Lean, informalized

Nielsen & Chuang · Chapter 2, Introduction to quantum mechanics · p. 105

Exercise 2.73 - Minimal ensembles of a density operator, and the weight each of their states must carry

Minimal ensembles for ρ\rho can contain any pure state in its support.

Hover over a blue or green term in the text, or a node in the graph, to find it in the other. Select it to read its declaration both in Lean and in words.

The Exercise

A source that emits the pure state |ψi⟩|\psi_i\rangle with probability pip_i prepares a mixed state ρ\rho, and many different ensembles {pi,|ψi⟩}\{p_i, |\psi_i\rangle\} prepare the same one. Those that use exactly as many pure states as the rank of ρ\rho are the minimal ensembles for ρ\rho. Take a pure state |ψ⟩|\psi\rangle in the support of ρ\rho. The exercise is to show that some minimal ensemble for ρ\rho contains |ψ⟩|\psi\rangle, and that in every such ensemble |ψ⟩|\psi\rangle carries the same probability, p=1/⟨ψ|ρ−1|ψ⟩p = 1/\langle\psi|\rho^{-1}|\psi\rangle, with ρ−1\rho^{-1} the inverse of ρ\rho on its support.

The library states both parts for a state ρ\rho of a quantum system, the second for every member |ψi⟩|\psi_i\rangle of every minimal ensemble for ρ\rho and not only for |ψ⟩|\psi\rangle.

States and Ensembles

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; it determines the projector |ψ⟩⟨ψ||\psi\rangle\langle\psi| onto |ψ⟩|\psi\rangle.

An ensemble {pi,|ψi⟩}i=1n\{p_i, |\psi_i\rangle\}_{i=1}^{n} of SS is a collection of nn pure states |ψi⟩|\psi_i\rangle of SS with weights pip_i that form a probability distribution, pi≥0p_i \geq 0 and ∑ipi=1\sum_i p_i = 1.✦ Its density operator is the mixture ∑i=1npi|ψi⟩⟨ψi|\sum_{i=1}^{n} p_i\, |\psi_i\rangle\langle\psi_i|(1)✦ of the projectors onto its states.

The support of a state ρ\rho of SS is the range of ρ\rho, the subspace of the state space of SS formed by the vectors ρ|φ⟩\rho|\varphi\rangle for |φ⟩|\varphi\rangle in that space.✦ A pure state |ψ⟩|\psi\rangle of SS is in the support of ρ\rho when it belongs to this subspace.✦ The rank of a state ρ\rho of SS is the dimension of its support, the number of nonzero eigenvalues of ρ\rho counted with multiplicity.✦

An ensemble {pi,|ψi⟩}i=1n\{p_i, |\psi_i\rangle\}_{i=1}^{n} of SS is a minimal ensemble for a state ρ\rho of SS when its density operator (1) is ρ\rho and its number of states nn is the rank of ρ\rho.✦

The Inverse on the Support

The pseudoinverse ρ−1\rho^{-1} of a state ρ\rho of SS is the operator on the state space of SS that inverts ρ\rho on its support: for an orthonormal basis of eigenvectors |ek⟩|e_k\rangle of ρ\rho, with ρ|ek⟩=λk|ek⟩\rho|e_k\rangle = \lambda_k|e_k\rangle, ρ−1=∑k:λk≠01λk|ek⟩⟨ek|.\rho^{-1} = \sum_{k \,:\, \lambda_k \neq 0} \frac{1}{\lambda_k}\, |e_k\rangle\langle e_k| .(2)✦ The expectation of the pseudoinverse in a pure state |ψ⟩|\psi\rangle of SS is the real number ⟨ψ|ρ−1|ψ⟩\langle\psi|\rho^{-1}|\psi\rangle.✦

The Claim

Theorem 1. Let ρ\rho be a state of a quantum system SS, of rank rr, and let |ψ⟩|\psi\rangle be a pure state of SS in the support of ρ\rho. Then there is a minimal ensemble {pi,|ψi⟩}i=1r\{p_i, |\psi_i\rangle\}_{i=1}^{r} for ρ\rho with |ψ1⟩=|ψ⟩|\psi_1\rangle = |\psi\rangle: ρ=p1|ψ⟩⟨ψ|+∑i=2rpi|ψi⟩⟨ψi|.\rho = p_1\, |\psi\rangle\langle\psi| + \sum_{i=2}^{r} p_i\, |\psi_i\rangle\langle\psi_i| .(3)✦

Theorem 2. Let ρ\rho be a state of a quantum system SS, and let {pi,|ψi⟩}\{p_i, |\psi_i\rangle\} be a minimal ensemble for ρ\rho. Then for every ii pi=1⟨ψi|ρ−1|ψi⟩.p_i = \frac{1}{\langle\psi_i|\rho^{-1}|\psi_i\rangle} .(4)✦

What Is Checked

Both theorems are stated over the states, pure states and ensembles of a quantum system SS, with minimality defined through the density operator (1) and the rank of ρ\rho, and the weights through the pseudoinverse (2).

The library proves that the projector |ψ⟩⟨ψ||\psi\rangle\langle\psi| onto a pure state is a state of SS, and that the density operator (1) of an ensemble is a state of SS: positive, as a sum of projectors with nonnegative weights, and of unit trace, since each projector has unit trace and the weights sum to one.