2^{7/2} p^{3/2} \sqrt{\log k} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \cdot
\sqrt{\lVert M \rVert_2^2 + \lVert N \rVert_2^2} \\
& \leq &
2^{4} p^{3/2} \sqrt{\log k} \cdot
\lVert M - N \rVert_2.
\end{eqnarray*}
Above, the first inequality follows from Equation~\ref{eq:derivative},
the second inequality is Cauchy-Schwarz and the last inequality
follows from \cite[Section~4]{Mirsky}.
By setting $j = 4$ in Step~I, we conclude that
the mean $\mu$ of $f$ under the Haar measure is less
than $2^{4p} k^{1-p}$.
We use the notation of Theorem~\ref{thm:maintdesign}.
Let $\lambda := k^{1 - p}$.
Define
\[
L_2 :=
2^{4p+3} p^{3/2}
\sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}},
\]
$p(i) := (i+4)^{2p-1}$ for all $i \in \mathbb{N}$.
Then $C \leq p^{2p}$.
Define the layers $\Omega_1, \Omega_2, \ldots, $ to be
the subsets
\[
\Omega_i :=
\left\{
M \in \mathbb{S}_{\mathbb{C}^{k^3}}:
\lVert M \rVert_{2p} \leq
2 (i+3)^{\frac{1}{2}} k^{\frac{1}{2p}-\frac{1}{2}}
\right\}.
\]
We will now show that $f$ restricted to $\Omega_i$ has local
Lipschitz constant at most $L_2 \sqrt{p(i)}$. Note that for any
$M \in \Omega_i$,
$
\lVert M \rVert_\infty \leq
2 (i+3)^{\frac{1}{2}} k^{\frac{1}{2p} - \frac{1}{2}}.
$
Let $B$ denote the number of singular values of $M$ larger than
$(i+3)^{\frac{1}{2}} k^{\frac{1}{3p} - \frac{1}{2}}$.
Let $b_1, \ldots b_k$ be the singular values of $M$ in descending order.
Then
\[
2^{2p} (i+3)^p k^{1-p} \geq
\lVert M \rVert_{2p}^{2p} \geq
\sum_{i=1}^B b_i^{2p} \geq
\left(\sum_{i=1}^B b_i^{2}\right)
(i+3)^{p-1} k^{\frac{5}{3} - \frac{2}{3p} - p},
\]
which gives
$
\sum_{i=1}^B b_i^{2} \leq
2^{2p} (i+3) k^{\frac{2}{3p} - \frac{2}{3}}.
$
Let $C$ denote the number of singular values of $N$ larger than
$(i+3)^{\frac{1}{2}} k^{\frac{1}{3p} - \frac{1}{2}}$.
Without loss of generality, $B \geq C$. Restricting $M$, $N$
to belong to $\Omega_i$, we get from Equation~\ref{eq:derivative} that
\begin{eqnarray*}
\lefteqn{
|f(M) - f(N)|
} \\
& = &
|
\mathrm{Tr}\, [
p(\mathrm{Sing}(M)^2) - p(\mathrm{Sing}(N)^2)
]
| \\
& \leq &
\sum_{i=1}^C |p(b_i^2) - p(c_i^2)| +
\sum_{i=C+1}^B |p(b_i^2) - p(c_i^2)| +
\sum_{i=B+1}^k |p(b_i^2) - p(c_i^2)| \\
& \leq &
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sum_{i=1}^C |b_i^2 - c_i^2| \\
& &
{} +
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sum_{i=C+1}^B |b_i^2 - c_i^2| \\
& &
{} +
8 p^{3/2} ((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\sum_{i=B+1}^k |p(b_i^2) - p(c_i^2)| \\
& \leq &
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=1}^C (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=1}^C (b_i + c_i)^2} \\
& &
{} +
8 p^{3/2} (5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=C+1}^B (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=C+1}^B (b_i + c_i)^2} \\
& &
{} +
8 p^{3/2} ((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\sqrt{\sum_{i=B+1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=B+1}^k (b_i + c_i)^2} \\
& \leq &
2^{7/2} p^{3/2}
(5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=1}^C (b_i^2 + c_i^2)} \\
& &
{} +
2^{7/2} p^{3/2}
(5 (i+3))^{p-1} \sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}}
\sqrt{\sum_{i=1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=C+1}^B (b_i^2 + c_i^2)} \\
& &
{} +
2^{7/2} p^{3/2}
((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\sqrt{\sum_{i=1}^k (b_i - c_i)^2} \cdot
\sqrt{\sum_{i=1}^k (b_i^2 + c_i^2)} \\
& \leq &
2^{4} p^{3/2} 2^p
5^{p-1} (i+3)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}} \cdot
k^{\frac{1}{3p} - \frac{1}{3}} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& &
{} +
2^{4} p^{3/2} 2^p
5^{p-1} (i+3)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot k^{2 - p - \frac{1}{p}} \cdot
k^{\frac{1}{3p} - \frac{1}{3}} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& &
{} +
2^{4} p^{3/2}
((i+4))^{p-1} \sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}}
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& \leq &
2^{6} p^{3/2} 2^p
5^{p-1} (i+4)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}} \cdot
\lVert \mathrm{Sing}(M) - \mathrm{Sing}(N) \rVert_2 \\
& \leq &
2^{4p+3} p^{3/2}
(i+4)^{p - \frac{1}{2}}
\sqrt{\log k} \cdot
k^{\frac{5}{3} - p - \frac{2}{3p}} \cdot
\lVert M - N \rVert_2.
\end{eqnarray*}
This completes the proof of the claim above that $f$ restricted to
$\Omega_i$ has local Lipschitz constant at most $L_2 \sqrt{p(i)}$.
Define $c_1 := 2^{8p+6} p^3 \hat{c}_1$,
$m = \hat{c}_1 k^{\frac{4}{3p}+\frac{5}{3}} (\log k)^{-1}$, $c_2 := 1$,
$c_3 := \ln 2$. By Step~I,
a Haar random subspace of dimension $m i$ lies in $\Omega_i$ with
probability at least $1 - c_2 e^{-c_3 m i}$.
Let
\[
0 \leq \epsilon <
k^{\hat{c}_1 k^{2p + 3} (\log k)^{-1/2}} <
\left(\frac{k^{1 - p}}{4 L_1}\right)^{2m}
\frac{k^{3 (2a-1) m} 5^{(2p-1)m} L_2^{2m}}{\alpha(f)^{2m}}.
\]
Theorem~\ref{thm:maintdesign}
tells us that there is a universal constant $\hat{c}_3$ such that
for
\[
m' :=
\hat{c}_3 k^{\frac{4}{3p} + \frac{5}{3}} \frac{\log \log k}{(\log k)^2},
\]
with probability at least
$1 - 2 \cdot 2^{-m'}$, a subspace $W$ of dimension $m'$ chosen from an
$\epsilon$-approximate $(2am)$-design $\nu$
satisfies
\[
f(M) = \mathrm{Tr}\, [p(M M^\dag)] <
2^{4p} k^{1-p} + k^{1-p} <
2^{4p+1} k^{1-p}
\]
for all $M \in W$. By Equation~\ref{eq:derivative}, this implies that
\[
\mathrm{Tr}\, [(M M^\dag)^p] <
\mathrm{Tr}\, [p(M M^\dag)] + 3 k^{1-p} <
2^{4p+3} k^{1-p}
\]
for all $M \in W$. In other words,
$
\lVert M \rVert_{2p}^2 < 2^7 k^{\frac{1}{p} - 1}
$
for all $M \in W$.
We shall now see how this result gives us a channel with strict
supermultiplicativity of the $\lVert \cdot \rVert_{1 \rightarrow p}$-norm
or equivalently, strict
subadditivity of minimum output R\'{e}nyi $p$-entropy for any $p > 1$.
\begin{theorem}
\label{thm:supermultiplicativity}
Let $k$ be a positive integer. Let $1 < p \leq 1.1$.
Let $W$ be a subspace of dimension
$
m' :=
\hat{c}_3 k^{\frac{4}{3p} + \frac{5}{3}} \frac{\log \log k}{(\log k)^2}
$
chosen with uniform probability from a
$k^{\hat{c}_1 k^{5} (\log k)^{-1/2}}$-approximate unitary
$(2^{11} \hat{c}_1 k^{5.1})$-design
from the Hilbert space $\mathbb{C}^{k^3}$, where $\hat{c}_1$, $\hat{c}_3$ are
universal constants. Let $\Phi$ be the channel with output dimension
$k$ corresponding to the
subspace $W$. Then with probability at least $1 - 2 \cdot 2^{-m'}$ over the
choice of $W$,
\[
\lVert \Phi \rVert_{1 \rightarrow p} \leq
2^{7} k^{\frac{1}{p} - 1},
~~~
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p} \geq
\hat{c}_3 k^{\frac{4}{3p} - \frac{4}{3}}.
\]
In other words,
$
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p} >
\lVert \Phi \rVert_{1 \rightarrow p} \cdot
\lVert \bar{\Phi} \rVert_{1 \rightarrow p} \cdot
$
for large enough $k$. For $p > 1.1$, the channel $\Phi$ obtained
for $p = 1.1$ suffices to show supermultiplicativity.
\end{theorem}
\begin{proof}
The input dimension of the channel $\Phi$ is $\dim W = m'$. The
Stinespring dilation
of the channel $\Phi$ is the $k^3 \times k^3$ unitary matrix
that defines the subspace $W$. The subspace $W$ is obtained by
taking the span of first $m'$ columns of the unitary matrix. This unitary
matrix is chosen uniformly at random from a
$k^{\hat{c}_1 k^{5} (\log k)^{-1/2}}$-approximate unitary
$(2^{11} \hat{c}_1 k^{5.1})$-design. Note that
$
2 a m < 2^{11} \hat{c}_1 k^{5.1},
$
$
\epsilon > k^{\hat{c}_1 k^{5} (\log k)^{-1/2}},
$
where $a$, $m$ and $\epsilon$ are defined in Step~III above.
Let $M$ be a unit $\ell_2$-norm vector in $\mathbb{C}^{k^3}$ rearranged as
a $k \times k^2$ matrix.
From Equation~\ref{eq:redefpnorm}, we get
\[
\lVert \Phi \rVert_{1 \rightarrow p} =
\max_{M \in W: \lVert M \rVert_2 = 1}
\lVert M \rVert_{2p}^2 \leq
2^{7} k^{\frac{1}{p} - 1}.
\]
From Fact~\ref{fact:maxeigenvalue},
\[
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p} \geq
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow \infty} \geq
\frac{m'}{k^3} =
\hat{c}_3 k^{\frac{4}{3p} - \frac{4}{3}}
\frac{\log \log k}{(\log k)^2} >
(\lVert \Phi \rVert_{1 \rightarrow p})^2
\]
for large enough $k$.
This shows the supermultiplicativity of the
$\lVert \cdot \rVert_{1 \rightarrow p}$-norm for $1 < p \leq 1.1$. For
$p > 1.1$, we use the fact that
$
\lVert \cdot \rVert_{1 \rightarrow \infty} \leq
\lVert \cdot \rVert_{1 \rightarrow p} \leq
\lVert \cdot \rVert_{1 \rightarrow 1.1}
$
to conclude the supermultiplicativity of
$\lVert \cdot \rVert_{1 \rightarrow p}$.
\end{proof}
Thus by setting $p = 1.1$, we see
that for large enough $n$ approximate unitary
$(n^{1.7} \log n)$-designs give
rise to channels
exhibiting strict subadditivity of minimum output R\'{e}nyi $p$-entropy
for any $p > 1$. Combined with the result of the previous section,
we can furthermore state
that for large enough $n$ approximate unitary
$(n^{1.7} \log n)$-designs give
rise to channels
exhibiting strict subadditivity of minimum output R\'{e}nyi $p$-entropy
for any $p \geq 1$.
\paragraph{Remarks:}
\noindent
1.\
In \cite{aubrun_szarek_werner_2010}, for channels obtained from
Haar random subspaces the lower bound on
$
\lVert \Phi \otimes \bar{\Phi} \rVert_{1 \rightarrow p}
$
was of the order of
$
k^{\frac{1}{p} - 1},
$
whereas in our work it is
of the order of
$
k^{\frac{4}{3p} - \frac{4}{3}},
$
for channels obtained from approximate $t$-designs.
Hence the counter example we get for additivity of minimum output
R\'{e}nyi $p$-entropy of quantum channels, when the channel is
chosen
from an approximate unitary $t$-design has weaker parameters
than the Haar random channels of
\cite{aubrun_szarek_werner_2010}. Nevertheless, our work
is the first partial derandomisation of a construction of quantum
channels violating additivity of minimum output
R\'{e}nyi $p$-entropy, since it is possible to uniformly
sample a unitary from an exact
$(n^{1.7} \log n)$-design
using of the order of
$
n^{1.7} (\log n)^2
$
random bits versus $\Omega(n^2)$ random bits required to choose
a Haar random unitary to constant precision.
\smallskip
\noindent
2.\
It is possible to do the above counterexample on a sphere in $\mathbb{C}^{k^2}$.
However
in that case the number of random bits required to choose a
unitary from an exact design is larger than $k^4 \log k$, which is what a
Haar random unitary would require!
