Theory of Quantum Information with Memory

15 Channels with Markovian memory

This chapter constructs and explores classical capacities for channels with ergodic and general Markov memories.

15.1 A brief review on Markov chains

We first review some concepts of a homogeneous discrete time Markov chain with a finite state space from probability theory. The review material presented in this section can be found in the research monograph by Norris [117].

A discrete-time Markov chain is a sequence of random variables ${\{{\mathit{X}}_{\mathit{n}}\}}_{\mathit{n}=0}^{+\infty }$ defined on a probability space $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{\Sigma }\mathrm{,}Pr\mathrm{)}$ and taking its values in a state space $\mathbb{I}$ (which is assumed to be a finite set) with the following Markov property:

$\begin{array}{rcl}Pr\mathrm{(}{\mathit{X}}_{\mathit{n}+1}& =& {\mathit{x}}_{\mathit{n}+1}\mid {\mathit{X}}_{\mathit{n}}={\mathit{x}}_{\mathit{n}}\mathrm{,}\dots \mathrm{,}{\mathit{X}}_1={\mathit{x}}_1\mathrm{,}{\mathit{X}}_0={\mathit{x}}_0\mathrm{)}\\ =& Pr\mathrm{(}{\mathit{X}}_{\mathit{n}+1}={\mathit{x}}_{\mathit{n}+1}\mid {\mathit{X}}_{\mathit{n}}={\mathit{x}}_{\mathit{n}}\mathrm{)}\mathrm{,}\end{array}$

for all $\mathit{n}\ge 0$ and for all ${\mathit{x}}_{\mathit{i}}\in \mathbb{I}\mathrm{,}\mathit{i}=0\mathrm{,}1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{n}$, if both ...