The Exercise

A source prepares a quantum system in the state $\rho_i$ with probability $p_i$; with the outcome $i$ forgotten, it emits the mixed state $\sum_i p_i \rho_i$. How close the output comes to a fixed state $\sigma$ of the same system can be measured in two ways with the fidelity $F$: by averaging the fidelities $F(\rho_i,\sigma)$ of the prepared states, or by the fidelity of the emitted mixture with $\sigma$. The exercise is to show that the average never exceeds the fidelity of the mixture: the fidelity is concave in its first entry.^✦

The library states the inequality for any finite family of states of one quantum system, weighted by any probabilities, and any state $\sigma$ of that system.

A Quantum System

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦

The mixture of states $\rho_i$ of $S$, indexed by $i$ in a finite set, with weights $p_i$ is the state of $S$ whose density operator is the convex combination $$\sum_i p_i \rho_i , \qquad p_i \ge 0 , \quad \sum_i p_i = 1 .$$(1)^✦

The Fidelity

The fidelity of two states $\rho$ and $\sigma$ of $S$ is the nonnegative real number $$F(\rho,\sigma) = \operatorname{tr} \sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}} ,$$(2)^✦ the trace taken over the state space of $S$. Here $\sqrt{\rho}\,\sigma\sqrt{\rho}$ is a positive operator, and each square root is the positive square root of a positive operator, taken by applying the square root to each of its eigenvalues.

The Claim

What Is Checked

The claim is stated over states of one quantum system $S$: the fixed state $\sigma$, the family $\rho_i$ with its weights, their mixture (1) and the fidelity (2). Of these constructions the library proves that the mixture is a state: a convex combination of density operators with nonnegative weights summing to one is again positive with unit trace.