\section{Conclusion}
\label{sec:conclusion}
In this paper we have shown that a unitary chosen from an approximate
unitary $n^{2/3}$-design leads
to a quantum channel with superadditive classical Holevo capacity. In the
process of coming up
with such a channel we developed two new technical tools viz.
stratified analysis
of a sphere in $\mathbb{C}^n$ for Haar measure and unitary designs
(Theorems~\ref{thm:mainHaar}, \ref{thm:maintdesign}),
and approximation of any continuous monotonic function by
a polynomial of moderate degree (Proposition~\ref{prop:poly}).
The stratified analysis for the Haar measure was used to
recover in a simple fashion Aubrun, Szarek and Werner's counterexample
\cite{aubrun_szarek_werner_2010_main} for
additivity of minimum output Von Neumann entropy. The stratified
analysis for unitary designs was used to prove
counterexamples for additivity of minimum output von Neumann entropy and
R\'{e}nyi $p$-entropy for $p > 1$, when the unitary Stinespring dilation
of the channel is chosen from approximate unitary $t$-design for
suitable values of $t$. Choosing a unitary from these $t$-designs requires
less random bits than choosing from the Haar measure.
However the value of $t$ required is much larger than what is known to
be efficiently implementable by quantum circuits.
We believe our work results in a better understanding of the interplay
between geometric functional analysis and additivity questions in
quantum information theory, and our technical tools will find applications
to other problems in quantum information theory.
Our work represents a step in the quest for an efficient explicit
channel violating additivity of minimum output von Neumann entropy. This
is the major open problem in the area. Another problem left open is
whether there is a single channel that violates additivity of
minimum output R\'{e}nyi $p$-entropy for all $p \geq 1$.
|
{
"timestamp": "2019-04-18T02:18:48",
"yymm": "1902",
"arxiv_id": "1902.10808",
"language": "en",
"url": "https://arxiv.org/abs/1902.10808"
}
|
\section{Introduction}
\label{intro}
Warped disks are expected to occur in a large number of astrophysical
situations \citep[e.g.][]{Pringle1981,Pringle1999,Kingetal2013}.
Warping may occur due to external torques from various
sources. Binaries can provide such a torque on a circumstellar disk
from an external binary component
\citep[e.g.][]{PT1995,Larwoodetal1996,LO2000,Martinetal2009,Martinetal2011}
or a torque on a circumbinary disk form an internal binary
\citep[e.g.][]{Facchinietal2013,Lodato2013,Martin2017}. Around spinning black
holes, general relativistic Lense--Thirring precession may cause
warping \citep{BP1975} in X--ray binaries
\citep[e.g.][]{SF1996,WP1999, Ogilvie2001, Martinetal2007} and around
supermassive black holes \citep[e.g.][]{Herrnstein1996,Martin2008}.
Disks in AGN and in binary X-ray sources may be warped by the effects
of radiation pressure \citep{Pringle1996,OD2001}. Disks may also be
warped by a misaligned magnetic field \citep[e.g.,][]{Lai1999} or a planet \citep[e.g.][]{Nealon2018}.
The evolution of the warp in a disk depends upon how the \cite{SS1973}
viscosity $\alpha$ parameter compares with the disk aspect ratio
$H/R$. If $\alpha>H/R$, then the warp propagates diffusively through
the effects of viscosity. If $\alpha<H/R$, then pressure forces drive
the evolution and the warp propagates as a bending wave that travels
at half the sound speed, $c_{\rm s}/2$ \citep{PL1995,Pringle1999,LO2002}. In this
case, the viscosity is too small to damp the wave locally. For a
recent review of warped disks, see \cite{Nixon2016}.
One-dimensional (based on radius) models of disk tilt evolution offer
advantages over multi-dimensional models. They permit tracking the
evolution over long timescales with much less computational effort
than multi-dimensional models. They are also easier to interpret
physically. On the other hand, their applicability is limited by the
simplifications made to reduce the dimensionality. In any case, such
models can be compared with multi-dimensional models to obtain more
physical insight.
A one-dimensional model should ideally conserve angular momentum and
be valid for arbitrary tilts and warps (the derivative of the tilt
with respect to the logarithm of the radius). In the viscous regime,
\cite{Pringle1992} developed a set of intuitively-based
one-dimensional equations that satisfy these conditions.
\cite{Ogilvie1999} extended this analysis by directly working with the
fluid equations and obtained one-dimensional equations that apply for
even large warps. This analysis showed that the \cite{Pringle1992}
equations are valid for small warps and small $\alpha$, $H/r < \alpha
\ll 1$, but arbitrary tilts, with the extension that the effective
viscosity coefficients are constrained by the internal fluid
dynamics.
In application to disks around young stars, the wave--like regime is
of importance. One-dimensional linear disk evolution equations for
this regime typically assume that the warp is small and ignore disk
surface density evolution \citep[e.g.,][]{PL1995,
LO2000}. \cite{Ogilvie2006} analyzed the nonlinear dynamics of free
warps (imposed by initial conditions) in the absence of viscous
density evolution. However, as found in \cite{Bateetal2000},
significant density evolution can occur as the tilt evolves.
We are interested in the case that the disk is in good radial
communication so that the level of warping is small, as should apply
to protostellar disks. Therefore a linear analysis is often valid. We
are interested in the case that the disk tilt and surface density
change over the course of its evolution.
The goal of this work is to find a formulation
which describes the disk evolution correctly in both regimes, and manages to
connect the two. In Section~\ref{equations} we present two sets of warped disk equations, one valid in the viscous regime and one valid in the wave--like regime.
In Section~\ref{general}, we follow along the lines of \cite{Pringle1992} to extend the linear tilt evolution equations to apply to arbitrary tilts and account for viscous density evolution.
In Section~\ref{numerical} we
numerically solve the equations for an initially warped disk around a
single central object in the absence of any external torques. We draw
our conclusions in Section~\ref{conc}.
\section{Warped disk equations in the two regimes}
\label{equations}
Currently there are two sets of warped disk equations that describe
mutually exclusive regimes, the wave--like regime with $\alpha < H/R$
and the diffusive (or viscous) regime with $\alpha > H/R$. We provide
an overview of these two sets of equations in this Section. We
describe the disk as consisting of a set circular rings with spherical
radius $R$. We assume that the disk is in near Keplerian
rotation. \footnote{ Thus our results do not apply to strongly
non--Keplerian flows such as those that occur in accretion disks
close to the event horizon of a black hole.} The rings rotate with
Keplerian angular frequency $\Omega(R)=\sqrt{G M/R^3}$ about a central
object of mass $M$ and have a surface density $\Sigma(R)$. The disk
extends from inner radius $R_{\rm in}$ to outer radius $R_{\rm
out}$. The angular momentum per unit area of each ring is
\begin{equation}
\bm{L}=\Sigma R^2 \Omega \bm{l},
\label{eqL}
\end{equation}
where $\bm{l}$ is a unit vector. We consider a locally isothermal disk with aspect ratio $H/R$, where $H$ is the disk scale height.
Angular momentum transport in an accretion disk is driven by turbulent
eddies with a maximum size $H$, and maximum speed the sound speed,
$c_{\rm s}=H \Omega$. The azimuthal shear viscosity has the standard
form
\begin{equation}
\nu_1 = \alpha_1 \left(\frac{H}{R}\right)^2 R^2 \Omega
\end{equation}
\citep{SS1973}, where $\alpha_1 \simeq \alpha$, for dimensionless
parameter $\alpha<1$. The vertical shear viscosity is
\begin{equation}
\nu_2 = \alpha_2 \left(\frac{H}{R}\right)^2 R^2 \Omega,
\end{equation}
\citep{PP1983}, where $\alpha_2\simeq 1/(2\alpha)$ in the linear approximation.
\subsection{Wave--like limit equations}
In the wave--like limit, $\alpha <H/R$, we assume that the surface density does not evolve, $\partial \Sigma/ \partial t=0$. The evolution of a warped disk is described by two equations
\begin{align}
\frac{\partial \bm{G}}{\partial t}+ \omega \, \bm{l}\times \bm{G} + \alpha \Omega \bm{G}=\frac{\Sigma H^2 R^3\Omega^3}{4}\frac{\partial \bm{l}}{\partial R}
\label{lo1}
\end{align}
and
\begin{align}
\Sigma R^2 \Omega \frac{\partial \bm{l} }{\partial t} =
\frac{1}{R}\frac{\partial \bm{G}}{\partial R}+\bm{T},
\label{lo2}
\end{align}
where $\bm{G}$ is the internal disk torque and $\bm{T}$ is the external torque on the disk \citep[see equations~12 and~13 in][]{LO2000}.
The apsidal precession frequency in the plane of the disk is
\begin{equation}
\omega =\frac{\Omega^2-\kappa^2}{2 \Omega}
\end{equation}
with epicyclic frequency $\kappa$. We solve equations~(\ref{lo1}) and~(\ref{lo2}) in Section~\ref{wl} to compare to our solution to the generalised equations in the wave--like limit.
\subsection{Viscous limit equations}
In the viscous limit, $\alpha > H/R$, a warped disk is described by the evolution equation
\begin{align}
\frac{\partial \bm{L} }{\partial t} = &
\frac{3}{R}\frac{\partial}{\partial R} \left[\frac{R^{1/2}}{\Sigma} \frac{\partial}{\partial R} \left(\nu_1 \Sigma R^{1/2} \right) \bm{L} \right] \cr
& +\frac{1}{R}\frac{\partial}{\partial R} \left[
\left( \nu_2 R^2 \left|\frac{\partial \bm{l}}{\partial R} \right|^2-\frac{3}{2}\nu_1 \right) \bm{L} \right] \cr
& +\frac{1}{R}\frac{\partial}{\partial R} \left[
\frac{1}{2}\nu_2 R |\bm{L}|\frac{\partial \bm{l}}{\partial R} \right]
+\bm{T}
\label{viscous}
\end{align}
\citep{Pringle1992}. We solve equation~(\ref{viscous}) in Section~\ref{diff} to compare to our solution to the generalised equations in the viscous regime.
\section{Generalised warped disk equations}
\label{general}
We now show how it is possible to combine the above equations into
a single set that are valid in both the wave--like and the diffusive warp
propagation regimes. We follow the methods of \cite{PP1983} and \cite{Pringle1992}. We note that this is not a first principles derivation of the
evolution equations \citep[cf.,][]{Ogilvie1999, Ogilvie2006}. Conservation of mass is expressed as
\begin{equation}
\frac{\partial \Sigma}{\partial t}+\frac{1}{R}\frac{\partial}{\partial R}(R\Sigma v_R)=0,
\label{mass}
\end{equation}
where $v_R$ is the radial velocity. Conservation of angular momentum gives us
\begin{equation}
\frac{\partial \bm{L}}{\partial t }+\frac{1}{R}\frac{\partial }{\partial R}(\Sigma v_R R^3 \Omega \bm{l})
=\frac{1}{R}\frac{\partial \bm{G}}{\partial R} +\bm{T},
\label{angmom}
\end{equation}
where $\bm{G}$ is the internal disk torque and $\bm{T}$ is the
external torque on the disk. In equation (\ref{angmom}), we have
included the second term on the LHS compared to equation~(\ref{lo2}), in the wave--like equations, in order to enforce conservation of angular momentum as the disk density evolves.
\begin{figure*}
\centering
\includegraphics[width=7.5cm]{wave1cincsig.eps}
\includegraphics[width=7.5cm]{full0.eps}
\includegraphics[width=7.5cm]{full1.eps}
\includegraphics[width=7.5cm]{full10.eps}
\caption{Evolution of an initially warped disk around a single object
with no external torque with $\alpha=0.01$ and $H/R=0.1$ (in the
wave--like regime). The upper panels show the inclination and the
lower panels the surface density. Top left: The wave--like
equations~(\ref{lo1}) and~(\ref{lo2}). The other panels solve the full equations~(\ref{main}) and~(\ref{flux}) with $\beta=0$ (top right), $\beta=1$ (bottom left) and $\beta=10$ (bottom right). The times shown are every
$10\,P_{\rm in}$.}
\label{wavelike}
\end{figure*}
We take the dot product of equation~(\ref{angmom}) with $\bm{l}$ and subtract $R^2 \Omega$ times equation~(\ref{mass}) to obtain an equation for the radial velocity
\begin{equation}
v_R=\frac{ \partial \bm{G}/\partial R \cdot \bm{l}}{R\Sigma \, d (R^2 \Omega)/ d R}.
\label{vr}
\end{equation}
We substitute the radial velocity equation~(\ref{vr}) into the conservation of mass equation~(\ref{mass}) to obtain an equation for the surface density evolution
\begin{align}
\frac{\partial \Sigma }{\partial t} =
- \frac{1}{R} \frac{\partial }{\partial R}\left[ \frac{\partial \bm{G}/\partial R \cdot \bm{l} }{ d (R^2 \Omega)/ d R}\right].
\end{align}
Further, we substitute the radial velocity equation~(\ref{vr}) into the conservation of angular momentum equation~(\ref{angmom}) to obtain an equation for the evolution of the angular momentum in the disk
\begin{align}
\frac{\partial \bm{L}}{\partial t} =
& -\frac{1}{R} \frac{\partial }{\partial R}\left[ \frac{(\partial \bm{G}/\partial R \cdot \bm{l} )}{\Sigma \, d (R^2 \Omega)/d R}\bm{L}\right]
+\frac{1}{R}\frac{\partial \bm{G}}{\partial R} +\bm{T}.
