The Exercise

A channel $\mathcal{E}$ takes the states of a quantum system $Q$ to states of $Q$. A sender picks a label $j$ with probability $p_j$, prepares a state $\rho_j$ and sends it through the channel, so that the receiver holds the output ensemble $\{p_j, \mathcal{E}(\rho_j)\}$. The exercise concerns the Holevo quantity $\chi(\mathcal{E})$ of the channel, taken over all input ensembles, and asks to show that mixed inputs are never needed: the same value comes out when the inputs are pure states, and again when there are at most $d^2$ of them, $d$ being the dimension of the state space of $Q$.

A Quantum System

A quantum system $Q$ is a physical object whose state space is a finite dimensional complex Hilbert space^✦, and the dimension $d$ of $Q$ is the dimension of that space.^✦ A state $\rho$ of $Q$, also called a density operator, is a positive operator of unit trace on the state space of $Q$; it is pure when it is a projector onto a unit vector and mixed otherwise.^✦ A pure state $\lvert\psi\rangle$ of $Q$ is a vector of unit norm, $\lVert\psi\rVert = 1$, in the state space of $Q$, fixed up to a global phase.^✦ A pure state $\lvert\psi\rangle$ of $Q$ determines the state $\lvert\psi\rangle\langle\psi\rvert$ of $Q$, the projector onto that vector.^✦

The von Neumann entropy of a state $\rho$ of $Q$ is $$S(\rho) = -\operatorname{tr}(\rho \log \rho),$$(1)^✦ where the logarithm is the natural one.

Ensembles

An ensemble $\{p_i, \rho_i\}$ of states of $Q$ assigns to each label $i$ of a finite set a state $\rho_i$ of $Q$ and a weight $p_i \ge 0$, with $\sum_i p_i = 1$. The mixture of an ensemble $\{p_i, \rho_i\}$ of states of $Q$ is the state $\bar\rho$ of $Q$ with density operator $$\bar\rho = \sum_i p_i \rho_i ,$$(2)^✦ The Holevo quantity $\chi(\{p_i, \rho_i\})$ of an ensemble $\{p_i, \rho_i\}$ of states of $Q$ is the real number given by the entropy of the mixture less the average entropy of the members, $$\chi(\{p_i, \rho_i\}) = S\Bigl(\sum_i p_i \rho_i\Bigr) - \sum_i p_i\, S(\rho_i).$$(3)^✦

The Channel

A quantum channel $\mathcal{E}$ on $Q$ is a map on the states of $Q$ that is the restriction of a completely positive, trace preserving linear map $\Phi$ on the operators of the state space of $Q$: $$\mathcal{E}(\rho) = \Phi(\rho) \ \text{for every state } \rho, \qquad \operatorname{tr}\Phi(A) = \operatorname{tr} A \ \text{for every operator } A.$$(4)^✦

Each input ensemble $\{p_j, \rho_j\}$ of states of $Q$, of any finite size, yields the output ensemble $\{p_j, \mathcal{E}(\rho_j)\}$ and with it a value of (3); these values form a set of real numbers.^✦ The Holevo quantity $\chi(\mathcal{E})$ of a quantum channel $\mathcal{E}$ on $Q$ is the supremum of this set, $$\chi(\mathcal{E}) = \sup_{\{p_j, \rho_j\}} \chi\bigl(\{p_j, \mathcal{E}(\rho_j)\}\bigr).$$(5)^✦

The Claim

The values of (3) for the outputs of finite ensembles $\{p_j, \psi_j\}$ of pure states form a subset of the values in (5)^✦, and those from ensembles of $n \le d^2$ pure states a subset of that.^✦

What Is Checked

The claim is stated over a quantum channel (4) on a system of dimension $d$ and over its Holevo quantity (5), the supremum over all finite input ensembles.