The Exercise

Two quantum systems $A$ and $B$ are prepared together by a source that draws a label $i$ from a finite set $\iota$ with probability $p_i$ and then prepares the pair in the state $\rho_i$. An observer who is not told the label describes the pair by the mixture of the $\rho_i$ with weights $p_i$. The exercise asks to show that the conditional entropy $S(A|B)$ is a concave function of the joint state of $A$ and $B$.^✦

A Quantum System

A quantum system $S$ is a physical object described by its state space $\mathcal{H}_S$, a finite dimensional complex Hilbert space.^✦ The composite system $A \otimes B$ of two quantum systems $A$ and $B$ is the quantum system whose state space is the tensor product of theirs, $$\mathcal{H}_{A \otimes B} = \mathcal{H}_A \otimes \mathcal{H}_B .$$(1)^✦

A state $\rho$ of $S$ is a positive operator of unit trace on $\mathcal{H}_S$.^✦ The mixture $\bar\rho$ of a family of states $\rho_i$ of $S$, indexed by the elements $i$ of a finite set $\iota$, with weights $p_i$ that are nonnegative and sum to one, is the state of $S$ whose density operator is the convex combination $$\bar\rho = \sum_{i \in \iota} p_i \, \rho_i .$$(2)^✦

Entropies

The von Neumann entropy $S(\rho)$ of a state $\rho$ of $S$ is the Shannon entropy of the eigenvalues of $\rho$, taken with the natural logarithm and so measured in nats: $$S(\rho) = -\operatorname{tr}(\rho \log \rho) .$$(3)^✦

The reduced state $\rho_B$ of a state $\rho$ of $A \otimes B$ is the state of $B$ obtained by tracing out $A$.^✦ In an orthonormal basis $\{|j\rangle\}$ of $\mathcal{H}_A$, with $I$ the identity operator on $\mathcal{H}_B$, $$\rho_B = \operatorname{tr}_A \rho = \sum_j \bigl(\langle j| \otimes I\bigr)\, \rho \,\bigl(|j\rangle \otimes I\bigr) .$$(4)^✦

The conditional entropy $S(A|B)_\rho$ of a state $\rho$ of $A \otimes B$ is the von Neumann entropy of $\rho$ minus that of its reduced state on $B$, $$S(A|B)_\rho = S(\rho) - S(\rho_B) .$$(5)^✦

The Claim

What Is Checked

The claim (6) is stated over states of the composite system (1), their mixture (2), and the conditional entropy (5), built from the von Neumann entropy (3) of the joint state and of its reduced state (4) on $B$.