The Exercise

Two states $\rho$ and $\sigma$ of the same quantum system are given, and the question is whether $\rho$ can be turned into $\sigma$ by local operations and classical communication (LOCC). Theorem 12.15 answers it from the eigenvalues of the two states alone.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space, of dimension $d$.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, a density operator, pure or mixed.^✦

The spectrum of a state $\rho$ of $S$ is the list $\lambda_1(\rho), \dots, \lambda_d(\rho)$ of the eigenvalues of $\rho$, each repeated according to its multiplicity. Its entries are nonnegative real numbers adding up to one.

Measurement and Conversion

A measurement $\{M_j\}$ on $S$ with finitely many outcomes $j$ is a family of operators $M_j$ on the state space of $S$, one for each outcome, satisfying the completeness relation, in which $M_j^\dagger$ is the adjoint of $M_j$ and $I$ the identity operator on the state space of $S$, $$\sum_j M_j^\dagger M_j = I.$$(1)^✦

A state $\rho$ of $S$ is convertible by LOCC to a state $\sigma$ of $S$, written $\rho \rightsquigarrow \sigma$, when there are a measurement $\{M_j\}$ on $S$ and probabilities $p_j \ge 0$ on its outcomes, with $\sum_j p_j = 1$, such that $$M_j \rho M_j^\dagger = p_j \, \sigma \qquad \text{for every outcome } j.$$(2)^✦

Majorization

Write $\lambda^{\downarrow}_1(\rho) \ge \dots \ge \lambda^{\downarrow}_d(\rho)$ for the spectrum of $\rho$ arranged in decreasing order. A state $\rho$ of $S$ is majorized by a state $\sigma$ of $S$, written $\rho \prec \sigma$, when the spectrum of $\rho$ is majorized by the spectrum of $\sigma$: $$\sum_{i=1}^{k} \lambda^{\downarrow}_i(\rho) \le \sum_{i=1}^{k} \lambda^{\downarrow}_i(\sigma) \qquad \text{for } k = 1, \dots, d.$$(3)^✦

For a qubit, with eigenvalues $\lambda \ge 1 - \lambda$ for $\rho$ and $\mu \ge 1 - \mu$ for $\sigma$, the condition is $\lambda \le \mu$. In any dimension the maximally mixed state, all of whose eigenvalues equal $1/d$, is majorized by every state, and every state is majorized by a pure state, a state of rank one.

The Claim

What Is Checked

The claim is stated over a quantum system $S$ and two of its states $\rho$ and $\sigma$, and over two relations between them: conversion by a single measurement with finitely many outcomes, (1) and (2), and majorization of their spectra, (3).

About these constructions the library proves that the density operator of every state is self adjoint.