\end{align}
Since the disk is in near--Keplerian rotation we take
$\Omega\propto R^{-3/2}$ and find
\begin{align}
\frac{\partial \Sigma}{\partial t}= -\frac{2}{R}\frac{\partial }{\partial R}\left[\frac{(\partial \bm{G}/\partial R \cdot \bm{l} )}{R\Omega}\right]
\label{densevol}
\end{align}
and
\begin{align}
\frac{\partial \bm{L} }{\partial t} =
-\frac{2}{R}\frac{\partial}{\partial R} \left[ \left(
\frac{( \partial \bm{G}/\partial R \cdot \bm{l})}{ \Sigma R\Omega} \right)\bm{L} \right] +
\frac{1}{R}\frac{\partial \bm{G}}{\partial R}+\bm{T}.
\label{main}
\end{align}
The key step to generalising the equations so that they are valid in
both diffusive and wave--like regimes is to now amend
equation~(\ref{lo1}) to read
\begin{align}
\frac{\partial \bm{G}}{\partial t}+ \omega \, \bm{l}\times \bm{G} &+
\alpha \Omega \bm{G}+\beta \Omega (\bm{G}\cdot \bm{l})\bm{l} = \cr
&\frac{\Sigma H^2
R^3\Omega^3}{4}\frac{\partial \bm{l}}{\partial R} - \frac{3}{2}
(\alpha+\beta) \nu_1 \Sigma R^2 \Omega^2 \bm{l}.
\label{flux}
\end{align}
To effect this generalisation we have found it necessary to introduce
two extra terms dependent on a new dimensionless parameter
$\beta$. The fourth term on the left hand side has the effect of
damping the component of disk torque $\bm{G}$ perpendicular to the
local disk plane. The addition of the final term on the right hand
side is to add an additional shear viscosity term. At this stage the
magnitude of $\beta$ is arbitrary, except that we shall require that
$\beta \gg \alpha$. We show the effects of different values for
$\beta$ in Section~\ref{numerical}.
\subsection{The new generalised equation in the two limits}
Equations~(\ref{main}), and~(\ref{flux}) provide a one-dimensional description of both the the disk surface density and the disk tilt. We now show that this generalised equation has the previous equations (Section~\ref{equations}) in both limits.
\begin{enumerate}
\item In the wave-like limit we have $\alpha < H/R \ll 1$. The
equations derived in this limit assumed that the surface density
did not change with time, because in this limit the wave-like warp
propagation happens on the shorter timescale than the viscous
evolution of the surface density. Thus, in this limit, the final
term on the right hand side of equation~(\ref{flux}) is
negligible. In addition the assumption that $\Sigma$ is independent
of time implies that $v_{\rm R} = 0$, unless there is an
external source of mass. Thus (equation~\ref{vr}) we may take
$\partial \bm{G}/\partial R \cdot \bm{l}=0$ and we may ignore the
fourth term on the left hand side. Given this,
equation~(\ref{flux}) now reduces to equation~(\ref{lo1}), as
required. We note, however, that the full solution to the new
equations allows for the evolution of the surface density also in
the wave-like regime. The degree to which the surface density
evolves depends on the magnitude of the new parameter $\beta$.
\item In the viscous limit $(\alpha > H/R)$, $\bm{G}$ evolves on a
viscous timescale and so $\partial \bm{G}/\partial t \ll \alpha
\Omega \bm{G}$ and we set $\partial \bm{G}/\partial t=0$.
Furthermore, we set $\omega=0$ provided that $\omega \ll \alpha
\Omega$ and we are left with
\begin{align}
\alpha \Omega \bm{G}+ &\beta \Omega (\bm{G}\cdot \bm{l})\bm{l} = \cr
&\frac{\Sigma H^2
R^3\Omega^3}{4}\frac{\partial \bm{l}}{\partial R} - \frac{3}{2}
(\alpha+\beta) \nu_1 \Sigma R^2 \Omega^2 \bm{l}.
\label{eq1}
\end{align}
We take the dot product of this with $\bm{l}$ to find an expression for
$\bm{G}\cdot \bm{l} $ and then substitute that into equation~(\ref{eq1}) to find
\begin{equation}
\bm{G}=\frac{1}{2}\nu_2 \Sigma R^3 \Omega \frac{\partial \bm{l}}{\partial R} - \frac{3}{2}\nu_1 \Sigma R^2 \Omega \bm{l}.
\label{viscouslimit}
\end{equation}
Substituting this equation for $\bm{G}$ into equation~(\ref{main}), we recover the viscous disk evolution equation~(\ref{viscous}), which is valid for $H/r < \alpha \ll 1$ \citep{Ogilvie1999}.
\end{enumerate}
\section{Numerical solutions}
\label{numerical}
We solve equations~(\ref{main}) and~(\ref{flux}) as an initial
value problem for $\bm{L}$, and $\bm{G}$ using finite differences.
The method is first--order explicit in time. We use Cartesian
coordinates and treat each component of the vectors separately. The
units in the code are defined with $G=M=1$, where $M$ is the mass of
the central object. The Keplerian orbital period at the inner disk
radius $R_{\rm in}=1$ is $P_{\rm in}=2\pi$. We take the boundary
conditions that $\bm{G}=\bm{0}$, $\Sigma=0$ and $\partial
\bm{l}/\partial R=0$ at $R=R_{\rm in}$ and $R=R_{\rm out}$. The
initial condition on $\bm{G}$ is always taken as
$\bm{G}(R,0)=\bm{0}$.
We consider the evolution of an initially warped disk around a single
central object. There is no external torque on the disk, so $\bm{T}=0$
and $\omega=0$. The disk extends from $R_{\rm in}=1$ up to $R_{\rm
out}=20$. We take the initial surface density of the disk to be
distributed as a simple power law with ends truncated at $R_{\rm in}$ and $R_{\rm out}$
\begin{equation}
\Sigma(R,0)=\Sigma_0 \left(\frac{R}{R_{\rm in}}\right)^{-1/2}\left[1-\left(\frac{R_{\rm in}}{R}\right)^{\frac{1}{2}}\right]\left[1-e^{R-R_{\rm out}}\right].
\end{equation}
The constant $\Sigma_0$ is arbitrary as the equations are linear in
$\Sigma$. Here we have scaled the total disk mass to be $0.001\,M$.
The first factor in brackets on the RHS is a power law that represents
a steady disk with $\nu_1 \Sigma \propto \,\rm const$ if mass is added
at the outer edge. The second and third factors enforce zero torque
($\Sigma=0$) inner and outer boundary conditions, respectively. Note
that the surface density is not in steady state since we do not add
material to the disk.
The initial tilt of the disk is described by
\begin{equation}
i(R,0)=10^\circ \,\left[\frac{1}{2}\tanh \left( \frac{R-R_{\rm warp}}{R_{\rm width}}\right)+\frac{1}{2}\right].
\end{equation}
Since the equations are linear in disk tilt, the normalisation of $i$ is arbitrary.
The disk has an inclination of zero at the inner disk edge, an
inclination of $10^\circ$ at the outer disk edge and a warp at radius $R_{\rm
warp}=10$ with a width of radius $R_{\rm width}=2$. There is no twist in
the disk. Since we do not have any torques to cause precession, the
disk remains untwisted throughout its evolution. Thus we consider only the inclination of
the disk and not the nodal precession angle.
\subsection{Wave--like propagation; $\alpha< H/R$}
\label{wl}
We consider the evolution of an initially warped disk with parameters
in the wave--like limit, $\alpha=0.01$ and $H/R=0.1$.
Figure~\ref{wavelike} shows the disk inclination and surface density
evolution for several cases. In the top left panel we solve the
wave--like warped disk equations~(\ref{lo1}) and~(\ref{lo2}) with a
fixed density distribution. The warp in the disk propagates both
inwards and outwards. The inwards propagating wave reflects off the
inner boundary and then begins to propagate outwards.
In the other panels of Fig.~\ref{wavelike} we solve the full disk
equations~(\ref{main}) and~(\ref{flux}) with different values for
$\beta$. In the top right panel we show the behaviour that occurs if
we do not introduce the parameter $\beta$. With $\beta=0$ the result
is that there appears to be unphysical evolution of the disk surface
density which occurs where the initial warp change was strongest, and
which continues long after the initial warp has propagated away.
The surface density anomaly shuld not keep growing at the position
of the initial tilt change, even when the tilt at that point has
evolved elsewhere. Furthermore, this behaviour is not seen in three
dimensional hydrodynamical simulations \citep[e.g.][]{Nealon2015}.
This unphysical behaviour was the reason for introducing the new
parameter $\beta$. The inclination evolution is very similar when we
solve the wave--like equations (top left panel) or the full equations
for $\beta \gtrsim 1$ (bottom panels). However, there is surface
density evolution when we solve the full equations and angular
momentum is conserved. The bottom two panels show that for $\beta
\gtrsim 1$, the surface density evolution is independent of the value
for $\beta$. There is slight difference between $\beta=1$ and
$\beta=10$ but we find no difference for even higher $\beta$ compared
to $\beta=10$.
\begin{figure}
\centering
\includegraphics[width=7.5cm]{diff10.eps}
\caption{Evolution of an initially warped disk around a single object
with no external torque with $\alpha=0.1$ and $H/R=0.01$ (in the
viscous regime). The upper panels show the inclination and the
lower panels the surface density.
The full equations are solved with $\beta=10$. The times shown are every
$500\,P_{\rm in}$ and as time advances the inclination at the inner
edge of the disk increases.}
\label{diffusive}
\end{figure}
\subsection{Diffusive warp propagation; $\alpha> H/R $}
\label{diff}
As a check, we consider the evolution of a disk with parameters in the
diffusive regime. We take $\alpha=0.1$ and
$H/R=0.01$. Fig.~\ref{diffusive} shows the disk inclination and
surface density evolution solving the full equations~(\ref{main})
and~(\ref{flux}) with $\beta=10$. We have also solved the diffusive equation~(\ref{viscous}) but find there is no difference between the two solutions and so we do not show this. In the diffusive regime there is no difference between solving the diffusive equations and the full equations that we have derived. The additional $\beta$ damping term has no effect in this limit.
\begin{figure}
\centering
\includegraphics[width=7.5cm]{int10.eps}
\caption{Evolution of an initially warped disk around a single object
with no external torque with $\alpha=0.1$ and $H/R=0.1$ (in the
intermediate regime). The full equations are solved with $\beta=10$. The upper panels show the inclination and the
lower panels the surface density. The times shown are every
$10\,P_{\rm in}$ and as time advances the inclination at the inner
edge of the disk increases.}
\label{intermediate}
\end{figure}
\subsection{Intermediate regime; $\alpha=H/R$}
Neither the wave-like equations nor the diffusive equations are able to model the evolution of a disk with $\alpha \approx H/R$. However, the full equations we have developed, equations~(\ref{main})
and~(\ref{flux}), can be used in this regime. Fig.~\ref{intermediate} shows the solution to the full equations with $\beta=10$ for a disk with $\alpha=0.1$ and $H/R=0.1$. The inner parts of the disk appear more diffusive in nature and the outer parts look more wave-like in the inclination evolution.
\section{Conclusions}
\label{conc}
We have introduced a new set of equations that describe the evolution of disk warp and of disk surface density in both low viscosity and high viscosity disks. We have shown that the two sets of equations agree with the equations for warp propagation previously derived in the two distinct regimes of low viscosity (wave-like warp propagation) and of high viscosity (diffusive warp propagation). In order to achieve this we have introduced a new dimensionless parameter $\beta$ which has the dominant effect of preventing unphysical evolution of surface density in the wave-like regime. We have not been able to determine the required magnitude of $\beta$ except to note that for $\beta\gg \alpha$ the unphysical evolution of surface density in the wave-like regime no longer occurs. In order to determine the value of $\beta$, and indeed to determine whether or not the new equations we present here provide an adequate description of warp evolution in general, it will be necessary to undertake a detailed analytic analysis \citep[c.f.][]{Ogilvie1999} and/or compare with detailed numerical simulations.
\section*{Acknowledgments}
RGM, SHL and AF acknowledge support from NASA through grant
NNX17AB96G. RN has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 681601). CJN is supported by the Science
and Technology Facilities Council (grant number ST/M005917/1).
\bibliographystyle{apj}
|
{
"timestamp": "2019-03-01T02:18:29",
"yymm": "1902",
"arxiv_id": "1902.11073",
"language": "en",
"url": "https://arxiv.org/abs/1902.11073"
}
|
\section{introduction}
Over the years there have been many attempts to solve some of the drawbacks of the Standard Model (SM) related to the presence of
a fundamental scalar boson (like the hierarchy problem, triviality, etc...). Some of the proposals along these lines are interesting
due to the fact that fundamental scalar bosons fit naturally into these models, as in supersymmetric models~\cite{su1,su2,su3}
and asymptotically safe SM extensions~\cite{sa00,sa11}. However, no signals of these theories have appeared so far. The Higgs particle found at the LHC~\cite{atlas,cms}
may be the first signal of a fundamental scalar boson, although the possibility that this boson is a composite one has not yet been discarded, and in this case some of the SM problems commented above may be alleviated.
Scalar bosons are essential to the mechanisms of chiral and gauge symmetry breaking in the SM, but it should be remembered that most
of what we have learned about the mechanisms of spontaneous symmetry breaking is based on the presence of composite or pair-correlated
scalar states, as happens in the Nambu-Jona-Lasinio model, QCD chiral symmetry breaking, and in the microscopic BCS theory of superconductivity.
