The Exercise

Two states $\rho_{AB}$ and $\sigma_{AB}$ of a composite system $AB$ are given. An experimenter with access to $A$ alone sees only the reduced states $\rho_A$ and $\sigma_A$, what remains of the two states once $B$ is discarded. Theorem 11.17 of Nielsen and Chuang compares the relative entropy of this pair with the relative entropy of the pair $\rho_{AB}$, $\sigma_{AB}$ on the whole system.

A Quantum System

A quantum system $A$ is a physical system whose state space $\mathcal{H}_A$ is a finite dimensional complex Hilbert space.^✦ The composite system $AB$ of two quantum systems $A$ and $B$ is the quantum system whose state space is the tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$.^✦

A state $\rho$ of a quantum system $A$ is a positive operator of unit trace on $\mathcal{H}_A$.^✦

The reduced state $\rho_A$ of a state $\rho_{AB}$ of $AB$ is the state of $A$ obtained by tracing out $B$ against an orthonormal basis $\{|j\rangle\}$ of $\mathcal{H}_B$, $$\rho_A = \operatorname{tr}_B(\rho_{AB}) = \sum_j (I_A \otimes \langle j|)\, \rho_{AB}\, (I_A \otimes |j\rangle),$$(1)^✦ where $I_A$ is the identity on $\mathcal{H}_A$.

Relative Entropy

The relative entropy $S(\rho\|\sigma)$ of a state $\rho$ with respect to a state $\sigma$ of the same quantum system is the real number $$S(\rho\|\sigma) = \operatorname{tr}(\rho\log\rho) - \operatorname{tr}(\rho\log\sigma).$$(2)^✦

The Claim

What Is Checked

The claim is stated over two states of the composite system $AB$ of arbitrary quantum systems $A$ and $B$, their reduced states (1) on $A$, and the relative entropy (2) of each pair.

The library proves that the partial trace of a density operator on $\mathcal{H}_A \otimes \mathcal{H}_B$ is a density operator on $\mathcal{H}_A$, so that $\rho_A$ and $\sigma_A$ are states of $A$ and the left side of (3) is the relative entropy of two states.