The Exercise

A source prepares a quantum system in one of finitely many states $\rho_i$, the state $\rho_i$ with probability $p_i$, and forgets which. The system is then in the mixture of the $\rho_i$. Compare it with a fixed state $\sigma$ of the same system. The exercise is to show that the mixture lies no farther from $\sigma$, in trace distance, than the states $\rho_i$ lie on average, with the weights $p_i$: the trace distance is convex in its first argument.^✦ The same bound holds in the second argument, for a fixed state $\rho$ and a mixture of states $\sigma_i$.^✦

A Quantum System

A quantum system $S$ is described 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 $S$, also called a density operator.^✦

Mixtures and Trace Distance

The mixture of a family of states $\rho_i$ of $S$, indexed by a finite set, with weights $p_i$ that are nonnegative and sum to one, is the state of $S$ whose density operator is the convex combination $$\sum_i p_i \rho_i .$$(1)^✦

The trace distance $D(\rho,\sigma)$ between two states $\rho$ and $\sigma$ of $S$ is half the trace norm of $\rho - \sigma$, the sum of the absolute values of its eigenvalues: $$D(\rho,\sigma) = \tfrac{1}{2} \operatorname{tr} \lvert \rho - \sigma \rvert .$$(2)^✦

The Claim

What Is Checked

The claim is stated over the mixture (1) of finitely many states of a quantum system $S$ and the trace distance (2) between two states of $S$.

The library proves that the mixture is a state: the convex combination (1) of density operators is again a positive operator of unit trace.