For instance, chiral symmetry breaking is promoted in QCD by a nontrivial vacuum expectation value of a fermion
bilinear operator and the role of the Higgs boson is played by the composite $\sigma$ meson. These types of gauge theory models, dubbed technicolor (TC), were proposed
40 years ago~\cite{wei,sus} and reviewed in Refs.~\cite{far,mira}. The many variations of these models continue to be studied~\cite{an,sa,sa1,sa2,sa3,sa4,be},
but no phenomenologically viable model has been found so far.
It is clear that building SM extensions in order to solve unknown questions (like the origin of the fermionic mass spectra), is easier when we
deal with fundamental scalar bosons, than when the spontaneous symmetry breaking
is promoted by composite scalars, even if we are far from solving the problems related to the existence of fundamental scalar bosons. The difficulty in models with
composite scalar boson resides in knowing the dynamics of the non-Abelian gauge theory responsible for their formation.
We may say that the root of most TC problems lies in the way the ordinary fermions acquire their masses, which is
shown in Fig.1, where an ordinary fermion $f$ couples to a technifermion $F$ mediated by an extended technicolor (ETC) boson.
\begin{figure}[h]
\centering
\vspace*{-0.5cm}
\includegraphics[scale=0.45]{figETC.eps}
\vspace{-0.25cm}
\caption[dummy0]{Ordinary fermion mass $f$ in ETC models}
\label{fig1}
\end{figure}
\noindent
\par Assuming a standard non-Abelian TC self-energy ($\Sigma_{\T}$) given by~\cite{lane2}
\be
\Sigma_{\T} (p^2) \propto \frac{\mu_{\tt{TC}}^3}{p^2} \(\frac{p}{\mu_{\tt{TC}}}\)^{\gamma_m},
\label{eq1}
\ee
where $\mu_{\tt{TC}}$ is the characteristic TC dynamical mass at zero momentum (of order the Fermi mass) and $\gamma_m$ is the anomalous mass dimension (which depends on the TC coupling constant, and for an asymptotically free theory has a small value), the ordinary
fermion mass turns out to be
\be
m_f \propto \frac{\mu_{\tt{TC}}^3}{M_{\E}^2},
\label{eq2}
\ee
where $M_{\E}$ is the ETC gauge boson mass. In order to explain the top-quark mass we need a small $M_{\E}$ value, and since ETC is one interaction that changes flavor, the simplest model that we can imagine will inevitably lead to flavor changing neutral currents incompatible with the experimental data (among other problems).
Solutions to the above dilemma seem to require a large $\gamma_m$ value~\cite{holdom} leading to a TC self-energy with a harder momentum behavior, and many models along these lines can be found in the literature~\cite{lane0,appel,yamawaki,aoki,appelquist,shro,kura,yama1,yama2,mira2,yama3,mira3,yama4}. In particular, we may quote the work of Takeuchi~\cite{takeuchi} where the TC Schwinger-Dyson equation (SDE) was solved with the introduction of an four-fermion \textit{ad hoc} interaction, which can lead to the following expression for the TC self-energy:
\be
\Sigma_{\T}(p^2\rightarrow\infty)\propto \ln^{-\delta} (p^2/\mu_{\tt{TC}}^2),
\label{eq3}
\ee
where $\delta$ is a function of the many parameters of the model. The Takeuchi solution, when dominated by the four-fermion interaction,
is not different from the behavior of the self-energy when a bare mass is introduced into the theory, or from irregular SDE solution~\cite{lane2}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{figETCfull.eps}
\vspace{-0.25cm}
\caption[dummy0]{The coupled system of SDEs for TC ($T\equiv$technifermion) and QCD ($Q\equiv$quark) including ETC and electroweak or other corrections. $G \,(g)$ indicates a technigluon (gluon).}
\label{fig2}
\end{figure}
Recently, we numerically solved the coupled TC [based on an $SU(2)$ group] and QCD gap equations~\cite{us1}, which are depicted in the Fig.2. It turned out that
both self-energies have the same asymptotic behavior as Eq.(\ref{eq3}). It is not difficult to understand the origin of such behavior. In Ref.~\cite{us2}
we analytically verified that the radiative corrections shown in Fig.2 act as an effective bare mass. In the case of ordinary quarks the second diagram ($b_2$)
on the right-hand of Fig.2 originates an effective mass due to TC condensation; on the other hand, the techniquarks obtain a tiny effective mass due to
QCD condensation [see diagram ($a_2$) in Fig.2], and an even larger mass due to the other diagrams [($a_3$) and ($a_4$)]. Therefore, the TC self-energy can be described by
\be
\Sigma_{\T}(p^2)\approx \mu_{\tt{TC}} \left[ 1+ \delta_1 \ln\left[(p^2+\mu^2_{\tt{TC}})/\mu^2_{\tt{TC}}\right] \right]^{-\delta_2} \,,
\label{eq4}
\ee
where $\delta_1$ and $\delta_2$ are parameters that will depend on the many possible SDE radiative corrections depicted in Fig.2; in particular, the dominant
correction to the technifermion masses will be generated by diagrams $(a_3)$ and $(a_4)$ of Fig.\ref{fig2}, and by diagram ($b_2$) in the case of ordinary fermion masses. We get a similar expression for ordinary quarks, and it should be noticed that the \textit{isolated} infrared TC and QCD self-energy
behavior is the traditional one [the one associated to the regular solution or Eq.(\ref{eq1})] with dynamical masses of order $\mu_{\tt{TC}} \approx O(1)$TeV and
$\mu_{\tt{QCD}}\approx 250$MeV, respectively, i.e., the coupled SDE system is a combination of the regular and irregular self-energy solutions~\cite{lane2}.
It is interesting to recall that such behavior is indeed that which minimizes the vacuum energy in gauge theories~\cite{us0}, and it is not different from Takeuchi's result
but rather originates from known interactions (QCD, for example).
The main consequence of the results of Refs. \cite{us1} and \cite{us2} [i.e., Eq.(\ref{eq4})] is that the dynamically generated masses will barely
depend on the ETC scale $M_\E$. In Ref.\cite{us1} we numerically verified that the ordinary quark masses behave as
\be
m_{\Q} \propto \lambda_E \mu_{\tt{TC}} [1+\kappa_1 \ln(M^2_{\E}/\mu_{\tt{TC}}^2)]^{-\kappa_2} \, ,
\label{eq5}
\ee
where $\lambda_E$ involves ETC couplings, a Casimir operator eigenvalue, and other constants, and $\kappa_i$ are related to the self-energies that enter in the calculation
of the generated masses, which is compatible with the quark mass computed with the help of Eq.(\ref{eq4}). Looking at Eq.(\ref{eq5}), it is clear that we can push the ETC scale up to the grand unification scale (or even the Planck scale) without large variations
of the $m_{\Q}$ values with $M_E$. It is also clear that the ordinary fermionic mass hierarchy will not arise from different $M_\E$ scales! The purpose of the present work it to
discuss how viable TC models can be built in this context, as well as to verify the phenomenological consequences of these models, and to show how that
they can be consistent with existing high-energy data.
It is important to note that the study of SDEs is very sophisticated, taking into account gluon-mass generation and possibly
confinement~\cite{g1,g2,g3,g4,g5} as well as complex vertex structures~\cite{g6,g7}. However, the solutions discussed in Refs.~\cite{us1,us2} and in this work are related to the asymptotic behavior produced by the effective mass of the coupled SDE, and are not affected by the infrared intricacies of the strongly interacting theories.
The paper is organized as follows. In Sec. II we present one specific TC model, which is just an example of the many models that can be built
along the lines described in that section. We discuss the fact that a horizontal symmetry is necessary in this scheme. In Sec. III we discuss how a composite scalar boson can be lighter than the typical composition scale
of the theory responsible for this particular state. In Sec. IV we determine the order of magnitude of pseudo-Goldstone masses. In Sec. V
we compare the value of the TC condensate in our model with the one expected in walking TC theories. Section VI contains a
brief discussion of possible experimental consequences of the models discussed in Sec. II, and in Sec. VII we discuss what can be expected
regarding the trilinear scalar coupling. Section VIII contains our conclusions.
\section{Building TC models}
In Ref.~\cite{us1} we briefly proposed one specific TC model, which will be detailed here. As will be discussed at the end of this section,
there is a large class of models that can be built along the
same lines as the model described here. The model discussed in Ref.~\cite{us1} is based on the following group structure
$$
SU(9)_U \otimes SU(3)_H \,\, ,
$$
where the $SU(9)_U$ group is a non-Abelian grand unified theory (GUT) containing the SM and a $SU(4)_{\tt{TC}}$ group. The $SU(3)_H$ group is a horizontal or family
symmetry that is important for generating the hierarchy of fermion masses.
There are several reasons for this particular choice.
First, the $SU(9)_U$ GUT will play the role of ETC, because the generated fermion masses will weakly depend on the GUT boson masses
(here acting as ``ETC"
boson masses) as
shown in Eq.(\ref{eq5}). This group also contains the standard $SU(5)_{\tiny \textsc{gg}}$ Georgi-Glashow GUT~\cite{gg}. Second, the $SU(4)_{\tt{TC}}$ group contained in the GUT will condense before QCD, generating an appropriate Fermi scale necessary
to break the electroweak group. Note that this choice is based on the most attractive channel (MAC) hypothesis~\cite{cor1,suss}, but it can be relaxed if the GUT breaking can
be promoted at very high energies, where even fundamental scalar bosons may be natural due to the presence of supersymmetry~\cite{su1,su2}. In this case we
could not neglect the possibility of a small TC group [perhaps $SU(2)$] that condenses at one mass scale larger than the QCD one. Third, the horizontal or family
symmetry is necessary to prevent the first- and second-generation ordinary fermions from coupling to TC. The third fermionic generation will obtain masses
due to diagrams like the one in Fig.1, and will be of order $\lambda_E \mu_{\tt{TC}}$, as described below.
The $SU(9)_U$ group has the following anomaly free fermionic representations~\cite{fra}:
\be
5\otimes[9,8]_i \oplus 1\otimes [9,2]_i \, ,
\ee
where $[\underline{8}]$ and $[\underline{2}]$ are antisymmetric under $SU(9)_U$, and $i=1,2,3$ is the horizontal index necessary for the replication
of the $SU(3)_H$ families. The decompositions of these representations under $SU(4)_{\tt{TC}}\otimes SU(5)_{\tiny \textsc{gg}}$ are
\br &&\hspace{-0.5cm}[\bf{9},\bf{2}]_i\nonumber\\
&&(1,10) = \left(\begin{array}{ccccc} 0 & \bar{u_{i}}_{B} & - \bar{u_{i}}_{Y} & -{u_{i}}_{R} & -{d_i}_{R} \\
-\bar{u_i}_{B} & 0 & \bar{u_i}_{R} & -{u_i}_{Y} & -{d_{i}}_{Y} \\ \bar{u_{i}}_{Y} & -\bar{u_{i}}_{R} & 0
& -{u_i}_{B} & -{d_{i}}_{B} \\ {u_i}_{R} & {u_i}_{Y} & {u_i}_{B} & 0 & \bar{e_i}\\
{d_i}_{R} & {d_i}_{Y} & {d_i}_{B} & -\bar{e_{i}} & 0\end{array}\right)\nonumber\\\nonumber \\ &&(4,5) =
\,\,\,\left(\begin{array}{c} {T_i}_{R} \\ {T_i}_{Y} \\ {T_i}_{B} \\ \bar{L_i}\\ \bar{N_i}
\end{array}\right)_{TC}\,\,\,,\,\,\,(\bar{6},1)= N_{i}\nonumber \\ \nonumber \\
&&\hspace{-0.5cm}[\bf{9},\bf{8}]_i\nonumber\\
&&(1,\bar{5}) =\,\,\, \left(\begin{array}{c} \bar{d_i}_{R} \\ \bar{d_i}_{Y} \\ \bar{d_i}_{B} \\
e_i \\ \nu_{e_i}
\end{array}\right)
\,\,\,\,\,\,(1,\bar{5}) = \left(\begin{array}{c} \bar{X}_{R_{k}} \\ \bar{X}_{Y_{k}} \\ \bar{X}_{B_{k}} \\
E_{k} \\ N_{E_{k}}
\end{array}\right)_i\nonumber\\ \nonumber \\\nonumber\\
&&(\bar{4},1)= \,\,\,\,\,\bar{T_i}_{\varepsilon}, L_i ,{N_i}_{L}. \nonumber \er
\noindent In the fermionic content of the above model, we identify the usual quarks as $Q = (u ,d)$, while $T$ corresponds to techniquarks and $(L , N)$ to technileptons, where $\varepsilon = R,Y,B$ is a color index, and $k=1...4$ indicates the generation number of exotic fermions that must be introduced in order to render the model anomaly free.
The $SU(3)_H$ quantum numbers must be assigned such that the quartet of technifermions that condenses in the MAC of the
product ${\bf \bar{4}\otimes 4}$ belongs to the ${\bf \overline{6}}$ representation of the horizontal group, whereas the QCD quark condensate (generated in the color product
${\bf \bar{3}\otimes 3}$), is formed in the triplet representation (${\bf 3}$) of $SU(3)_H$. This is nothing else than the horizontal symmetry scheme
with fundamental scalar bosons proposed in Refs.~\cite{h1,h2,h3,h4}, and it leads to a quark mass matrix in the horizontal group basis of the form
\br
m_q =\left(\begin{array}{ccc} 0 & m_1 & 0\\ m_1^* & 0 & 0 \\
0 & 0 & m_3
\end{array}\right),
\label{eq6} \er
\noindent where $m_1$ and $m_3$ indicate the first- and third-generation quark masses.
