Quantum Markov Process on a Lattice

TL;DR

This paper develops a framework for quantum Markov processes on lattice phase spaces, introducing novel operators and extending the space to define positive distribution functions for quantum state evolution.

Contribution

It presents a systematic approach to defining Weyl and Fano operators on lattice phase space, including the introduction of ghost variables for symmetric treatment of lattices.

Findings

Defined a Wigner function on lattice phase space.

Extended the space with a dichotomic variable to obtain positive distributions.

Established the existence of a quantum Markov process on the extended space.

Abstract

We develop a systematic description of Weyl and Fano operators on a lattice phase space. Introducing the so-called ghost variable even on an odd lattice, odd and even lattices can be treated in a symmetric way. The Wigner function is defined using these operators on the quantum phase space, which can be interpreted as a spin phase space. If we extend the space with a dichotomic variable, a positive distribution function can be defined on the new space. It is shown that there exits a quantum Markov process on the extended space which describes the time evolution of the distribution function.