Lean, informalized

Nielsen & Chuang · Chapter 12, Quantum information theory · p. 597

Exercise 12.33 - One measurement whose outcomes fix the values of all commuting observables at once

Commuting observables have a classical joint distribution.

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The Exercise

A system SS is prepared in a state ρ\rho, and AjA_j, for jj in a finite index set JJ, are observables of SS that commute with one another. Each has an expectation value in ρ\rho. The exercise is to show that, because the observables commute, these expectation values come from a single classical probability distribution.

One measurement on SS suffices. Its outcome ω\omega occurs with probability p(ω)p(\omega) and comes with a pure state |ψω⟩|\psi_\omega\rangle that is an eigenstate of every AjA_j at once, so the outcome fixes a value xj(ω)x_j(\omega) of each observable, its eigenvalue. The expectation value of AjA_j is the mean of xjx_j under pp.

A Quantum System

A quantum system SS is a physical system whose state space is a finite dimensional complex Hilbert space.✦ A state ρ\rho of SS is a positive operator of unit trace on the state space of SS.✦ A pure state |ψ⟩|\psi\rangle of SS is a unit vector in the state space of SS, ‖ψ‖=1\lVert \psi \rVert = 1.✦

An observable AA on SS is a self adjoint operator on the state space of SS, A†=AA^\dagger = A.✦ The expectation value of an observable AA on SS in a state ρ\rho of SS is the real number ⟨A⟩ρ=Tr⁡(ρA).\langle A \rangle_\rho = \operatorname{Tr}(\rho A).(1)✦

Two observables AA and BB on SS commute when their operators commute, [A,B]=AB−BA=0.[A, B] = AB - BA = 0.(2)✦ A pure state |ψ⟩|\psi\rangle of SS is an eigenstate of an observable AA on SS with eigenvalue μ\mu, a real number, when A|ψ⟩=μ|ψ⟩.A|\psi\rangle = \mu\,|\psi\rangle.(3)✦

Measurements

A measurement on SS with a finite set of outcomes Ω\Omega is a family of measurement operators MωM_\omega on the state space of SS, one for each outcome ω∈Ω\omega \in \Omega, that satisfies the completeness relation ∑ω∈ΩMω†Mω=I.\sum_{\omega \in \Omega} M_\omega^\dagger M_\omega = I.(4)✦ The positive operator valued measure (POVM) of the measurement assigns to the outcome ω\omega the effect Eω=Mω†MωE_\omega = M_\omega^\dagger M_\omega.✦ The probability p(ω)p(\omega) of the outcome ω\omega, when the measurement is performed on a state ρ\rho of SS, is given by the Born rule, p(ω)=Tr⁡(MωρMω†)=Tr⁡(Eωρ).p(\omega) = \operatorname{Tr}(M_\omega \rho M_\omega^\dagger) = \operatorname{Tr}(E_\omega \rho).(5)✦

The Claim

Theorem 1. Let SS be a quantum system in a state ρ\rho, and let AjA_j, j∈Jj \in J, be a finite family of observables on SS that commute pairwise, [Ai,Aj]=0[A_i, A_j] = 0 for all i,j∈Ji, j \in J. Then there are a finite set of outcomes Ω\Omega, a measurement on SS with outcomes in Ω\Omega, real numbers xj(ω)x_j(\omega) and pure states |ψω⟩|\psi_\omega\rangle of SS such that the Born probabilities p(ω)p(\omega) of the measurement in the state ρ\rho satisfy, for every j∈Jj \in J and ω∈Ω\omega \in \Omega, p(ω)≥0,∑ω∈Ωp(ω)=1,Aj|ψω⟩=xj(ω)|ψω⟩,∑ω∈Ωp(ω)xj(ω)=⟨Aj⟩ρ.\begin{aligned} &p(\omega) \ge 0, \qquad \sum_{\omega \in \Omega} p(\omega) = 1, \qquad A_j|\psi_\omega\rangle = x_j(\omega)\,|\psi_\omega\rangle, \\ &\sum_{\omega \in \Omega} p(\omega)\, x_j(\omega) = \langle A_j \rangle_\rho . \end{aligned}(6)✦

What Is Checked

The claim rests on the notions of commuting observables (2), eigenstate (3), expectation value (1), completeness relation (4) and Born probabilities (5).