It is instructive to show the diagrams that lead to the different masses shown in Eq.(\ref{eq6}). For instance, let us assume that
$m_q$ is the mass matrix of charge $2/3$ quarks, where $m_3$ would be related to the top-quark mass. The diagrams responsible for
this mass are shown in Fig.3.
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{masstop2.eps}
\caption[dummy0]{Diagrams contributing to the top-quark mass.}
\label{fig3}
\end{figure}
\par In this figure the technifermions $T$ and $L$ [that condense in the ${\bf \overline{6}}$ of $SU(3)_H$] give masses to the $t$ quark whose interaction is mediated by one $SU(9)$ gauge boson. Apart from the logarithmic term appearing in Eq.(\ref{eq5}) this mass is
\be
m_3\approx 2 \lambda_9 \mu_{\tt{TC}} \, ,
\label{eq7}
\ee
where we can assume that $\lambda_9 \approx 0.1$, is the product of the $SU(9)$ coupling constant times some Casimir operator eigenvalue, the factor $2$ accounts both diagrams of Fig.3, and $\mu_{\tt{TC}}$ can be assumed to be of $O(1)$TeV. The $SU(9)$ interaction is playing
the role of the ETC interaction. These naive assumptions
will lead to a top-quark mass of approximately $200$GeV. The logarithmic term appearing in Eq.(\ref{eq5}) [and
neglected in Eq.(\ref{eq7})] slightly decreases the value of our rough estimate.
Note that the first and second charge $2/3$ quarks do not couple directly to the techniquarks
due to the different $SU(3)_H$ quantum numbers, and at this level they remain massless.
We can now see how the first-generation fermions obtain their masses. In Fig.4 we show the diagrams that are responsible for the $u$-quark mass.
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{massup2.eps}
\caption[dummy0]{Diagrams contributing to the light-quarks masses.}
\label{fig4}
\end{figure}
This quark does not couple to techniquarks at leading order, but does couple to other ordinary quark and itself due to the
bosons of the unified theory and the horizontal one. Its mass can be approximated from Eq.(\ref{eq5}) [as we did to obtain Eq.(\ref{eq7})] and
is given by
\be
m_1\approx \lambda_5 \mu_{\tt{QCD}} \, ,
\label{eq8}
\ee
where we can assume naively that the $SU(5)_{\tiny \textsc{gg}}$ factor $\lambda_5 \approx 0.1$ and $\mu_{\tt{QCD}}\approx 200$MeV, which gives
a mass of order $20$MeV. Here we do not introduce a factor of $2$ in Eq.(\ref{eq8}) due to the presence of the two diagrams in Fig.4, because the $c$-quark condensate (in the second diagram of Fig.4) may be smaller than the $u$ and $d$ condensates\footnote{Note that the self-energy and the condensate values are intimately connected, i.e., one is basically an integral of the other.
The $c$-quark self-energy appearing in Fig.4 will involve the same type of integral as the $c$-quark condensate.
It is known that the introduction of heavy quark masses act to diminish
the condensate value or the amount of chiral symmetry breaking ~\cite{x1}. For example, it has been determined for the $s$-quark that
$\<\bar{s}s\>/\<\bar{u}u\>=0.6\pm 0.1$ ~\cite{x2,x3}.
In Ref.~\cite{sa5} the same effect of a heavy fermion mass (e.g., fermion loops) was also observed as a factor that lowers the composite Higgs boson mass. Therefore, the second diagram of Fig.4 is
expected to have a smaller effect in the calculation of the first-generation quark masses.}.
In Eqs.(\ref{eq7}) and (\ref{eq8}) we probably overestimated the results when we neglected the logarithmic dependence on
the unified or ``ETC" boson masses. These are very simple calculations. To obtain better estimates we must solve the
coupled SDE and obtain good fits to the self-energies, which would give us reasonable values for the parameters $\delta_1$ and $\delta_2$ in the
approximate expression of Eq.(\ref{eq4}). It is clear that this is far beyond the scope of this work.
The mass of the second quark generation will necessarily involve the horizontal symmetry, where the coupling to techniquarks will appear
only at two-loop order. The $c$-quark mass will be generated by diagrams like the ones shown in Fig.5,
\begin{figure}[h]
\centering
\includegraphics[scale=0.55]{masscharm.eps}
\caption[dummy0]{Diagrams contributing to the $c$-quark mass.}
\label{fig5}
\end{figure}
and it is expected to be $1$ order of magnitude below the typical mass of the third quark generation, due to an extra factor $\lambda_{3H}\approx 0.1$ that contains the $SU(3)_H$ coupling constant. In this way, we verify that the horizontal or family symmetry is fundamental to generate a quark mass
matrix with the Fritzsch texture~\cite{f1,f2}
\br
m_q =\left(\begin{array}{ccc} 0 & m_1 & 0\\ m_1^* & 0 & m_2 \\
0 & m_2^* & m_3
\end{array}\right),
\label{eq9} \er
which has several good qualities of the experimentally known quark mass matrix.
Lepton masses will appear in the same way as quark masses. The $\tau$ lepton is the only one that will couple with techniquarks at leading order, due to the appropriate choice of quantum numbers of the horizontal symmetry. As a consequence, the mass matrix for the leptonic sector is similar to the one described above, although lepton masses should be naturally smaller
than quark masses, because quarks end up coupling to two different condensates and a larger number of diagrams contribute to their masses. It is not difficult to verify
the different number of SDEs between quarks and leptons that can be generated with the Feynman rules of the model described here.
We have not discussed the $SU(9)_U$ and horizontal symmetry breaking, which we just assume to happens at the unification scale $\Lambda_{{}_{SU(9)}}$,
which can possibly be
naturally promoted by fundamental scalar bosons. The breaking of the GUT symmetry can also be used to produce a larger splitting in the third fermionic generation.
For instance, if in the $SU(9)_U$ breaking (besides the Standard model interactions and the TC one) we
leave an extra $U(1)$ interaction, we could have quantum numbers such that only the top quark would be allowed to couple to the TC condensate at leading order.
In fact, the splitting ($S_{(t-b)}$) between the $t$ and $b$ quarks
\be
S_{(t-b)}= \frac{m_b}{m_t}\approx \frac{1}{40} \, ,
\ee
is quite large, and it is interesting that the $b$ quark and the $\tau$ lepton could couple at a larger order in the coupling constant [possibly $(\alpha_9^2)$], which could be accomplished by
this remaining $U(1)$ interaction that we referred to above. More sophisticated models in which large fermionic mass splittings and even neutrino masses can be generated
were presented in Refs.~\cite{as1,as2,as3,as4,as5}.
At this point, we hope that we have made clear the necessity of introducing a horizontal or family symmetry. It is necessary to prevent
the first and second generations of ordinary fermions from obtaining large masses that couple to TC at leading order. This symmetry can be a local one, but a
global symmetry is not necessarily discarded. If the family symmetry is local, its breaking can also happen at very high energies and (again) may even be
promoted by fundamental scalars at the GUT or Planck scale, producing feeble effects at lower energies.
When building a TC model the existence of
grand unification is also welcome. For example, in the model described here
a $SU(5)_{\tiny \textsc{gg}}$ gauge boson interaction is fundamental to give the electron a mass, which appears due to the electron coupling to the
first-generation quark, with exactly the same interaction that may mediate proton decay in the $SU(5)_{\tiny \textsc{gg}}$ theory. There are more diagrams
contributing to the first-generation quark masses than there are for the electron mass, which may explain why leptons are less massive than quarks.
Concerning the possible class of models presented here, it is also clear that a full and precise determination of the mass spectra is quite complex. Once a GUT involving the SM and TC is proposed, we also have to choose
the horizontal symmetry. The coupled SDE of such a model has to be solved by determining all self-energies with their specific infrared and ultraviolet expressions.
Of course, simple estimates can be made by approximating the calculation of each specific fermion mass diagram, by the product of the dynamical mass involved in
the diagram (TC or QCD) with the respective coupling constants and Casimir operator eigenvalues, as performed in Eqs.(\ref{eq7}) and (\ref{eq8}) where a logarithmic
term was neglected.
\section{Scalar mass}
The common lore about theories with a composite scalar boson is that its mass should be of the order of the dynamical mass scale that forms such particle.
This concept is related to the work of Nambu-Jona-Lasinio~\cite{nl} and was also discussed for the $\sigma$ meson in QCD~\cite{ds}, where the scalar composite mass appearing in one strongly interacting theory is given by
\be
m_\sigma = 2 \mu_{\tt{QCD}}\,\, .
\label{eq10}
\ee
Equation (\ref{eq10}) comes from the fact that at leading order the SDE for the quark propagator is similar to the homogeneous Bethe-Salpeter equation (BSE)
for a massless pseudoscalar bound state $\Phi_{BS}^P (p,q)|_{q \rightarrow 0}$ (the pion), and a scalar p-wave bound state $\Phi_{BS}^S (p,q)|_{q^2 = 4 \mu^2 }$
[the sigma meson or the $f_0(500)$~\cite{pdg}], i.e.,
\be
\Sigma (p^2) \approx \Phi_{BS}^P (p,q)|_{q \rightarrow 0} \approx \Phi_{BS}^S (p,q)|_{q^2 = 4 \mu_{\tt{QCD}}^2 }\,\,.
\label{eq11}
\ee
Equation (\ref{eq11}) tells us that in QCD the $\sigma$ meson must have a mass $2\mu_{\tt{QCD}}\approx 500$MeV. In TC we should expect a scalar boson with a mass of $2$TeV, which
is clearly not the case for the observed Higgs boson~\cite{atlas,cms}
There are two subtle points concerning the result of Eq.(\ref{eq10}) and the determination of the scalar composite mass. The first one is that Eq.(\ref{eq10})
was determined using the homogeneous BSE. There is nothing wrong with this. However this gives the right result if the fermionic self-energy that
enters into the BSE is a soft one. When the self-energy decreases slowly [as in Eq.(\ref{eq4})] the scalar mass is modified by the
normalization condition of the inhomogeneous BSE. This modification lowers the composite scalar mass as a consequence
of Eq.(\ref{eq4}).
The second point about Eq.(\ref{eq10}) that we would like to note is not exactly about the equation itself, but rather about the values of the
dynamical QCD and TC mass scales that arise at such a scale. The QCD dynamical mass scale is usually extracted from the hadronic spectra;
for instance, it is expected to be $1/3$ of the nucleon mass or $1/2$ of the sigma meson mass. However, it is not currently clear how much this spectra
is affected by gluons (or technigluons in the TC case) and mixing among different particles. These points will be discussed in the
following subsections.
\subsection{Normalization condition and the scalar mass}
The BSE normalization condition in the case of a non-Abelian gauge theory is given by \cite{lane2}
\br
2\imath q_{\mu}= \imath^2\!\!\int d^4\!p\, Tr\left\{{\cal P}(p,p+ q)\left[\frac{\partial}{\partial q^{\mu}}F(p,q)\right]{\cal P}(p, p+ q) \right\}\nonumber \\
-\imath^2\!\!\int d^4\!pd^4\!k \,Tr\left\{{\cal P}(k,k + q)\left[\frac{\partial}{\partial q^{\mu}}K'(p,k,q)\right]{\cal P}(p, p+ q)\right\} \nonumber
\label{eq11a}
\er
where
$$
K'(p,k,q) = \frac{1}{(2\pi)^4}K(p,k,q) \,\,\, ,
$$
$$
F(p,q) = \frac{1}{(2\pi)^4}S^{-1}(p+q) S^{-1}(p) \,\,\, ,
$$
where ${\cal P}(p, p + q)$ is a solution of the homogeneous BSE and $K(p,k,q)$ is the fermion-antifermion scattering kernel
in the ladder approximation. When the internal momentum $q_{\mu} \rightarrow 0$, the wave function ${\cal P}(p, p + q)$ can be determined only through
the knowledge of the fermionic propagator:
\be
{\cal P}(p) = S(p)\gamma_{5}\frac{\Sigma(p)}{F_{\Pi}}S(p) \,\, ,
\ee
\noindent where $\Sigma (p)$ will describe the technifermion self-energy and it should be noticed that $F_{\Pi}$ describes the technipion decay constant associated with $n_{d}$ technifermion doublets. If we identify $\Sigma(p^2) \equiv \mu_{\tt{TC}} f(p^2)$ we can write the normalization condition in the rainbow approximation as
\br
&&2i\left(\frac{F_{\pi}}{\mu_{\tt{TC}}}\right)^2 q_{\mu} = \frac{i^2}{(2\pi)^4}\times \nonumber \\
&& \left[\int d^4\!p\, Tr{\Big \{}S(p)f(p)\gamma_{5}S(p )\left[\frac{\partial}{\partial q^{\mu}}S^{-1}(p + q) S^{-1}(p)\right]\right. \nonumber \\
&& \left. S(p)f(p)\gamma_{5}S(p){\Big \}} + \frac{i^2}{(2\pi)^4}\int d^4\!pd^4\!k \,Tr{\Big \{} S(k)\right. \nonumber \\
&& \left.f(k)\gamma_{5}S(k)\left[\frac{\partial}{\partial q^{\mu}}K(p,k,q)\right]S(p)f(p)\gamma_{5}S(p){\Big \}}\right]. \nonumber \\
\label{eq11b}
\er
\par Equation (\ref{eq11b}) is quite complicated, but it can be separated into two parts:
\be
2i\left(\frac{F_{\Pi}}{\mu_{\tt{TC}}}\right)^2 q_{\mu} = I_\mu^{0} + I_\mu^{K} \,\,\, ,
\label{eq14a}
\ee
corresponding, respectively, to the two integrals on the right-hand side of Eq.(\ref{eq11b}). The fermion propagator given by
$S(p) = {1}/[{\not{\!\!p} - \Sigma(p)}]$ can be written as
\be
\frac{\partial}{\partial q^{\mu}}S^{-1}(p + q) = \gamma_{\mu} - \frac{\partial}{\partial q^{\mu}}\Sigma(p+q) \,\,\, ,
\ee
\noindent and the term $ \frac{\partial}{\partial q^{\mu}}\Sigma(p+q)$ in the above expression may be written as
\be
\frac{\partial \Sigma(p+q) }{\partial q^{\mu}} = (p + q)_{\mu} \frac{d\Sigma(Q^2)}{dQ^2}
\ee
\noindent where $Q^2 = (p + q)_{\mu}(p + q)^\mu$. Considering the angle approximation we transform the term $\frac{d\Sigma(Q^2)}{dQ^2}$ as
\be
\frac{d\Sigma(Q^2)}{dQ^2} = \frac{d\Sigma(p^2)}{dp^2}\Theta(p^2 - q^2) + \frac{d\Sigma(q^2)}{dq^2}\Theta(q^2 - p^2)
\ee
where $\Theta$ is the Heaviside step function. We can finally contract Eq.(\ref{eq14a}) with $q^\mu$ and compute it at $q^2=M_H^2$ in order to obtain
\br
M_{H}^2 = 4&&\mu_{\tt{TC}}^2{\Big\{}\frac{n_{f}N_{TC}}{8\pi^2}\int d^2\!p\frac{f^2(p)\Sigma(p)}{(p^2 + \Sigma^2(p))^2} \times \nonumber \\
&& \times \left(-p^2\frac{d\Sigma(p)}{dp^2} \right) \left(\frac{\mu_{\tt{TC}}}{F_{\Pi}}\right)^2 + \nonumber \\
&& + \,\,I^{K}(q^2 = M_H^2,f(p,k),g_{TC}^2(p,k)) {\Big \}},
\label{eq11c}
\er
where $n_f$ is the number of technifermions, $N_{TC}$ is the number of technicolors and $g_{TC}$ is the technicolor coupling constant.
