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quant-ph/0605096
|
Quantum Information and Entropy
|
quant-ph cs.IT math.IT
|
Thermodynamic entropy is not an entirely satisfactory measure of information
of a quantum state. This entropy for an unknown pure state is zero, although
repeated measurements on copies of such a pure state do communicate
information. In view of this, we propose a new measure for the informational
entropy of a quantum state that includes information in the pure states and the
thermodynamic entropy. The origin of information is explained in terms of an
interplay between unitary and non-unitary evolution. Such complementarity is
also at the basis of the so-called interaction-free measurement.
|
quant-ph/0607111
|
`Plausibilities of plausibilities': an approach through circumstances
|
quant-ph cs.AI
|
Probability-like parameters appearing in some statistical models, and their
prior distributions, are reinterpreted through the notion of `circumstance', a
term which stands for any piece of knowledge that is useful in assigning a
probability and that satisfies some additional logical properties. The idea,
which can be traced to Laplace and Jaynes, is that the usual inferential
reasonings about the probability-like parameters of a statistical model can be
conceived as reasonings about equivalence classes of `circumstances' - viz.,
real or hypothetical pieces of knowledge, like e.g. physical hypotheses, that
are useful in assigning a probability and satisfy some additional logical
properties - that are uniquely indexed by the probability distributions they
lead to.
|
quant-ph/0609117
|
Quantum Pattern Retrieval by Qubit Networks with Hebb Interactions
|
quant-ph cond-mat.dis-nn cs.NE
|
Qubit networks with long-range interactions inspired by the Hebb rule can be
used as quantum associative memories. Starting from a uniform superposition,
the unitary evolution generated by these interactions drives the network
through a quantum phase transition at a critical computation time, after which
ferromagnetic order guarantees that a measurement retrieves the stored memory.
The maximum memory capacity p of these qubit networks is reached at a memory
density p/n=1.
|
quant-ph/0609229
|
Ergodic Classical-Quantum Channels: Structure and Coding Theorems
|
quant-ph cs.IT math-ph math.IT math.MP
|
We consider ergodic causal classical-quantum channels (cq-channels) which
additionally have a decaying input memory. In the first part we develop some
structural properties of ergodic cq-channels and provide equivalent conditions
for ergodicity. In the second part we prove the coding theorem with weak
converse for causal ergodic cq-channels with decaying input memory. Our proof
is based on the possibility to introduce joint input-output state for the
cq-channels and an application of the Shannon-McMillan theorem for ergodic
quantum states. In the last part of the paper it is shown how this result
implies coding theorem for the classical capacity of a class of causal ergodic
quantum channels.
|
quant-ph/0610153
|
Subsystem Codes
|
quant-ph cs.IT math.IT
|
We investigate various aspects of operator quantum error-correcting codes or,
as we prefer to call them, subsystem codes. We give various methods to derive
subsystem codes from classical codes. We give a proof for the existence of
subsystem codes using a counting argument similar to the quantum
Gilbert-Varshamov bound. We derive linear programming bounds and other upper
bounds. We answer the question whether or not there exist
[[n,n-2d+2,r>0,d]]<sub>q</sub> subsystem codes. Finally, we compare stabilizer
and subsystem codes with respect to the required number of syndrome qudits.
|
quant-ph/0610200
|
Quantum List Decoding of Classical Block Codes of Polynomially Small
Rate from Quantumly Corrupted Codewords
|
quant-ph cs.CC cs.IT math.IT
|
Given a classical error-correcting block code, the task of quantum list
decoding is to produce from any quantumly corrupted codeword a short list
containing all messages whose codewords exhibit high "presence" in the
quantumly corrupted codeword. Efficient quantum list decoders have been used to
prove a quantum hardcore property of classical codes. However, the code rates
of all known families of efficiently quantum list-decodable codes are,
unfortunately, too small for other practical applications. To improve those
known code rates, we prove that a specific code family of polynomially small
code rate over a fixed code alphabet, obtained by concatenating generalized
Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an
efficient quantum list-decoding algorithm if its codewords have relatively high
codeword presence in a given quantumly corrupted codeword. As an immediate
application, we use the quantum list decodability of this code family to solve
a certain form of quantum search problems in polynomial time. When the codeword
presence becomes smaller, in contrast, we show that the quantum list
decodability of generalized Reed-Solomon codes with high confidence is closely
related to the efficient solvability of the following two problems: the noisy
polynomial interpolation problem and the bounded distance vector problem.
