Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the existence of the KMS spectral gap in Gaussian quantum Markov semigroups

Zheng Li

TL;DR

This work provides a complete criterion for the existence of the KMS spectral gap in Gaussian quantum Markov semigroups, showing that the gap depends only on noise operators and the distribution of inverse temperatures across modes. By reducing the problem to finite-dimensional drift-diffusion matrices, the authors derive explicit KMS and GNS gap matrices and establish that the KMS gap exists if and only if certain kernel intersections $\ker U_n \cap \ker V_n$ vanish for all temperature blocks. They also prove that a GNS spectral gap implies a KMS spectral gap, unifying the two notions under Gaussian dynamics. The results are illustrated with a boson-chain model, demonstrating both existence and non-existence scenarios depending on the inverse-temperature structure, and highlighting the practical relevance for stability and relaxation properties in open quantum systems.

Abstract

In arXiv:2405.04947, it was shown that the GNS spectral gap of a Gaussian quantum Markovian generator is strictly positive if and only if there exists a maximal number of linearly independent noise operators, under the assumption that the generated semigroup admits a unique faithful normal invariant state. In this paper, we provide a necessary and sufficient condition for the existence of the KMS spectral gap, which also depends only on the noise operators of the generator. We further show that the existence of the GNS spectral gap implies the existence of the KMS spectral gap.

On the existence of the KMS spectral gap in Gaussian quantum Markov semigroups

TL;DR

This work provides a complete criterion for the existence of the KMS spectral gap in Gaussian quantum Markov semigroups, showing that the gap depends only on noise operators and the distribution of inverse temperatures across modes. By reducing the problem to finite-dimensional drift-diffusion matrices, the authors derive explicit KMS and GNS gap matrices and establish that the KMS gap exists if and only if certain kernel intersections vanish for all temperature blocks. They also prove that a GNS spectral gap implies a KMS spectral gap, unifying the two notions under Gaussian dynamics. The results are illustrated with a boson-chain model, demonstrating both existence and non-existence scenarios depending on the inverse-temperature structure, and highlighting the practical relevance for stability and relaxation properties in open quantum systems.

Abstract

In arXiv:2405.04947, it was shown that the GNS spectral gap of a Gaussian quantum Markovian generator is strictly positive if and only if there exists a maximal number of linearly independent noise operators, under the assumption that the generated semigroup admits a unique faithful normal invariant state. In this paper, we provide a necessary and sufficient condition for the existence of the KMS spectral gap, which also depends only on the noise operators of the generator. We further show that the existence of the GNS spectral gap implies the existence of the KMS spectral gap.
Paper Structure (10 sections, 15 theorems, 141 equations)

This paper contains 10 sections, 15 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Gaussian Quantum Markov Semigroups
  3. Spectral Gaps as Eigenvalues of Certain Matrices
  4. KMS Embedding
  5. GNS Embedding
  6. Existence of the KMS Spectral Gap
  7. Application to a Boson Chain Model
  8. Matrix Identification of Real-linear Operators
  9. Standardization and Invariance of Spectral Gaps
  10. Drift Matrix of the Dual Gaussian Quantum Markov Semigroups

Key Result

Theorem 2.2

We have where $Z, C$ are real-linear operators on $\mathbb{C}^d$ given by

Theorems & Definitions (34)

  • Remark 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Corollary 3.4
  • ...and 24 more