An expression similar to Eq.(\ref{eq11c}) was already obtained by us in Ref.~\cite{usx}. In that work we just assumed (in a totally \textit{ad hoc} fashion)
a hard momentum behavior for the TC self-energy. The calculation here will differ not only in the origin of the self-energy but also in the approach
we follow to determine the value of $M_H$. Considering the work of Ref.~\cite{us2}
it becomes evident that the behavior of $M_{H}$ is a result that will fundamentally depend on the boundary conditions satisfied by the
coupled system described in Fig.2. In Eq.(\ref{eq11c}) the UV behavior of the term
\be
{\rm \bf (UV)}\,\,\,\,\,\,\,lim_{{}_{{}_{\hspace*{-0.5cm} p^2 \to \Lambda^2}}}\!\!\!-p^2\frac{d\Sigma(p)}{dp^2} ,
\label{newuv}
\ee
\noindent will be affected by the effective mass generated by the diagrams $(a_2)$, $(a_3)$, and $(a_4)$ in Fig.2. In Ref.~\cite{us2} we verified that
the UV behavior of the term in Eq.(\ref{newuv}) is modified as $\alpha_E$ is different or equal to zero, and we shall comment on this term later.
We compute $M_H$ by numerically solving the
differential coupled equations shown in Eqs.(11) and (12) of Ref.~\cite{us2} , fitting the resulting solutions (all fits with $R^2=0.98$), and inserting the fits into Eq.(\ref{eq11c}). We consider the TC gauge groups $SU(2)_{TC}$, $SU(3)_{TC}$ and $SU(4)_{TC}$, with $n_f=5$ fermions in the fundamental representation, $\mu_{\tt{TC}}=1$TeV, and use the MAC hypothesis to constrain the TC gauge coupling and Casimir eigenvalue. Hereafter, we follow
Refs.~\cite{us1,us2} and use a Casimir eigenvalue $C_E=1$ and gauge coupling constant $\alpha_E = 0.032$, which are quantities related to the ETC gauge theory.
Our results for $M_H$ are shown in Table I, where we can see that the normalization condition lowers the scalar mass by a factor of $O(1/10)$. The results are consistent with those of Ref.~\cite{usx} obtained with the naive assumption of an irregular solution for the TC self-energy. Therefore, the effect
of radiative corrections in coupled SDEs involving a TC theory act in order to produce a scalar composite boson with a mass compatible with that of
the observed Higgs boson.
\begin{table}[htbp]
\centering
\begin{tabular}{ccc}\hline \hline
SU(N) & $n_f$ & $M_H$(GeV) \\ \hline
2 & 5 & 105.3 \\
3 & 5 & 141.5 \\
4 & 5 & 148.8 \\
\hline \hline
\end{tabular}
\caption{The last column contains the composite scalar mass determined through Eq.(\ref{eq11c}), where we used the TC self-energy obtained by solving the coupled SDE system. The different factors and couplings of the gap equations are described in the text. }
\label{tbl:IneqFP}
\end{table}
\subsection{Dynamical mass scales and mixing}
The most precise quantity to constrain the dynamical mass scale in the QCD case is the
pion decay constant, which is a function of the quark self-energy. In the TC case the technipion decay constant is
related to the $W$ and $Z$ gauge boson masses. However, in both cases that quantity depends on the dynamical mass scale as well as the functional
expression for the self-energy. Therefore, we have some freedom in pinpointing the dynamical mass scale. Even the numerical determination of the self-energy
through SDE solutions includes the introduction of a cutoff and specific approximations. We conclude that the calculation of the scalar boson mass
depends on the functional form of the self-energy and on the dynamical mass scale. It is curious that in the past the scalar boson mass
was considered in order to constrain the dynamical mass scale, i.e., in QCD the scalar $\sigma$ meson mass has led to the usual value $\mu_{\tt{QCD}}\approx 250$MeV, which is also approximately
the value of the QCD mass scale ($\Lambda_{\tt{QCD}}$). The problem is that the result of Eq.(\ref{eq10}) is modified not only by the inhomogeneous
BSE condition, butalso by many other effects as we discuss in the following.
The dynamical QCD mass scale is also thought to be related to the nucleon mass, but even this is not certain since we do not know how much gluons
contribute to the nucleon mass~\cite{lorce}. It is also not yet clear how much of the sigma meson mass comes from mixing with heavier
quark-antiquark scalars and with glueballs~\cite{mi1,mi2,mi3,mi4,mi5,mi6}, and the same is true if we just exchange QCD with TC, which means that
the scales $\mu_{\tt{QCD}}$ and $\mu_{\tt{TC}}$ may be smaller than usually thought, leading to a smaller scalar
composite mass (i.e., the $\sigma$ and the ``Higgs" mass). The scalar mass can also be modified by the effect of radiative loop corrections due to
the presence of heavy fermions, as described in Ref.~\cite{sa5}.
These are not the only effects that modify the scalar mass and lead to a new relation between the scalar mass and the dynamical mass scale. There is still another effect that is intimately related to the type of dynamical
symmetry breaking model that we discussed in the previous section.
In Sec. II we discussed a model with two composite scalar states responsible for the chiral (and gauge) symmetry breaking: the scalars
belonging to the ${\bf \overline{6}}$ and ${\bf 3}$ representations of the horizontal group formed by technifermions and
quarks, respectively. The different scalars may mix among themselves due to electroweak or other interactions, as already pointed out in Ref.~\cite{us1}.
An order-of-magnitude estimate of these mixing diagrams is quite lengthy, but the most important
fact is that the scalar coupling to the electroweak bosons is going to be enhanced, when compared to this coupling
calculated when the TC self-energy is soft. Note that this effective coupling happens when scalars and $W$ bosons couple through a ordinary fermion or technifermion
loop. The $W$ coupling to fermions is the SM one, while the scalar composite coupling to ordinary fermions was shown by Carpenter \textit{et al}.~\cite{ca1,ca2} to be
proportional to $\frac{g_w}{2M_W}\Sigma$, where $\Sigma$ is the fermionic self-energy, which now is a slowly decreasing function of momentum and enhances
the effective coupling. If we denote a composite scalar by $\phi$, it is possible to show that the $\phi\phi WW$ effective coupling will
be proportional to~\cite{us3}
\be
\Gamma_{\phi\phi WW}\propto \frac{g_W^4\delta^{ab}}{M_W^2}\frac{g^{\mu\nu}}{32\pi^2}\int dq^2 \frac{\Sigma_\phi^2}{q^2},
\label{eq12}
\ee
where $\Sigma_\phi$ has to be substituted by the TC or QCD self-energy depending on which fermion is involved in the composite scalar. Of course, the
complete calculation of the mixing diagrams is quite model dependent, but, as commented in Ref.~\cite{us1}, the origin of this mixing is another
way to see how a full Fritzsch matrix pattern of fermion masses can be generated in the type of model that we are proposing here. It is due to
this type of coupling that the second-generation fermion masses are generated in models with fundamental scalar bosons~\cite{h1,h2,h3,h4}. Finally, in
the context where all SM symmetry breaking is promoted by composite scalars we cannot even say how much of the $\sigma$ [r $f_0 (500)$] meson mass is due to a possible mixing with a composite Higgs boson.
\section{Pseudo-Goldstone bosons}
In the condensation of the $SU(4)_{\tt{TC}}$ group a large number of Goldstone bosons are formed. Even if we consider other TC groups, only
three of the Goldstone bosons are absorbed in the SM gauge breaking, and regardless of the theory we may end up with several light composite
states resulting from the chiral symmetry breaking of the strong sector.
These pseudo-Goldstone bosons (or technipions) in the model
of Sec. II may have different quantum numbers.
They may be colored bosons $~\bar{Q}\gamma_5\lambda^a Q$, where $\lambda^a$ is a color group generator, charged bosons $~\bar{L}\gamma_5 Q$ and
neutral pseudo-Goldstone bosons $~\bar{N}\gamma_5 N$. These bosons receive masses through radiative corrections, and we will verify that,
as a consequence of the logarithmic TC self-energy, they will be heavier than usually thought, which is desired in view of the
stringent limits on light technipions~\cite{scs1}.
In Ref.~\cite{us1} we briefly commented that the technipion masses ($m_\Pi$) are enhanced in comparison with models where the TC self-energy does
not have the form of Eq.(\ref{eq4}). One of the arguments is quite simple: the technifermions obtain an effective mass ($m_F$) of several GeV
through diagrams ($a_3$) and ($a_4$) of Fig.2. Note that in our case the condensation effect is not soft, and the calculation of these diagrams will result in
a mass that is not different from those of the third ordinary fermionic family. In particular, in our model there will be several contributions
to these types of diagrams. Even the neutral technifermion $N$ will receive contributions from TC condensation mediated by the electroweak $Z$ boson,
and from QCD condensation due to $SU(9)$ GUT bosons. These masses, apart from small logarithmic terms, will be roughly of order
\be
m_F \approx \sum_{i}\lambda_i \mu_{\tt{TC}} \, ,
\label{eq13}
\ee
where $\lambda_i$ represents the product of some coupling constant times Casimir operator eigenvalue contained in any diagram of the type ($a_3$) or ($a_4$)
contributing to the technifermion mass. For the colored and charged technifermions we cannot even discard a mass as heavy or higher than the top-quark
mass. These masses will generate rather heavy technipions as can be verified using the Gell-Mann-Oakes-Renner relation
\be
m_\Pi^2 \approx m_F \frac{\<{ \bar{\psi}_T}\psi_T\>}{2F_{\Pi}^2} \, ,
\label{eq14}
\ee
where $\<{ \bar{\psi}_T}\psi_T\>$ is the TC condensate and $F_{\Pi}$ is the technipion decay constant. With $m_F$ of order of several GeV and standard values
for the condensate and technipion decay constant the technipion masses turn out to be of order of $100$ GeV or higher, as discussed in Ref.~\cite{us1}.
Another way to see that technipion masses are enhanced through the calculation of a diagram that was already shown in Ref.~\cite{us1}
(see Fig.4 of that reference). Any radiative boson exchange within a technipion modifying its mass will necessarily involve the technipion vertex
connecting it to technifermions ($\Gamma_{\Pi F}$). However this vertex is proportional to the technipion wave function $\Phi_{BS}^\Pi (p,q)$,
which at leading order is also related to the TC self-energy as
\be
\left.\Phi_{BS}^\Pi (p,q)\right|_{q\rightarrow 0} \approx \Sigma_T (p^2) \, ,
\label{eq15}
\ee
which is responsible for an enhancement of this radiative correction. An order-of-magnitude calculation of such a diagram was presented in Ref.~\cite{us1},
and we will comment later on the phenomenology of technipions with masses that are not very different from that of the Higgs boson.
\section{TC condensate}
In the previous section and throughout this work we have commented about the different condensates (TC and QCD), and it is interesting to make a connection between
the several studies about the TC condensate value based on walking TC~\cite{yama} and the one we are discussing here. The TC condensate at one high energy
scale $\Lambda$ is related to its value at another scale $\mu$ by
\be
\<{ \bar{\psi}_T}\psi_T\>_{\Lambda} = Z^{-1}_m\<{ \bar{\psi}_T}\psi_T\>_\mu \, ,
\ee
where $Z^{-1}_m$ is a renormalization constant which is given by
$$
Z^{-1}_m \sim \left(\frac{\Lambda}{\mu}\right)^{\gamma_m} \, ,
$$
where $\gamma_m$ is the condensate operator anomalous dimension.
