Theory of Quantum Information with Memory

16 Channels with long-term memory

Datta and Dolas [31] (see also Dorlas and Morgan [42] and Datta–Suhov–Dorlas [35]) investigated the product state classical capacity as well as entanglement assisted classical capacity of a convex combination of memoryless channels that are the subjects of exposition in this chapter.

16.1 The model

The model obtained in [31], [42] and [35] for quantum channel with long term memory is described below.

Given a collection of M memoryless channels ${\{{\mathrm{\Phi }}_{\mathit{i}}\}}_{\mathit{i}=1}^{\mathit{M}}$ with common input Hilbert space ${\mathbb{H}}_{\mathit{A}}$ and the common output Hilbert space ${\mathbb{H}}_{\mathit{B}}$, a convex combination (or average) of these channels is defined by the map

(16.1)$\begin{array}{c}\mathrm{\Phi }\mathrm{(}{\mathit{\rho }}_{\mathit{A}}\mathrm{)}=\sum _{\mathit{i}=1}^{\mathit{M}}{\mathit{p}}_{\mathit{i}}{\mathrm{\Phi }}_{\mathit{i}}\mathrm{(}{\mathit{\rho }}_{\mathit{A}}\mathrm{)}\mathrm{,}\forall {\mathit{\rho }}_{\mathit{A}}\in \mathcal{S}\mathrm{(}{\mathbb{H}}_{\mathit{A}}\mathrm{)}\mathrm{,}\end{array}$

where ${\{{\mathit{p}}_{\mathit{i}}\}}_{\mathit{i}=1}^{\mathit{M}}$ is probability distribution over choices of memoryless ...