量子力学19

In the previous lecture we discussed the diamond norm,its connection to the problem of distin-guishing admissible super-operators,and some of its basic properties.Recall that for X and Y complex Euclidean spaces andΦ∈T(X,Y)a super-operator,we have

Φ ⋄=max (Φ⊗1L(X))(uv∗) 1:u,v∈S(X⊗X) .

In this lecture we will discuss two alternate ways in which this norm may be characterized.

19.1Maximum Output Fidelity Characterization

Suppose X and Y are complex Euclidean spaces andΦ,Ψ∈T(X,Y)are completely positive(but not necessarily trace-preserving)super-operators.Let us define the maximum outputfidelity ofΦandΨas

F max(Φ,Ψ)=max{F(Φ(ρ),Ψ(ξ)):ρ,ξ∈D(X)}.

In other words,this is the maximumfidelity between an output ofΦand an output ofΨ,ranging over all pairs of density operator inputs.Ourfirst characterization of the diamond norm is based on the maximum outputfidelity,and is given by the following theorem.

Theorem19.1.Let X and Y be complex Euclidean spaces and letΦ∈T(X,Y)be an arbitrary super-operator.Suppose further that Z is a complex Euclidean space and A,B∈L(X,Y⊗Z)satisfy

Φ(X)=Tr Z(AXB∗)

for all X∈L(X).Then,for completely positive super-operatorsΨA,ΨB∈T(X,Z)defined as

ΨA(X)=Tr Y(AXA∗),

ΨB(X)=Tr Y(BXB∗),

for all X∈L(X),we have Φ ⋄=F max(ΨA,ΨB).

Remark19.2.Note that it is the space Y that is traced-out in the definition ofΨA andΨB,rather than the space Z.

To prove this theorem,we will begin with the following lemma that establishes a simple rela-tionship between thefidelity and the trace norm.(This appeared as a problem on problem set1.) Lemma19.3.Let X and Y be complex Euclidean spaces and let P,Q∈Pos(X).Assume that u,v∈X⊗Y are arbitrary purifications of P and Q,respectively,meaning that Tr Y(uu∗)=P and Tr Y(vv∗)=Q. Then

F(P,Q)= Tr X(uv∗) 1.

144

ε2max

P∈A

Q∈B

Tr(P+ε1)Tr(Q+ε1),

for any R that does not satisfy this bound will surely not produce a value for expression(19.1)that is smaller than the choice R=1.Any R that satisfies these properties is certainly contained K for the choice

K=

1

α

R1/2(P+ε1)R1/2:P∈A ∩D(X),

D= 1