It is possible to compare the condensate values for a theory where the anomalous dimension is perturbative and small at high energy,
i.e. $\gamma_m \to 0$ and the one with a nontrivial large anomalous dimension, for instance, in the extreme walking case where $\gamma_m \to 2$.
We can define the following ratio that measures the difference between condensates in the walking and nonwalking regimes:
\be
R_w = \frac{\<{ \bar{\psi}_T}\psi_T\>_{\Lambda}^{\gamma_m \to 2}}{\<{ \bar{\psi}_T}\psi_T\>_{\Lambda}^{\gamma_m \to 0}} \, ,
\label{rw}
\ee
Considering these extreme cases this ratio is proportional to
\be
\left. R_w\right|_{\gamma_m \to 2} \approx \left( \frac{\Lambda}{\mu}\right)^{2} \, ,
\label{rw2}
\ee
and this expression serves as an indicator of how much the theory is modified by the nontrivial anomalous dimension. This kind of relation can also be used to
verify how radiative corrections appearing in Fig.2 change the TC behavior.
The UV boundary conditions of the differential TC gap equations modified by the radiative corrections (as can be seen in Ref.~\cite{us2}) are given by
\be
\left. p^2\frac{d\Sigma(p)}{dp^2}\right|_{\Lambda \to \infty} = -a\int^{\Lambda^2}_0 dk^2\frac{\Sigma(k)}{k^2 + \Sigma^2(p)} \, ,
\ee
where $a$ is a factor involving the gauge coupling constant and Casimir operator eigenvalue related to the interaction that induces the radiative correction [e.g., constants related to one of the diagrams $(a_2)$, $(a_3)$ or
$(a_4)$ in Fig.2]. On the other hand, we recall that in an $SU(N)$ gauge theory the condensate can be represented by
\be
\<{ \bar{\psi}_T}\psi_T\>_{\Lambda} = -\frac{N}{4\pi^2}\int^{\Lambda^2}_0 dk^2\frac{\Sigma(k)}{k^2 + \Sigma^2(k)} \, .
\ee
These relations allow us to redefine the ratio shown in Eq.(\ref{rw}) where the condensate values are determined with and without radiative corrections, i.e., when they
are calculated with the coupled SDE system ($\alpha_E \neq 0$ ) and with the values of the isolated condensates ($\alpha_E = 0$),
\be
R_w^{rad.cor.} = \frac{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda}}{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E =0}_{\Lambda }}
\approx
\frac{\left. p^2\frac{d\Sigma(p)}{dp^2}\right|^{\alpha_E\neq 0}_{\Lambda \rightarrow\infty} }{\left. p^2\frac{d\Sigma(p)}{dp^2}\right|^{\alpha_E=0}_{\Lambda \rightarrow \infty}} \, .
\label{rwcor}
\ee
We computed Eq.(\ref{rwcor}) by considering the solutions of the coupled and isolated SDE system in the case of the $SU(3)$ TC group, with
$\mu =1$ TeV, $\alpha_E =0.032$, $\alpha_{{_{TC}}} = 0.87$ and $C_{{}_{TC}} = 4/3 $. The self-energies were obtained in terms of
the variable $x = p^2/\mu^2$ for each ETC scale $M_E$, and the condensates were integrated from $x=10^2$ up to the UV cutoff $ x_\Lambda= \Lambda^2/\mu^2 \sim 10^7 $.
The ratio $R_w^{rad.cor.}$ was fitted with $R^2=0.999$ in the form $a_1[ln(M^2_E/\mu^2)]^{a_2}$ and the result is
\be
R_w^{rad.cor.} \propto 7.87 \times 10^6[ln(M^2_E/\mu^2)]^{-4.3} \, .
\label{eq14g}
\ee
If we consider the value of our cutoff ($\Lambda^2/\mu^2 = 10^7$), we can verify that the effect of the radiative correction is not exactly that of
the extreme walking case shown in Eq.(\ref{rw2}), but it is still quite large. We again see that the effect of radiative corrections is not that
different from the effect of the \textit{ad hoc} four-fermion interactions determined by Takeuchi~\cite{takeuchi}. Moreover, if we compute
the generated quark mass ($m_Q$) as a function of the TC condensate we obtain
\be
m_Q \approx \frac{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda}}{\Lambda^2} \approx C [ln(M^2_E/\mu^2)]^{-\kappa_2},
\label{eq14j}
\ee
where the constant $C \sim O(\mu)$. This behavior is consistent with that of Eq.(\ref{eq5}).
\section{Experimental constraints}
\subsection{$S$ parameter}
The $S$ parameter provides an important test for new physics beyond the Standard Model~\cite{pt}. This parameter can be described by the absorptive
part of the vector-vector minus axial-vector-axial-vector vacuum polarization in the following form in the case of a TC model with
new composite vector and axial-vector mesons with masses $M_V$ and $M_A$ and respective decay constants $F_V$ and $F_A$~\cite{pt}:
\be
S=4\int_0^\infty \frac{ds}{s} Im \overline{\Pi}(s)=4\pi \left[ \frac{F_V^2}{M_V^2}-\frac{F_A^2}{M_A^2} \right] \, .
\label{eqss}
\ee
An interesting analysis of the $S$ parameter in TC theories was performed in Ref.~\cite{asan} with the use of the Weinberg sum rules,
where the case of a conformal theory was considered. In our case, we have a TC model which is just a scaled QCD theory,
with effective masses due to the different SDE contributions shown in Fig.2, besides its dynamical mass of $O(1)$ TeV.
There is no reason to expect modifications of Eq.(\ref{eqss}) for this type of theory, as well as the simple extension to TC of the first and second Weinberg sum rules,
which are respectively
\be
F_V^2 - F_A^2 = F_\Pi^2 \, ,
\label{1sm}
\ee
and
\be
F_V^2 M_V^2 - F_A^2 M_A^2 =0 \, ,
\label{2sm}
\ee
which lead to
\be
S=4\pi F_\Pi^2 \left[ \frac{1}{M_V^2}+\frac{1}{M_A^2} \right] \, .
\label{sfin}
\ee
We can also apply the result of vector meson dominance to Eq.(\ref{sfin})~\cite{wei2}, implying that $M_A^2 = 2 M_V^2$. This relation
is not exact even in QCD, but by considering it we are at most overestimating the $S$ parameter, which is now be given by
\be
S\approx \frac{6\pi F_\Pi^2}{M_V^2} \, .
\label{s2}
\ee
The TC technipion decay constant is usually assumed to be $F_\Pi \approx 246$GeV.
To determine the value of $S$ shown in Eq.(\ref{s2}) we must have one
estimate of the vector-meson mass. It should be remembered that the vector-boson mass is quite large only due to the spin-spin part
of the hyperfine interactions. We can determine the vector-boson mass by using the hyperfine splitting calculation performed in the
heavy quarkonium context in Ref.~\cite{ei}
\be
M(^3S_1)-M(^1S_0)\approx \frac{8}{9} {\bar{g}}^2(0) \frac{|\psi (0)|^2}{\mu^2} \, ,
\label{eqvc}
\ee
where $M(^3S_1)$ and $M(^1S_0)$ describe the masses of vector and scalar lighter bosons, respectively. In Eq.(\ref{eqvc}), $|\psi (0)|^2$ is the meson wave
function at the origin, describing a vector boson formed by techniquarks with dynamical mass $\mu_{\tt{TC}}$. Equation (\ref{eqvc}) seems to be reasonable
even when the vector-boson constituents are light~\cite{sc}.
We make the following assumptions: 1) The TC theory has an infrared frozen coupling constant ${\bar{g}}^2(0)/4\pi \approx 0.5$, whose value can be similar to several determinations of this quantity in the QCD case (see, for instance, Ref.~\cite{usf}), 2) The lightest TC scalar boson has the same mass as the Higgs
boson found at the LHC, i.e., $M(^1S_0)=125$GeV, 3) The wave function is approximated by $|\psi (0)|^2 \approx \mu_{\tt{TC}}^3\approx 1$TeV$^3$, consistent
with the other BSE wave functions proportional to the dynamical fermion mass (see Eq.(\ref{eq11})). As a consequence, we obtain
a vector-boson mass $M_V\approx 5.71$TeV, leading to
\be
S \approx 0.035 \, ,
\label{sfu}
\ee
whose value has probably been overestimated but is still consistent with the experimental data ($S=0.02\pm 0.07$)~\cite{pdg}.
\subsection{Horizontal symmetry}
A necessary condition for the type of model that we are proposing here is the presence of the horizontal (or family) symmetry. This symmetry can be local, and
it is only necessary to enforce the connection between the TC sector and the third ordinary fermionic generation, i.e., the $t$ and $b$ quarks, the $\tau$, and
its neutrino. This symmetry in general leads to flavor violations at an undesirable level; however, in the scheme proposed here the masses of the horizontal
gauge bosons can be quite heavy, affecting only logarithmic corrections to the fermion masses, and not producing significant tree-level reactions
that may be severely constrained by the experimental data. On the other hand there are hints of $B$ decay anomalies~\cite{b1,b2,b3,b4,b5} which, if confirmed, could also set a mass scale for our horizontal symmetry.
One of the anomalies in $B$ decays appears in the measurement of the ratio between the branching fractions of the processes $B^0 \rightarrow K^{*0}\mu^+\mu^-$ and
$B^0 \rightarrow K^{*0}e^+e^-$, which in the small dilepton invariant mass region is given by
\be
R (K^*)=\frac{B^0 \rightarrow K^{*0}\mu^+\mu^-}{B^0 \rightarrow K^{*0}e^+e^-}= 0.66 {\substack{+0.11 \\ -0.07}}\pm 0.03 \, ,
\label{bdec}
\ee
which is around $2.2$ standard deviations away from the SM expectation.
If such deviation is confirmed in the future, it could be explained by a current-current interaction described by the following effective Lagrangian:
\be
L_h \propto \alpha_h \frac{\lambda_{bs}C^{\mu\mu}}{M_h^2} (\overline{s}\gamma_\nu P_L b)(\overline{\mu}\gamma^\nu \mu) \, ,
\label{lb}
\ee
where $\alpha_h$ is the horizontal gauge coupling, $\lambda_{bs}$ are mixing angles, $M_h$ is the horizontal gauge boson mass, and $C^{\mu\mu}$ is a Wilson
coefficient. If we naively assume the results of the $SU(3)_h$ horizontal model of Ref.~\cite{alo} for these several constants, we can roughly estimate that
$M_h$ should be greater than $10$TeV. However, this is only a guess because (as said repeatedly in the previous sections) the horizontal gauge boson
can be quite heavy, and this scale can be set to these masses only if the anomalies remain discrepant with the SM expectation. Otherwise, the
dependence on the factor $1/M_h^2$ in all observables of this kind will lessen experimental constraints originated from horizontal symmetries.
There are other possible flavor-changing neutral currents induced by the horizontal symmetry. For instance, the effective Lagrangian
\be
L_h \propto \alpha_h \frac{\lambda_{sd}}{M_h^2} (\overline{s}_L\gamma_\nu d_L)(\overline{s}_R\gamma^\nu d_R) \, ,
\label{sd}
\ee
is induced by one-gauge-boson exchange and contributes to the $K^0 - \bar{K}^0$ transition, which for $\lambda \approx 1/20$ requires $M_h \geq 200$TeV ~\cite{die}.
This contribution can be easily evaded in our type of model simply by increasing the horizontal gauge boson mass scale, which will not affect the mechanism
of ordinary fermion mass generation. Therefore, a careful scrutiny of the gauge symmetry
breaking of the horizontal group will only be necessary if the $B$ decay anomaly is confirmed.
\subsection{Technipion masses}
The LHC collaborations already have enough data to constrain the existence of light technipions~\cite{scs1}. Due to the fact that the technifermions acquire masses
of $O(100)$ GeV, the resulting pseudo-Goldstone bosons [i.e., those generated in the chiral breaking of the $SU(4)_{\tt{TC}}$ TC gauge group discussed in Sec. II]
may be heavier than the SM Higgs boson. Moreover, due to the choice of the horizontal symmetry quantum numbers the technipions will mainly couple to the third ordinary fermionic family,
i.e., $t$ and $b$ quarks and the $\tau$ lepton, in such a way that may easily evade the limits found in Ref.~\cite{scs1} obtained from data on the SM Higgs boson decaying
into $\gamma \gamma$ and $\tau^+ \tau^-$.
The colored and charged technipions will be quite heavy and are produced along with $t$ and $b$ quarks. In the case of the decay into $b$ quarks the branching ratio
may be reduced by a possible small coupling between this quark and the technipion, which will happen through the exchange of a rather heavy gauge boson, and
their signal could easily be buried in the background. This leaves us with the lightest technipions, which should be the
neutral ones ($\bar{N}\gamma_5 N$). In this case a neutral technipion may be produced through vector-boson fusion and decay through the weak $ZZ$ channel.
The discussion of the TC condensate in Sec. V can be used to estimate the neutral technipion mass ($m_\Pi$) in a different way than in Ref.~\cite{us1}.
