Markov triplets on CCR-algebras

TL;DR

This paper provides a detailed analysis of Markov triplets within CCR-algebras, focusing on quasi-free states, their entropy, and the quantum Markov property, connecting quantum and classical Gaussian cases.

Contribution

It offers explicit computations and characterizations of quantum Markov triplets in CCR-algebras, linking entropy, block matrices, and classical Gaussian Markov properties.

Findings

Characterization of quantum Markov triplets via block matrices

Explicit computation of von Neumann entropy for quasi-free states

Connection between quantum and classical Gaussian Markov triplets

Abstract

The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasi-free states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasi-free state is described by the representing block matrix. The proof is based on results on the statistical sufficiency in the quantum case. The relation to classical Gaussian Markov triplets is also described.