Theory of Quantum Information with Memory

9 Channel output entropies

In the following, let ${\mathbb{H}}_{\mathit{A}}$, ${\mathbb{H}}_{\mathit{B}}$ and ${\mathbb{H}}_{\mathit{E}}$ be separable complex Hilbert spaces that represent correspondingly input system A, output system B of a quantum channel $\mathrm{\Phi }:\mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{A}}\mathrm{)}\to \mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{B}}\mathrm{)}$ and environment system E that is interacting with A or B.

The purpose of this chapter is to explore various properties of the output entropy ${\mathit{H}}_{\mathrm{\Phi }}\mathrm{(}·\mathrm{)}:=\mathit{H}\mathrm{(}\mathrm{\Phi }\mathrm{(}·\mathrm{)}\mathrm{)}:\mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{A}}\mathrm{)}\to [0\mathrm{,}+\infty ]$ of the map $\mathrm{\Phi }\mathrm{(}\mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{A}}\mathrm{)}\mathrm{)}\subseteq \mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{B}}\mathrm{)}$, where $\mathrm{\Phi }:\mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{A}}\mathrm{)}\to \mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{B}}\mathrm{)}$ is a quantum channel. The properties of the output von Neumann entropy ${\mathit{H}}_{\mathrm{\Phi }}\mathrm{(}·\mathrm{)}$ of a quantum channel Φ reveal an important characteristic of this channel used in the study of its information properties.

9.1 Monotonicity of output relative entropy and Petz’s theorem

The main results in this section consist of the monotonicity of channel ...