The Exercise

Alice chooses a value $x$ with probability $p_x$ and prepares a quantum system $Q$ in the state $\rho_x$; Bob receives $Q$ and measures it. With $X$ for Alice’s choice and $Y$ for Bob’s outcome, the mutual information $H(X:Y)$ is what the measurement tells Bob about $X$. The Holevo bound limits this information by the Holevo quantity of Alice’s ensemble.

The problem asks for three properties of the Holevo quantity along with the bound.

States and Entropy

A quantum system $Q$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $A \otimes B$ of two systems $A$ and $B$ is the system whose state space is the tensor product of their state spaces^✦, and the qudit with $d$ levels is the system whose state space is $\mathbb{C}^d$.^✦

A state $\rho$ of $Q$ is a density operator on the state space of $Q$, a positive operator of unit trace.^✦ A pure state $\omega$ of $Q$ is a unit vector in the state space of $Q$^✦; it defines the state $|\omega\rangle\langle\omega|$, the projector onto that vector.^✦ For a level $y$ of the qudit, the computational basis state $|y\rangle$ is the pure state whose vector is the $y$th standard basis vector of $\mathbb{C}^d$.^✦

The product state $\rho \otimes \sigma$ of a state $\rho$ of $A$ and a state $\sigma$ of $B$ is the state of $A \otimes B$ whose density operator is the tensor product of the two.^✦ The partial trace $\operatorname{Tr}_B \rho$ of a state $\rho$ of $A \otimes B$ is the reduced state of $A$ obtained by tracing out $B$.^✦

An ensemble $\{p_x, \rho_x\}$ of states of $Q$ is a probability distribution $p_x$ on a finite set, with a state $\rho_x$ of $Q$ for each $x$. The average state $\rho$ of an ensemble $\{p_x, \rho_x\}$ of states of $Q$ is the mixture, the state of $Q$ with density operator $$\rho = \sum_x p_x \rho_x .$$(1)^✦ The von Neumann entropy of a state $\rho$ is $S(\rho) = -\operatorname{tr}(\rho \log \rho)$, the Shannon entropy of its eigenvalues, measured in nats.^✦ The Holevo quantity $\chi(\{p_x, \rho_x\})$ of an ensemble $\{p_x, \rho_x\}$ of states of $Q$ is the real number $$\chi(\{p_x, \rho_x\}) = S\Bigl(\sum_x p_x \rho_x\Bigr) - \sum_x p_x\, S(\rho_x).$$(2)^✦

Operations and Measurements

A unitary $U$ of $Q$ is an operator on the state space of $Q$ with $U^\dagger U = U U^\dagger = 1$^✦; under it a state $\sigma$ of $Q$ evolves into $U \sigma U^\dagger$.^✦ For a second system $C$, the ancilla, a unitary $U$ of $Q \otimes C$ and a pure state $\omega$ of $C$, the dilated channel $\mathcal{E}_{U,\omega}$ on $Q$ sends a state $\rho$ of $Q$ to $$\mathcal{E}_{U,\omega}(\rho) = \operatorname{Tr}_C\bigl(U (\rho \otimes |\omega\rangle\langle\omega|) U^\dagger\bigr).$$(3)^✦

A measurement on $Q$ with a finite set of outcomes is a family of measurement operators $M_y$ on the state space of $Q$, one for each outcome $y$, with $\sum_y M_y^\dagger M_y = 1$.^✦ Its positive operator valued measure (POVM) is the family of effects $E_y = M_y^\dagger M_y$^✦, and the probability of the outcome $y$ on a state $\rho$ of $Q$ is given by the Born rule, $$p(y \mid \rho) = \operatorname{Tr}(M_y \rho M_y^\dagger) = \operatorname{Tr}(E_y \rho).$$(4)^✦

With the $d$ outcomes numbered $0, \dots, d-1$, the outcome state of $\rho$ is the state of the qudit with $d$ levels that records the outcome: $$\mathcal{M}(\rho) = \sum_y p(y \mid \rho)\, |y\rangle\langle y| .$$(5)^✦

For an ensemble $\{p_x, \rho_x\}$ of states of $Q$, the measurement gives $X$ and $Y$ the joint distribution $$p(x, y) = p_x\, p(y \mid \rho_x),$$(6) and the mutual information $H(X:Y)$ of $X$ and $Y$ is the mutual information of this joint distribution.

The Claim

What Is Checked

The four statements are about the Holevo quantity (2) of an ensemble, built on the mixture (1) and the von Neumann entropy. The first is stated over the dilated channel (3) and the third over the partial trace; the second and fourth are stated over a measurement, through its Born probabilities (4), its outcome state (5) and the joint distribution (6).

About these constructions the library proves that the Born probabilities of a measurement on a state are nonnegative^✦ and sum to one^✦, so that the outcome state is a mixture of the basis states $|y\rangle\langle y|$, and that the effects $E_y$ are positive with sum the identity. It proves that a mixture of states, the product of two states, the projector onto a unit vector, the evolved state $U \sigma U^\dagger$ and a partial trace are density operators, so the dilated channel sends states of $Q$ to states of $Q$.