The Exercise

Two sources prepare copies of a quantum system $S$. Each draws a label $i$ from the same finite set $I$: the first draws $i$ with probability $p_i$ and emits $S$ in the state $\rho_i$, the second draws $i$ with probability $q_i$ and emits $S$ in the state $\sigma_i$. To a receiver who is not told the label, the first source emits a mixture of the states $\rho_i$ with weights $p_i$, and the second a mixture of the $\sigma_i$ with weights $q_i$. Theorem 9.7 compares the fidelity $F$, the overlap measure between two states, of these two mixtures with the fidelities of the pairs $\rho_i$, $\sigma_i$ prepared under the same label.

States, Mixtures and Fidelity

A quantum system $S$ is a physical system whose state space $\mathcal{H}_S$ is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on $\mathcal{H}_S$.^✦

The mixture of a family of states $\rho_i$ of $S$, indexed by a finite set $I$, with weights $p_i \ge 0$ summing to one, is the state of $S$ with density operator $$\sum_{i \in I} p_i \rho_i .$$(1)^✦

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)^✦ where each square root is the positive square root of a positive operator on $\mathcal{H}_S$, obtained by taking the square root of each of its eigenvalues.

The Claim

The theorem is the strong concavity of the fidelity: the fidelity of two mixtures is bounded below by a sum over the pairs of the fidelities $F(\rho_i, \sigma_i)$, weighted by $\sqrt{p_i q_i}$.^✦

What Is Checked

The claim is stated over the states of a quantum system $S$, their fidelity (2), and the mixtures (1) of two families of states indexed by one finite set. About these constructions the library proves that a convex combination of density operators, with nonnegative weights summing to one, is again positive with unit trace, so that each mixture is a state of $S$.