The Exercise

Take a quantum system whose state space has finite dimension $d$ , and finitely many unitaries $U_i$ on it. Pick one of them at random, $U_i$ with probability $p_i$ , and conjugate an operator $A$ by it, $A \mapsto U_i A U_i^\dagger$ . The first part of the exercise asks for unitaries and probabilities whose average conjugation replaces every operator $A$ by the multiple of the identity with the same trace. The second part is about the von Neumann entropy: the maximally mixed state, the identity operator $I$ divided by the dimension, has the largest entropy among the states of the system, and it is the only state that does.

The library states the first part for operators on any finite dimensional Hilbert space of nonzero dimension, and the second for the states of a quantum system.

A Quantum System

A quantum system $Q$ is a physical system whose state space is a finite dimensional complex Hilbert space; $d$ denotes its dimension.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$ .^✦ When the state space of $Q$ is not zero, the maximally mixed state of $Q$ is the state whose operator is the identity $I$ divided by the dimension, $I/d$ .^✦

Entropy

The Shannon entropy of a probability distribution $p = (p_i)$ on a finite set of outcomes is $H(p) = -\sum_i p_i \log p_i ,$ (1)^✦ with the natural logarithm, so that entropies are measured in nats. The von Neumann entropy of a Hermitian operator $T$ on a finite dimensional Hilbert space is the Shannon entropy (1) of its eigenvalues, counted with multiplicity.^✦

The von Neumann entropy $S(\rho)$ of a state $\rho$ of $Q$ is that of its density operator.^✦ With $\lambda_1, \dots, \lambda_d$ the eigenvalues of $\rho$ , $S(\rho) = -\sum_{i=1}^{d} \lambda_i \log \lambda_i = -\operatorname{tr}(\rho \log \rho) .$ (2)^✦

The Claim

Theorem 1. Let $E$ be a finite dimensional Hilbert space of dimension $d \ge 1$ . There are finitely many unitaries $U_1, \dots, U_n$ on $E$ and probabilities $p_1, \dots, p_n \ge 0$ with $\sum_i p_i = 1$ such that every operator $A$ on $E$ satisfies $\sum_{i=1}^{n} p_i \, U_i A U_i^\dagger = \frac{\operatorname{tr} A}{d} \, I .$ (3)^✦

Theorem 2. Let $Q$ be a quantum system whose state space has dimension $d \ge 1$ , and let $\rho$ be a state of $Q$ . Then $S(\sigma) \le S(\rho) \ \text{for every state $\sigma$ of $Q$} \quad \Longleftrightarrow \quad \rho = \frac{I}{d} .$ (4)^✦

What Is Checked

The first statement is over unitaries and operators on a finite dimensional Hilbert space; the second is over a quantum system $Q$ , its states, their von Neumann entropy (2) built from the Shannon entropy (1), and the maximally mixed state $I/d$ . The library proves that $I/d$ is a density operator, positive with unit trace, so that the maximally mixed state is a state of $Q$ .