The Exercise

Alice has a finite collection of states $\rho_x$ of one quantum system $Q$. She draws the label $x$ at random with probability $p_x$, prepares $\rho_x$ and hands the system to Bob, who knows the states and the probabilities but not the label drawn. Bob performs a measurement on $Q$ and reads an outcome $y$. With $X$ the label and $Y$ the outcome, the mutual information $H(X{:}Y)$ measures what Bob has learned about Alice’s choice. Theorem 12.1 bounds it, for every measurement Bob may perform, by a quantity built from the states and the probabilities alone.

A Quantum System

A quantum system $Q$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on the state space of $Q$.^✦

The mixture of states $\rho_x$ of $Q$, indexed by $x$ in a finite set, with weights $p_x$ that are nonnegative and sum to one, is the state $\rho$ of $Q$ whose density operator is the convex combination $$\rho = \sum_x p_x \rho_x .$$(1)^✦ It is the state Bob assigns to the system he receives, not knowing $x$.

The von Neumann entropy of a state $\rho$ of $Q$ is $$S(\rho) = -\operatorname{tr}(\rho \log \rho) ,$$(2)^✦ the Shannon entropy of the eigenvalues of $\rho$, measured in nats.

The Measurement

A measurement on $Q$ with outcomes $y = 1, \ldots, r$ is a family of measurement operators $M_y$ on the state space of $Q$, one for each outcome, that satisfy the completeness relation $$\sum_y M_y^\dagger M_y = I .$$(3)^✦ The positive operators $E_y = M_y^\dagger M_y$, one for each outcome $y$, are the effects of the measurement; together they form its positive operator-valued measure (POVM). When the measurement is performed on a state $\rho$ of $Q$, the outcome $y$ occurs with the probability given by the Born rule, $$p(y) = \operatorname{tr}(M_y \rho M_y^\dagger) = \operatorname{tr}(E_y \rho) .$$(4)^✦

The Claim

When Bob performs the measurement on the system Alice prepared, the label $x$ and the outcome $y$ occur together with probability $$p(x, y) = p_x \operatorname{tr}(E_y \rho_x) .$$(5)

What Is Checked

The claim is stated over a quantum system $Q$, states of $Q$ with their mixture (1) and their von Neumann entropy (2), and a measurement on $Q$ whose outcome probabilities are the Born probabilities (4) of its POVM.

About these constructions the library shows that a mixture of states with nonnegative weights summing to one is again a state, positive with unit trace, and that the effects $E_y = M_y^\dagger M_y$ of a measurement are positive and sum to the identity by the completeness relation (3). For every state, the Born probabilities (4) therefore form a probability distribution over the outcomes.