As considered in Eq.(\ref{eq14j}) the neutral technifermion mass ($m_N$) in terms of the TC condensate generated by diagram $(a_4)$ of Fig.2 is given by
\be
m_N \sim \frac{\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda}}{\Lambda^2} \, .
\ee
The above equation together with Eq.(\ref{eq14}) leads to the following estimate of the neutral technipion mass
\be
m_\Pi^2 \approx \frac{(\<{\bar{\psi}_T}\psi_T\>^{\alpha_E \neq 0}_{\Lambda})^2}{2F_{\Pi}^2\Lambda^2} \, .
\ee
Assuming $SU(3)_{{TC}}$ as the TC gauge group, $\<{\bar{\psi}_T}\psi_T\>^{\alpha_E =0} \sim \mu^3$ with $\mu = 1$ TeV,
$R_w^{rad.cor.} \approx 7.87 \times 10^6[ln(M^2_E/\mu^2)]^{-4.3}$ defined and appearing in Eqs.(\ref{rwcor}) and (\ref{eq14g}), we obtain
\be
m_\Pi \sim 160 \,\,\, GeV ,
\ee
which is a rough estimate for the smallest pseudo-Goldstone mass of our type of model, which has not yet been eliminated by the LHC data~\cite{scs1}.
The fact that in our type of model the technifermions couple preferentially to the third fermionic family and obtain a large effective mass due ETC interactions, and that their other couplings to ordinary fermions are always diminished by the exchange of a very heavy horizontal or GUT gauge boson makes the search for pseudo-Goldstone signals quite difficult. The main hope for detecting technipions may be the resonant production of the lightest neutral technipion and its decay into neutral weak bosons.
\section{Scalar boson trilinear coupling}
As already pointed out many years ago~\cite{ebo}, the measurement of the Higgs boson trilinear coupling is fundamental to determining the nature of
this particle. If the Higgs boson is a composite particle its trilinear coupling may deviate from the SM value of a fundamental scalar boson,
and its measurement can even provide a signal of the underlying theory forming the composite state~\cite{doff}.
In TC or any composite scalar model the scalar trilinear coupling is determined through its coupling to fermions.
Using Ward identities, we can show that the couplings of the scalar boson to fermions are \cite{ca2}
\be
G^{\sl a} (p+q,p) = -\imath \frac{g_{W}}{2M_{W}}
\left[\tau^{\sl a}\Sigma(p)P_R - \Sigma(p+q)\tau^{\sl a} P_L \right]
\label{fsc}
\ee
where $P_{R,L} = \frac{1}{2} (1 \pm \gamma_5 )$, $\tau^{\sl a}$ is a $SU(2)$
matrix, and $\Sigma$ is the fermionic self-energy in weak-isodoublet
space. As in Ref.\cite{ca2}, we assume that the scalar composite Higgs
boson coupling to the fermionic self-energy is saturated by the
top quark. We also do not differentiate between the two fermion momenta $p$ and $p+q$ since, in all situations of interest,
$\Sigma(p+q)\approx \Sigma(p)$. Therefore, the coupling between a composite Higgs boson and fermions at large momenta is given by
\be
\lambda_{{}_{Hff}}(p)\equiv G(p,p) \sim -\frac{g_{W}}{2M_{W}}\Sigma(p^2).
\label{fh}
\ee
The trilinear coupling of the composite scalar boson is determined by the diagram shown in Fig.6.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{acop3H.eps}
\vspace{0.3cm}
\caption{The dominant contribution to the trilinear scalar coupling. The blobs in this figure represent the coupling of the composite scalar boson to fermions. The double lines represent
the composite scalar boson.}
\label{fig6}
\end{center}
\end{figure}
Assuming that the coupling of the scalar boson to the fermions is given by Eq.(\ref{fh}), we find that
\be
\lambda_{{}_{3H}} = \frac{3g^3_{W}}{64\pi^2}\left(\frac{3n_{F}}{M^3_{W}}\right)\int^{M^2_E}_{0}\frac{\Sigma^4(p^2)p^4dp^2}{(p^2 + \Sigma^2(p^2))^3}.
\label{tri}
\ee
\noindent where $n_{F}$ is the number of technifermions included in the model.
The SM trilinear scalar coupling value, according to the normalization of Ref.~\cite{malt}, is
\be
\lambda_{SM} = \frac{M^2_H}{2v^2}.
\label{norml}
\ee
Combined with the above normalization, the trilinear coupling of Eq.(\ref{tri}) leads to the following scalar trilinear coupling $\lambda$:
\be
\lambda = \frac{1}{6v}\lambda_{{}_{3H}} .
\label{nor2}
\ee
Considering Eqs.(\ref{tri}) and (\ref{nor2}), $v=F_{{}_{\Pi}}$, and the relation
$$
M^2_{W} = \frac{g^2_W F^2_{{}_{\Pi}}}{4}
$$
we obtain for the trilinear coupling
\be
\lambda = \frac{1}{16\pi^2}\left(\frac{3n_{F}}{F^4_{{}_{\Pi}}}\right)\int^{M^2_E}_{0}\frac{\Sigma^4(p^2)p^4dp^2}{(p^2 + \Sigma^2(p^2))^3},
\label{tri3}
\ee
which is the trilinear scalar composite coupling that can be compared to the SM coupling of Eq.(\ref{norml}).
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.45]{acopl3H2.eps}
\vspace{-2cm}
\caption{Experimental limits on the scalar boson trilinear coupling, and curves of the trilinear coupling value (\ref{tri3})) in the case of a composite scalar boson. }
\label{fig7}
\end{center}
\end{figure}
Using the results for the TC self-energy obtained in Ref.~\cite{us2} and Sec. III, which is dominated by diagrams ($a_1$) and ($a_4$) of Fig.2, we compute the trilinear coupling presented in Eq.(\ref{tri3}). A comparison of the trilinear composite coupling with the SM one is shown in Fig.7. The composite trilinear coupling does differ from the SM one,
but only a small amount. In Fig. 7 we also show the current LHC limits on this coupling obtained in Ref.~\cite{malt} from the $(b\bar{b}\gamma\gamma)$ signal, whose values are
$\lambda < -1.3\lambda_{SM} = -0.169$ (red region) and $\lambda > 8.7\lambda_{SM} = 1.13$ (green region). Figure 7 remind us that the actual result
for the scalar trilinear coupling does vary with $M_E$, and this variation should appear when the coupled gap equations are solved taking into account the
running of the ETC gauge coupling constant. Of course, this will introduce only a small variation in the curves of that figure.
The white region is not excluded yet, and this large region shows how difficult it is to differentiate one composite scalar boson from a fundamental one by just observing the specific coupling.
\section{Conclusions}
In Refs.~\cite{us1,us2} we called attention to the fact that the self-energies of strongly interacting theories are modified when we consider coupled SDEs including radiative
corrections. The effect of the radiative corrections is not very different from the \textit{ad hoc} introduction of effective four-fermion interactions, as verified many
years ago by Takeuchi~\cite{takeuchi}, and it leads to self-energies that decrease logarithmically with the momentum. This effect was reviewed in the Introduction of this work, where it was made clear that the usual TC model building has to be modified, where the ordinary fermion mass hierarchy is not related to different ETC gauge boson masses.
The presence of a horizontal symmetry is mandatory in the type of models envisaged in Sec. II. This symmetry is necessary to give masses to only the third generation of
ordinary fermions at leading order. The model discussed in Sec. II is based on the non-Abelian gauge group structure $SU(9)_U \otimes SU(3)_H$, where the $SU(9)_U$
group contains the SM, an $SU(5)_{\tiny \textsc{gg}}$ Georgy-Glashow GUT~\cite{gg}, and a $SU(4)_{\tt{TC}}$ group. The $SU(3)_H$ horizontal symmetry was introduced in
such a way that their fermionic quantum numbers allow only the third fermionic generation to be coupled to the technifermions. The other fermions remain massless at
leading order. However, the first-generation fermions obtain their masses due to the coupling with QCD, which also has a slowly decreasing self-energy. This is the most
interesting fact of our model: the different fermionic mass scales are dictated by the different strong interactions present in the model! We have shown some of the diagrams that
generate the different masses, and made rough estimates of their masses. We believe that a large number of theories can be built along the lines of the model of
Sec. II. Precise determinations of fermion masses in this type of model will demand a lengthy determination of SDE coupled equations, where different self-energies
can be fitted by equations like Eq.(\ref{eq5}).
The fact that the ETC interactions can be pushed to very high energies apparently seems to open
a path for a plethora of TC models capable of describing the ordinary fermionic mass spectra.
The determination of fermion masses will involve a delicate balance of different gauge group
theories for TC, ETC (or GUT), and horizontal symmetry. The ordinary fermion mass matrix calculation will involve the knowledge
of specific Casimir eigenvalues, which will depend on the different fermionic representations of the different gauge groups.
It will also involve the different coupling constant values of these theories at different scales, and the far more demanding
solutions of the coupled system of Schwinger-Dyson equations even with a minimum of approximations. Therefore, while
a new frontier arise, generic combination of gauge theories and
respective fermionic representations will not be able to explain the known fermionic spectra, meaning that an enormous engineering effort
will be necessary for a \textit{precise} calculation of ordinary fermion masses.
In Sec, III we discussed how the composite scalar boson may have a mass lighter than the characteristic mass scale of the theory that forms the composite particle.
This could explain how the observed Higgs boson mass, if composite, is smaller than the Fermi mass scale. Perhaps the most important factor regarding the mass value of the scalar
composite resides in the normalization
condition of the inhomogeneous BSE, which has to be taken into account when the self-energy is hard and not decaying as $1/p^2$. The normalization condition,
as shown by the results presented in Table I, is enough to lower the scalar mass by a factor of $1/10$. However, we have listed many other effects
that may also lower the scalar composite mass.
Section IV contains a brief discussion about pseudo-Goldstone boson masses. It is just a complementary discussion to the one already presented in Refs.~\cite{us1,us2},
indicating that their masses should be of the order of or higher than that of the observed Higgs boson. Moreover, the pseudo-Goldstone bosons couple at leading order
only to the third-generation fermions, which is another fact that will complicate their experimental observation.
In Sec. V we computed the TC condensate in the coupled SDE scenario. This calculation serves as a comparison with the enhancement that appears in the TC
condensate in walking TC theories. Although the mechanism is totally different, i.e., here the gauge theory is just a running theory, there is also one enhancement in the
condensates as a result of a logarithmically decreasing self-energy with the momentum. Again, it is possible to verify that the effect is not qualitatively different
from the \textit{ad hoc} inclusion of a four-fermion interaction, which is replaced by genuine radiative corrections of known interactions.
In Sec. VI we commented on possible experimental constraints on this type of model. The main point is that the ETC gauge boson masses may be pushed to very high
energies and unnatural flavor-changing events will be absent. The $S$ parameter will be of the expected order, and should not differ from the case of TC as a scaled QCD theory. Complementing the discussion of Sec. IV with what was presented in Sec. V, we estimated pseudo-Goldstone masses and verified that they cannot yet be seen
at the LHC according the analysis of Ref.~\cite{scs1}.
In Sec. VII we computed the trilinear scalar coupling and verified that a signal of compositeness is far from being observed with the present data~\cite{malt},
and this coupling does not differ by a large amount from the SM value in the case of a fundamental scalar boson. Finally, we may say that in the scenario
presented in this work there is a possibility that the SM gauge symmetry breaking promoted dynamically by composite scalar bosons is still alive.
\section*{Acknowledgments}
This research was partially supported by the Conselho Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico (CNPq)
under the grants 302663/2016-9 (A.D.) and 302884/2014 (A.A.N.).
|
{
"timestamp": "2019-03-18T01:18:32",
"yymm": "1902",
"arxiv_id": "1902.11072",
"language": "en",
"url": "https://arxiv.org/abs/1902.11072"
}
|
"\\section{Introduction}\n\nDeep neural networks have been evolved to a general-purpose machine lear(...TRUNCATED)
| {"timestamp":"2019-03-01T02:01:57","yymm":"1902","arxiv_id":"1902.10758","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nAbdominal Aortic Aneurysm (AAA), an enlargement of the abdominal aorta wit(...TRUNCATED)
| {"timestamp":"2019-03-01T02:19:10","yymm":"1902","arxiv_id":"1902.11089","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nData commonly come in the form of ranking in preference survey such as vot(...TRUNCATED)
| {"timestamp":"2019-03-01T02:12:49","yymm":"1902","arxiv_id":"1902.10963","language":"en","url":"http(...TRUNCATED)
|
"\\section{Summary \\label{S5}}\n\nWe present the determination of the\ninterstellar magnesium abund(...TRUNCATED)
| {"timestamp":"2019-03-01T02:18:43","yymm":"1902","arxiv_id":"1902.11083","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\label{sec:intro}\n\nAccreting black holes are found to exist across a wi(...TRUNCATED)
| {"timestamp":"2019-03-01T02:05:17","yymm":"1902","arxiv_id":"1902.10833","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nConstituent parsing is a core task in natural language processing (\\tex(...TRUNCATED)
| {"timestamp":"2019-03-01T02:14:38","yymm":"1902","arxiv_id":"1902.10985","language":"en","url":"http(...TRUNCATED)
|
"\n\n\\section{Introduction}\nBuilding an AI capable of inferring the 3D structure and pose of an ob(...TRUNCATED)
| {"timestamp":"2019-03-01T02:05:55","yymm":"1902","arxiv_id":"1902.10840","language":"en","url":"http(...TRUNCATED)
|
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