The Exercise

Two quantum systems $A$ and $B$ are prepared apart from each other, $A$ in a state $\rho$ and $B$ in a state $\sigma$, so that the pair carries no correlation and is described by the tensor product $\rho \otimes \sigma$ of the two density operators. The exercise asks how the von Neumann entropy of the pair is related to the entropies of its two parts.

Systems and States

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

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, acting as $\rho$ on the first factor and as $\sigma$ on the second: $$(\rho \otimes \sigma)\bigl(|a\rangle \otimes |b\rangle\bigr) = \rho|a\rangle \otimes \sigma|b\rangle , \qquad |a\rangle \in \mathcal{H}_A ,\ |b\rangle \in \mathcal{H}_B .$$(2)^✦

The Entropy of a State

The von Neumann entropy of a state $\rho$ of $A$ is the real number $$S(\rho) = -\operatorname{tr}(\rho \ln \rho) = -\sum_i \lambda_i \ln \lambda_i ,$$(3)^✦ the Shannon entropy of the nonzero eigenvalues $\lambda_i$ of $\rho$. The logarithm is natural, so entropy is measured in nats.

The Claim

What Is Checked

The claim is stated over any two quantum systems $A$ and $B$ and any states $\rho$ of $A$ and $\sigma$ of $B$.

About these constructions the library proves that the tensor product of two density operators is again positive with unit trace, so that $\rho \otimes \sigma$ is a state of $A \otimes B$.