The Exercise

A device prepares one of finitely many states $\rho_i$ of a quantum system, choosing $\rho_i$ with probability $p_i$ and keeping no record of $i$; what it emits is a mixture of the $\rho_i$. A second device, with the same probabilities, prepares states $\sigma_i$ of the same system. The exercise asks to show that the fidelity between the outputs of the two devices is at least the average, with weights $p_i$, of the fidelities between $\rho_i$ and $\sigma_i$.^✦ This is the joint concavity of the fidelity.

States and Fidelity

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

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)^✦ where the square root of a positive operator $A$, here $\rho$ and the positive operator $\sqrt{\rho}\,\sigma\sqrt{\rho}$, is the positive operator with the same eigenvectors as $A$ and the square roots of its eigenvalues.

The mixture of states $\rho_i$ of $S$, labelled by a finite set of indices $i$, with weights $p_i$ that are nonnegative and sum to one, is the state of $S$, a positive operator of unit trace, given by the convex combination $$\sum_i p_i \rho_i .$$(2)^✦

The Claim

What Is Checked

The claim is stated over the fidelity (1) of states of a finite dimensional system, taken between the pairs $\rho_i$, $\sigma_i$ and between their mixtures (2) with a common set of weights.

About these constructions the library proves that a mixture is again a state: a convex combination of density operators, with nonnegative weights summing to one, is a positive operator of unit trace.