On Marginal Markov Processes of Quantum Quadratic Stochastic Processes

TL;DR

This paper introduces two marginal Markov processes associated with quantum quadratic stochastic processes on von Neumann algebras, proving their uniqueness in determining the original process and exploring ergodic relations between them.

Contribution

It establishes the unique determination of quantum quadratic stochastic processes by their marginal Markov processes and analyzes their ergodic properties.

Findings

Two marginal Markov processes are defined on von Neumann algebras.

The marginal processes uniquely determine the quantum quadratic stochastic process.

Ergodic relations between the marginal processes are established.

Abstract

In the paper it is defined two marginal Markov processes on von Neumann algebras $\cm$ and $\cm\o\cm$, respectively, corresponding to given quantum quadratic stochastic process (q.q.s.p.). It is proved that such marginal processes uniquely determines the q.q.s.p. Moreover, certain ergodic relations between them are established as well.