Factorization of stochastic maps using the Stinespring representations

TL;DR

This paper proves that certain stochastic maps in non-commutative probability spaces can be factorized using Stinespring representations, under specific modular conditions involving anti-unitary operators.

Contribution

It introduces a new factorization result for stochastic maps using Stinespring representations and modular relations with anti-unitary operators.

Findings

Stochastic maps with adjoint admit a factorization.

Factorization relies on the existence of anti-unitary operators satisfying modular relations.

Provides a mathematical framework for analyzing stochastic maps in non-commutative probability.

Abstract

In this work, we investigate the existence of a factorization for a unital completely positive map, between non-commutative probability space which do not change the expectation values of the events. These maps are called in literature stochastic maps. Using the Stinespring representations of completely positive map and assuming the existence of anti-unitary operator on Hilbert space related to these representations which satisfying some modular relations, we prove that stochastic maps with adjoint, admits a factorization.