The decoherence-free subalgebra of Gaussian Quantum Markov Semigroups

TL;DR

This paper presents a method to identify the decoherence-free subalgebra of Gaussian quantum Markov semigroups, revealing its structure as a type I von Neumann algebra characterized by specific dimensions.

Contribution

It introduces a novel approach to determine the decoherence-free subalgebra in Gaussian quantum Markov semigroups and characterizes its algebraic structure explicitly.

Findings

The decoherence-free subalgebra is a type I von Neumann algebra.

The structure is determined by two natural numbers, $d_c$ and $d_f$.

Applications and examples illustrate the theoretical results.

Abstract

We demonstrate a method for finding the decoherence-subalgebra $\mathcal{N}(\mathcal{T})$ of a Gaussian quantum Markov semigroup on the von Neumann algebra $\mathcal{B}(\Gamma(\mathbb{C}^d))$ of all bounded operator on the Fock space $\Gamma(\mathbb{C}^d)$ on $\mathbb{C}^d$. We show that $\mathcal{N}(\mathcal{T})$ is a type I von Neumann algebra $L^\infty(\mathbb{R}^{d_c};\mathbb{C})\bar{\otimes}\mathcal{B}(\Gamma(\mathbb{C}^{d_f}))$ determined, up to unitary equivalence, by two natural numbers $d_c,d_f\leq d$. This result is illustrated by some applications and examples.