Most of this lecture will be devoted to establishing characterizations of quantum operations,bad on different ways of reprenting super-operators.First,though,we will discuss one more impor-tant fact about measurements,known as Naimark’s Theorem,that relates general measurements to projective measurements.
5.1Naimark’s Theorem
A measurement{P a:a∈Γ}on a complex Euclidean space X is said to be a projective(or von Neu-mann)measurement if it is the ca that each measurement operator P a is an orthogonal projection operator on X.For such a measurement,it necessarily holds that P a P b=0for a=b.When we refer to a measurement with respect to some orthonormal basis{x a:a∈Γ}of X,it is meant that the measurement is given by{P a:a∈Γ},where P a=x a x∗a for each a∈Γ.
There is a n in which no generality is lost in considering only projective measurements: every(general)measurement on a given register X can be realized as a projective measurement on a pair of registers(X,Y),provided that Y is large enough and initialized to a known pure state. This fact will be easily established once we have proved Naimark’s Theorem,which is as follows. Theorem5.1(Naimark’s Theorem).Let X be a complex Euclidean space,let{P a:a∈Γ}⊂Pos(X) be a measurement,and let Y=CΓ.Then there exists a linear isometry A∈U(X,X⊗Y)such that
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P a=A∗(1X⊗E a,a)A
20以内的加法表for every a∈Γ.
Proof.Define A∈L(X,X⊗Y)as
王寒冰
√
A=∑
a∈Γ
B j.
4.(The Choi-Jamiołkowski reprentation.)It holds that
J(Φ)=年夜饭的作文
k
∑
j=1
vec(A j)vec(B j)∗.
B j vec(X)
for each j=1,...,k and every X∈L(X).Finally,the equivalence between items1and4follows from the expression
J(Φ)=(Φ⊗1L(X))(vec(1X)vec(1X)∗)
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(A j⊗1X)vec(1X)=vec(A j)and vec(1X)∗(B∗j⊗1X)=vec(B j)∗
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Various facts may be derived from the above proposition.For instance,it follows that every super-operatorΦ∈T(X,Y)has a Kraus reprentation in which k=rank(J(Φ))≤dim(X⊗Y), and similarly that every suchΦhas a Stinespring reprentation in which dim(Z)=rank(J(Φ)).
5.3Characterization of quantum operations
Now we are ready to characterize quantum operations in terms of their Choi-Jamiołkowski,Kraus, and Stinespring reprentations.(The natural reprentation does not happen to help us with respect to the particular characterizations.This is not surprising,becau it esntially throws away the operator structure of the input and output of super-operators.)We will begin with a characterization of completely positive super-operators in terms of the reprentations. Theorem5.3.For every super-operatorΦ∈T(X,Y),the following are equivalent:
1.Φis completely positive.
2.Φ⊗1L(X)is positive.
3.J(Φ)∈Pos(Y⊗X).
4.There exists a positive integer k and operators A1,...,A k∈L(X,Y)such that
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Φ(X)=
k
∑
i=1
A i XA∗i(5.4)
for all X∈L(X).
5.Item4holds for k=rank(J(Φ)).
6.There exists a complex Euclidean space Z and an operator A∈L(X,Y⊗Z)such that
Φ(X)=Tr Z(AXA∗)
for all X∈L(X).
7.Item6holds for Z having dimension equal to the rank of J(Φ).
