The Exercise

A source prepares a quantum system $S$ in one of finitely many states $\rho_j$ , choosing $\rho_j$ with probability $p_j$ . Every state it prepares then goes through a process $\mathcal E$ that turns a state of $S$ into a state of $S$ , and one asks how well the process preserves what the source sends. The figure of merit is the ensemble average fidelity $\bar F$ of $\mathcal E$ on the source. The exercise is to show that $\bar F = 1$ exactly when $\mathcal E$ leaves unchanged every state the source emits with nonzero probability.^✦

The library states this for an arbitrary quantum system $S$ and an arbitrary map $\mathcal E$ from states of $S$ to states of $S$ .

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$ ; pure and mixed states are both states in this sense.^✦

The fidelity of two states $\rho$ and $\sigma$ of $S$ is the nonnegative real number $F(\rho,\sigma) = \operatorname{tr}\sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}},$ (1)^✦ the trace taken over the state space of $S$ , where the square root of a positive operator is obtained by taking the square root of each of its eigenvalues.

The Ensemble Average Fidelity

Let $\rho_j$ be states of $S$ with real weights $p_j$ , the index $j$ running over a finite set. The ensemble average fidelity $\bar F$ of a map $\mathcal E$ from states of $S$ to states of $S$ , on the states $\rho_j$ with weights $p_j$ , is the real number $\bar F = \sum_j p_j \, F\bigl(\rho_j, \mathcal E(\rho_j)\bigr)^2 .$ (2)^✦

The Claim

A state of zero weight drops out of (2), so the condition on $\mathcal E$ concerns only the states the source actually emits.

Theorem 1. Let $S$ be a quantum system, let $\rho_j$ be states of $S$ indexed by a finite set, and let $p_j$ be nonnegative real numbers with $\sum_j p_j = 1$ . Let $\mathcal E$ be a map from states of $S$ to states of $S$ . Then $\bar F = 1 \quad\text{if and only if}\quad \mathcal E(\rho_j) = \rho_j \ \text{for every $j$ with $p_j > 0$.}$ (3)^✦

What Is Checked

The claim is stated over the ensemble average fidelity (2) of a map $\mathcal E$ on the states of a quantum system $S$ , for finitely many states carrying probability weights, and through it over the fidelity (1) of each state $\rho_j$ with its image $\mathcal E(\rho_j)$ , itself a state of $S$ .

What the library proves about these constructions is the equivalence (3), in both directions, for every such source and every such map.