Moreover, assuming that NP is not included in BQP, we also prove that no
efficient quantum list decoder exists for the generalized Reed-Solomon codes.
|
quant-ph/0611167
|
Continuous Variable Quantum Cryptography using Two-Way Quantum
Communication
|
quant-ph cs.CR cs.IT math.IT physics.optics
|
Quantum cryptography has been recently extended to continuous variable
systems, e.g., the bosonic modes of the electromagnetic field. In particular,
several cryptographic protocols have been proposed and experimentally
implemented using bosonic modes with Gaussian statistics. Such protocols have
shown the possibility of reaching very high secret-key rates, even in the
presence of strong losses in the quantum communication channel. Despite this
robustness to loss, their security can be affected by more general attacks
where extra Gaussian noise is introduced by the eavesdropper. In this general
scenario we show a "hardware solution" for enhancing the security thresholds of
these protocols. This is possible by extending them to a two-way quantum
communication where subsequent uses of the quantum channel are suitably
combined. In the resulting two-way schemes, one of the honest parties assists
the secret encoding of the other with the chance of a non-trivial superadditive
enhancement of the security thresholds. Such results enable the extension of
quantum cryptography to more complex quantum communications.
|
quant-ph/0612052
|
Deciding whether a quantum state has secret correlations is an
NP-complete problem
|
quant-ph cs.IT math.IT
|
From the NP-hardness of the quantum separability problem and the relation
between bipartite entanglement and the secret key correlations, it is shown
that the problem deciding whether a given quantum state has secret correlations
in it or not is in NP-complete.
|
quant-ph/0612155
|
A father protocol for quantum broadcast channels
|
quant-ph cs.IT math.IT
|
A new protocol for quantum broadcast channels based on the fully quantum
Slepian-Wolf protocol is presented. The protocol yields an achievable rate
region for entanglement-assisted transmission of quantum information through a
quantum broadcast channel that can be considered the quantum analogue of
Marton's region for classical broadcast channels. The protocol can be adapted
to yield achievable rate regions for unassisted quantum communication and for
entanglement-assisted classical communication; in the case of unassisted
transmission, the region we obtain has no independent constraint on the sum
rate, only on the individual transmission rates. Regularized versions of all
three rate regions are provably optimal.
|
quant-ph/0701020
|
Quantum Quasi-Cyclic LDPC Codes
|
quant-ph cs.IT math-ph math.CO math.IT math.MP
|
In this paper, a construction of a pair of "regular" quasi-cyclic LDPC codes
as ingredient codes for a quantum error-correcting code is proposed. That is,
we find quantum regular LDPC codes with various weight distributions.
Furthermore our proposed codes have lots of variations for length, code rate.
These codes are obtained by a descrete mathematical characterization for model
matrices of quasi-cyclic LDPC codes.
Our proposed codes achieve a bounded distance decoding (BDD) bound, or known
as VG bound, and achieve a lower bound of the code length.
|
quant-ph/0701037
|
Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller
Codes
|
quant-ph cs.IT math.IT
|
Convolutional stabilizer codes promise to make quantum communication more
reliable with attractive online encoding and decoding algorithms. This paper
introduces a new approach to convolutional stabilizer codes based on direct
limit constructions. Two families of quantum convolutional codes are derived
from generalized Reed-Solomon codes and from Reed- Muller codes. A Singleton
bound for pure convolutional stabilizer codes is given.
|
quant-ph/0701168
|
Using quantum key distribution for cryptographic purposes: a survey
|
quant-ph cs.CR cs.IT math.IT
|
The appealing feature of quantum key distribution (QKD), from a cryptographic
viewpoint, is the ability to prove the information-theoretic security (ITS) of
the established keys. As a key establishment primitive, QKD however does not
provide a standalone security service in its own: the secret keys established
by QKD are in general then used by a subsequent cryptographic applications for
which the requirements, the context of use and the security properties can
vary. It is therefore important, in the perspective of integrating QKD in
security infrastructures, to analyze how QKD can be combined with other
cryptographic primitives. The purpose of this survey article, which is mostly
centered on European research results, is to contribute to such an analysis. We
first review and compare the properties of the existing key establishment
techniques, QKD being one of them. We then study more specifically two generic
scenarios related to the practical use of QKD in cryptographic infrastructures:
1) using QKD as a key renewal technique for a symmetric cipher over a
point-to-point link; 2) using QKD in a network containing many users with the
objective of offering any-to-any key establishment service. We discuss the
constraints as well as the potential interest of using QKD in these contexts.
