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defined to be
ψ(t) by the Schro¨dinger equation
| i P(a)=P (X =x )
ρ a
∂ ψ(t)
(1) i | i =H ψ(t) , i=√ 1. = ψ Q ψ =Tr(Q ψ ψ ), a=1,2,....
∂t | i − h | a | i a | ih |
Alternatively a quantum system can be described With the probability we derive the expectation un-
by a density operator (or density matrix). A density der pure state ψ ,
| i
operator ρ is an operator on H which (1) is self- E ψ(X)= x aP(a)= x a ψ Q a ψ
adjoint; (2) is semi-positive definite; (3) has unit h | | i
Xa Xa
trace [i.e., Tr(ρ)=1]. Following the convention in
= ψ X ψ =Tr(X ψ ψ ).
quantum information science, we reserve notation ρ
h | | i | ih |
forstate,densityoperatorordensitymatrix.Astate Note the difference between an observable X which
is often classified as a pure state or an ensemble of isaHermitian matrix andits measurementresult X
pure states. A pure state is a unit vector ψ in , which is a real-valued random variable.
| i H
which corresponds to a density operator ρ= ψ ψ , Measuring observable X will alter the state of
| ih |
and an ensemble of pure states corresponds to the the quantum system (Kiefer (2004); von Neumann
5
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
(1955)). Ifthe quantumsystem is preparedwith ini- J Q ψ ψ Q
a j j a
tial state ψ , the state of the system after the mea- = P(j a) | ih |
surement| rei sult x is defined to be Xj=1 | P(a |j)
a
(4) Q a |ψ i . = J p Q a |ψ j ihψ j |Q a = Q aρQ a .
P(a) j P(a) Tr(Q ρ)
Xj=1 a
p
For an ensemble state with density operator ρ
See Holevo (1982), Parthasarathy (1992) and Saku-
given by (2), if the quantum state is ψ , the prob-
| j i rai and Napolitano (2010).
ability that result x occurs is
a
2.4 Quantum Statistics
P(a j)= ψ Q ψ =Tr(Q ψ ψ ).
j a j a j j
| h | | i | ih |
For a given quantum system, it is very important
Applyingthelawoftotalprobability,weobtainthat
but difficult to know its state. If we do not know in
understateρ,theprobabilitythatx occursisequal
a advance the state of the quantum system, we may
to
infer the quantum state by the measurement results
J
ofsomeobservablesobtainedfromthequantumsys-
P(a)=P (X =x )= p P(a j)
ρ a j | tem and show that a certain state has been created.
Xj=1
In statistical terminology, we want to estimate den-
J sity matrix ρ based on measurements on an often
= p Tr(Q ψ ψ )=Tr(Q ρ). large number of systems which are identically pre-
j a j j a
| ih |
Xj=1 paredinthestateρ.Thatis,aftermeasuringobserv-
ables on some identical quantum systems, we can
The expectation of X under state ρ,
make statistical inference about probability distri-
p p
bution P of the measurements and thus indirectly
E [X]= x P [X =x ]= x Tr(Q ρ) ρ
ρ a ρ a a a aboutdensitymatrixρ.Intheliteratureofquantum
Xa=1 Xa=1
physics, quantum tomography is referred to as the
=tr(Xρ), reconstructionoftheunderlyingdensitymatrixρby
probing identically prepared quantum systems from
and variance
some different angles. Specifically, suppose that we
Var ρ[X]=tr[X2ρ] (tr[Xρ])2. perform measurements of observables on identically
−
prepared quantum systems in an unknown state ρ
We may derive the density operator of the quan-
andobtainmeasurementresultsX ,...,X .Assume
tum system after obtaining the measurement re- 1 n
that ρ is known up to some unknown parameter θ;
sult x by conditional probability arguments as fol-
a
then X ,...,X are i.i.d. observations with distri-
lows. If the quantum system is in pure state ψ 1 n
j
| i butions P which depend on θ. This gives a quan-
before the measurement, the quantum state after ρ
tum parametric statistical model. We may then de-
measurement result x has occurred is
a
fine quantum likelihood and Fisher quantum infor-
Q ψ
ψa = a | j i . mation and establish quantum point estimation and
| ji P(a j) quantumhypothesistestingtheory.Alternativelywe
|
p may model ρ nonparametrically by assumingthat ρ
If the quantum state is ρ before the measurement,