The Exercise

A quantum system with a finite dimensional state space $\mathcal{H}$ is prepared in a state $\rho$ and passed through a noisy device. With probability $p$ the device acts as a quantum operation $\mathcal{E}_1$, and otherwise as a second operation $\mathcal{E}_2$. Exercise 12.13 asks to show that the entropy exchange is concave in the operation: the entropy exchange of $\rho$ under the device is at least the average, with weights $p$ and $1-p$, of its entropy exchanges under $\mathcal{E}_1$ and under $\mathcal{E}_2$.^✦

A Quantum System

A density operator $\rho$ on $\mathcal{H}$ is a positive operator on $\mathcal{H}$ of trace one; the density operators are the states of the system.^✦

A quantum operation $\mathcal{E}$ on $\mathcal{H}$ is given by its operation elements, operators $E_i$ on $\mathcal{H}$ indexed by a finite set, and it is trace preserving when $$\sum_i E_i^\dagger E_i = I .$$(1)

For a weight $p$ and two operations on $\mathcal{H}$, $\mathcal{E}_1$ with elements $E_i$ and $\mathcal{E}_2$ with elements $F_k$, the mixture $p\,\mathcal{E}_1 + (1-p)\,\mathcal{E}_2$ is the operation whose elements are the $E_i$ scaled by $\sqrt{p}$ together with the $F_k$ scaled by $\sqrt{1-p}$: $$\bigl\{\sqrt{p}\,E_i\bigr\}_i \,\cup\, \bigl\{\sqrt{1-p}\,F_k\bigr\}_k .$$(2)^✦

Entropy Exchange

The Shannon entropy of a distribution $\lambda$ on a finite set is $$H(\lambda) = -\sum_m \lambda_m \log \lambda_m ,$$(3)^✦ measured in nats. The von Neumann entropy $S(T)$ of a Hermitian operator $T$ on a finite dimensional inner product space of dimension $n$ is the Shannon entropy of its eigenvalues $\lambda_1, \dots, \lambda_n$, counted with multiplicity; for an operator that is not Hermitian it is set to zero.^✦

For an operator $\rho$ on $\mathcal{H}$ and a quantum operation $\mathcal{E}$ with elements $E_i$, the matrix $W$ is the complex square matrix indexed by the operation elements, with entries $$W_{ij} = \operatorname{tr}\bigl(E_i\, \rho\, E_j^\dagger\bigr) .$$(4)^✦ The entropy exchange $S(\rho, \mathcal{E})$ of $\rho$ under $\mathcal{E}$ is the von Neumann entropy of $W$, taken as an operator on the complex Euclidean space with one basis vector for each operation element: $$S(\rho, \mathcal{E}) = S(W) .$$(5)^✦

The Claim

What Is Checked

The claim is stated over a density operator $\rho$ on a finite dimensional Hilbert space, two families of operation elements each satisfying (1), a weight $p$ between $0$ and $1$, the mixture (2), and the entropy exchange (5), built from the matrix (4), the von Neumann entropy and the Shannon entropy (3).

What the library proves about these constructions is the inequality (6), for every such state, pair of trace preserving operations and weight.