We finally give an overview of challenges relative to the development of QKD
technology that also constitute potential avenues for cryptographic research.
|
quant-ph/0702005
|
A decoupling approach to the quantum capacity
|
quant-ph cs.IT math.IT
|
We give a short proof that the coherent information is an achievable rate for
the transmission of quantum information through a noisy quantum channel. Our
method is to produce random codes by performing a unitarily covariant
projective measurement on a typical subspace of a tensor power state. We show
that, provided the rank of each measurement operator is sufficiently small, the
transmitted data will with high probability be decoupled from the channel's
environment. We also show that our construction leads to random codes whose
average input is close to a product state and outline a modification yielding
unitarily invariant ensembles of maximally entangled codes.
|
quant-ph/0702072
|
Markovian Entanglement Networks
|
quant-ph cs.AI
|
Graphical models of probabilistic dependencies have been extensively
investigated in the context of classical uncertainty. However, in some domains
(most notably, in computational physics and quantum computing) the nature of
the relevant uncertainty is non-classical, and the laws of classical
probability theory are superseded by those of quantum mechanics. In this paper
we introduce Markovian Entanglement Networks (MEN), a novel class of graphical
representations of quantum-mechanical dependencies in the context of such
non-classical systems. MEN are the quantum-mechanical analogue of Markovian
Networks, a family of undirected graphical representations which, in the
classical domain, exploit a notion of conditional independence among
subsystems.
After defining a notion of conditional independence appropriate to our domain
(conditional separability), we prove that the conditional separabilities
induced by a quantum-mechanical wave function are effectively reflected in the
graphical structure of MEN. Specifically, we show that for any wave function
there exists a MEN which is a perfect map of its conditional separabilities.
Next, we show how the graphical structure of MEN can be used to effectively
classify the pure states of three-qubit systems. We also demonstrate that, in
large systems, exploiting conditional independencies may dramatically reduce
the computational burden of various inference tasks. In principle, the
graph-theoretic representation of conditional independencies afforded by MEN
may not only facilitate the classical simulation of quantum systems, but also
provide a guide to the efficient design and complexity analysis of quantum
algorithms and circuits.
|
quant-ph/0703112
|
Graphs, Quadratic Forms, and Quantum Codes
|
quant-ph cs.IT math.IT
|
We show that any stabilizer code over a finite field is equivalent to a
graphical quantum code. Furthermore we prove that a graphical quantum code over
a finite field is a stabilizer code. The technique used in the proof
establishes a new connection between quantum codes and quadratic forms. We
provide some simple examples to illustrate our results.
|
quant-ph/0703113
|
Quantum Convolutional BCH Codes
|
quant-ph cs.IT math.IT
|
Quantum convolutional codes can be used to protect a sequence of qubits of
arbitrary length against decoherence. We introduce two new families of quantum
convolutional codes. Our construction is based on an algebraic method which
allows to construct classical convolutional codes from block codes, in
particular BCH codes. These codes have the property that they contain their
Euclidean, respectively Hermitian, dual codes. Hence, they can be used to
define quantum convolutional codes by the stabilizer code construction. We
compute BCH-like bounds on the free distances which can be controlled as in the
case of block codes, and establish that the codes have non-catastrophic
encoders.
|
quant-ph/0703181
|
Quantum Block and Convolutional Codes from Self-orthogonal Product Codes
|
quant-ph cs.IT math.IT
|
We present a construction of self-orthogonal codes using product codes. From
the resulting codes, one can construct both block quantum error-correcting
codes and quantum convolutional codes. We show that from the examples of
convolutional codes found, we can derive ordinary quantum error-correcting
codes using tail-biting with parameters [[42N,24N,3]]_2. While it is known that
the product construction cannot improve the rate in the classical case, we show
that this can happen for quantum codes: we show that a code [[15,7,3]]_2 is
obtained by the product of a code [[5,1,3]]_2 with a suitable code.
