Lean, informalized

Nielsen & Chuang · Chapter 9, Distance measures for quantum information · p. 423

Problem 9.3 - A lower bound on the entanglement fidelity of a state from how well the pure states of its support survive

Fidelity on the support bounds the entanglement fidelity.

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The Exercise

A quantum system QQ in a mixed state ρ\rho passes through a noisy process ℰ\mathcal{E}, a channel or a memory that acts on QQ alone. The process is known to be good on pure states, though only on those in the support of ρ\rho: each such |ψ⟩|\psi\rangle comes out as a state whose overlap with |ψ⟩|\psi\rangle is at least 1−η1-\eta. The problem asks what this guarantees for ρ\rho itself, entanglement with other systems included: a lower bound on the entanglement fidelity of ρ\rho under ℰ\mathcal{E} in terms of η\eta.

A State and an Operation

The state space of QQ is a finite-dimensional complex Hilbert space HH. The state ρ\rho of QQ is a density operator on HH given by an orthonormal basis {|i⟩}\{|i\rangle\} of HH and probabilities pi≥0p_i \ge 0 with ∑ipi=1\sum_i p_i = 1: ρ=∑ipi|i⟩⟨i|.\rho = \sum_i p_i \, |i\rangle\langle i| .(1) The support of ρ\rho is the subspace of HH spanned by the |i⟩|i\rangle with pi>0p_i > 0.

A trace-preserving quantum operation ℰ\mathcal{E} on QQ is given by a finite set of operation elements EjE_j, linear operators on HH that satisfy the completeness relation; it takes an operator σ\sigma on HH to ℰ(σ)=∑jEjσEj†,∑jEj†Ej=I.\mathcal{E}(\sigma) = \sum_j E_j \sigma E_j^\dagger , \qquad \sum_j E_j^\dagger E_j = I .(2)

The entanglement fidelity F(ρ,ℰ)F(\rho,\mathcal{E}) of ρ\rho under ℰ\mathcal{E} is the number that measures how well the entanglement between QQ and a reference system purifying ρ\rho survives when ℰ\mathcal{E} acts on QQ.

The Claim

Theorem 1. Let ρ\rho be a state of QQ as in (1), let ℰ\mathcal{E} be a trace-preserving quantum operation on QQ with operation elements EjE_j as in (2), and let η\eta be a real number. Suppose that every pure state |ψ⟩|\psi\rangle in the support of ρ\rho satisfies ⟨ψ|ℰ(|ψ⟩⟨ψ|)|ψ⟩≥1−η.\langle\psi|\,\mathcal{E}\bigl(|\psi\rangle\langle\psi|\bigr)\,|\psi\rangle \ge 1-\eta .(3) Then the entanglement fidelity of ρ\rho under ℰ\mathcal{E} satisfies F(ρ,ℰ)≥1−3η2.F(\rho,\mathcal{E}) \ge 1 - \frac{3\eta}{2} .(4)✦

What Is Checked

The claim is stated over every state of a finite-dimensional system written in an orthonormal basis as in (1), every trace-preserving operation given by finitely many operation elements as in (2), and every real number η\eta. About these constructions the library proves Theorem 1.