The Exercise

A source prepares a quantum system in the state $\rho$ with probability $p$ and in the state $\sigma$ otherwise. Someone who receives the system without being told which state was prepared assigns it the mixture of the two, and the exercise is to show that the von Neumann entropy is concave: the entropy of the mixture is at least the average of the two entropies, weighted by $p$ and $1-p$. The route goes through the entropy of the mixture as a function of the weight.

A Quantum System

A quantum system $Q$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$.^✦

The von Neumann entropy $S(\rho)$ of a state $\rho$ of $Q$ is the real number $$S(\rho) = -\operatorname{tr}(\rho \log \rho),$$(1)^✦ the Shannon entropy of the eigenvalues of $\rho$, measured in nats.

Mixtures and Their Entropy

For a real number $p$ with $0 \le p \le 1$, the mixture of two states $\rho$ and $\sigma$ of $Q$ with weight $p$ is the state of $Q$ given by the convex combination $$p\rho + (1-p)\sigma .$$(2)^✦

The entropy along the line through $\rho$ and $\sigma$ is the function $f$ that assigns to each real number $p$ the von Neumann entropy of the operator $p\rho + (1-p)\sigma$, $$f(p) = S\bigl(p\rho + (1-p)\sigma\bigr).$$(3)^✦ Between $0$ and $1$ it is the entropy of the mixture with weight $p$, with $f(1) = S(\rho)$ and $f(0) = S(\sigma)$.

The Claim

A state $\rho$ of $Q$ is strictly positive when every eigenvalue of $\rho$ is greater than zero.

What Is Checked

The three claims are stated over the states of a quantum system $Q$, their von Neumann entropy (1), the mixture (2) of two of them and the entropy $f$ (3) along the line through them.