|
quant-ph/0703182
|
Constructions of Quantum Convolutional Codes
|
quant-ph cs.IT math.IT
|
We address the problems of constructing quantum convolutional codes (QCCs)
and of encoding them. The first construction is a CSS-type construction which
allows us to find QCCs of rate 2/4. The second construction yields a quantum
convolutional code by applying a product code construction to an arbitrary
classical convolutional code and an arbitrary quantum block code. We show that
the resulting codes have highly structured and efficient encoders. Furthermore,
we show that the resulting quantum circuits have finite depth, independent of
the lengths of the input stream, and show that this depth is polynomial in the
degree and frame size of the code.
|
quant-ph/9703022
|
Reversibility and Adiabatic Computation: Trading Time and Space for
Energy
|
quant-ph cs.CC cs.CE cs.DS
|
Future miniaturization and mobilization of computing devices requires energy
parsimonious `adiabatic' computation. This is contingent on logical
reversibility of computation. An example is the idea of quantum computations
which are reversible except for the irreversible observation steps. We propose
to study quantitatively the exchange of computational resources like time and
space for irreversibility in computations. Reversible simulations of
irreversible computations are memory intensive. Such (polynomial time)
simulations are analysed here in terms of `reversible' pebble games. We show
that Bennett's pebbling strategy uses least additional space for the greatest
number of simulated steps. We derive a trade-off for storage space versus
irreversible erasure. Next we consider reversible computation itself. An
alternative proof is provided for the precise expression of the ultimate
irreversibility cost of an otherwise reversible computation without
restrictions on time and space use. A time-irreversibility trade-off hierarchy
in the exponential time region is exhibited. Finally, extreme
time-irreversibility trade-offs for reversible computations in the thoroughly
unrealistic range of computable versus noncomputable time-bounds are given.
|
quant-ph/9802028
|
Analogue Quantum Computers for Data Analysis
|
quant-ph cs.CV
|
Analogue computers use continuous properties of physical system for modeling.
In the paper is described possibility of modeling by analogue quantum computers
for some model of data analysis. It is analogue associative memory and a formal
neural network. A particularity of the models is combination of continuous
internal processes with discrete set of output states. The modeling of the
system by classical analogue computers was offered long times ago, but now it
is not very effectively in comparison with modern digital computers. The
application of quantum analogue modelling looks quite possible for modern level
of technology and it may be more effective than digital one, because number of
element may be about Avogadro number (N=6.0E23).
|
quant-ph/9809081
|
Concatenating Decoherence Free Subspaces with Quantum Error Correcting
Codes
|
quant-ph cs.IT math-ph math.IT math.MP
|
An operator sum representation is derived for a decoherence-free subspace
(DFS) and used to (i) show that DFSs are the class of quantum error correcting
codes (QECCs) with fixed, unitary recovery operators, and (ii) find explicit
representations for the Kraus operators of collective decoherence. We
demonstrate how this can be used to construct a concatenated DFS-QECC code
which protects against collective decoherence perturbed by independent
decoherence. The code yields an error threshold which depends only on the
perturbing independent decoherence rate.
|
quant-ph/9907009
|
The importance of quantum decoherence in brain processes
|
quant-ph cond-mat.dis-nn cs.NE physics.bio-ph q-bio
|
Based on a calculation of neural decoherence rates, we argue that that the
degrees of freedom of the human brain that relate to cognitive processes should
be thought of as a classical rather than quantum system, i.e., that there is
nothing fundamentally wrong with the current classical approach to neural
network simulations. We find that the decoherence timescales ~10^{-13}-10^{-20}
seconds are typically much shorter than the relevant dynamical timescales
(~0.001-0.1 seconds), both for regular neuron firing and for kink-like
polarization excitations in microtubules. This conclusion disagrees with
suggestions by Penrose and others that the brain acts as a quantum computer,
and that quantum coherence is related to consciousness in a fundamental way.
|
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