Quantum Information Theory Problem Set 4

Quantum Information Theory Problem Set 4

Spring 2008 Prof. R. Renner

Problem 4.1 Trace distance

The trace distance between two states given by density matricesρ, σ ∈ S(H) is defined as δ(ρ, σ) = 1

2tr|ρ−σ|. (1)

Alternatively we may write

δ(ρ, σ) = max

P tr(P(ρ−σ)), (2)

where we maximize over all projectorsP onto a subspace ofH.

The following lemma can be used to show the equality of the two definitions:

Lemma 1. Given two quantum states ρ, σ ∈ S(H), there exist two positive operators S, R ∈ P(H) with orthogonal support such that ρ−σ=R−S.

a) Prove Lemma 1.

b) Show that the two definitions (1) and (2) are equal.

Problem 4.2 Trace distance of pure states

Find a simple expression for the trace distance of two pure states δ(|φi,|ψi).

Problem 4.3 Purification

We are given a stateρA∈ S(HA) and a decompositionρA=P

xλxρ^x_Awithλx≥0 andP

xλx= 1.

a) We can always find a decomposition, such that ρ^x_A =|a_xiha_x|_A is pure. Show that |Ψi = P

x

√λ_x|a_xiA⊗ |b_xiB is a purification for any orthonormal basis{ |b_xiB}_x of HB.

*b) ForρA and any purification|ΦiofρA onH_A⊗ H_B, find a POVM{M_B^x}_x onH_B, such that λx= tr(|ΦihΦ|(1^A⊗M_B^x)) and ρ^x_A= trB(|ΦihΦ|(1^A⊗M_B^x))

λx

. (3)

In this picture λx is the probability of measuring x and ρ^x_A is the state after such a measurement.