Linking Classical and Quantum Stochastic Processes

TL;DR

This paper introduces a map linking classical stochastic matrices to qubit quantum channels, preserving key properties and providing a framework to relate classical and quantum stochastic processes.

Contribution

It defines a novel spectrum-preserving map between classical and quantum processes using the Bloch tetrahedron, extending the classical-quantum correspondence.

Findings

The map preserves the spectrum of processes.

It maintains the composition of classical and quantum processes.

Potential for generalization to higher dimensions is discussed.

Abstract

We define a map which relates four dimensional classical stochastic matrices to qubit quantum channels. The map preserves the spectrum and the composition of processes. To do this we introduce the concept of Bloch tetrahedron which plays the same role in the classical context as the Bloch sphere in the quantum context. A similar map is also induced between dynamical generators of classical and quantum stochastic processes. Possibilities for generalization to arbitrary dimensions are also discussed.