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Trace distance ergodicity for quantum Markov semigroups

Lorenzo Bertini, Alberto De Sole, Gustavo Posta

Abstract

We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.

Trace distance ergodicity for quantum Markov semigroups

Abstract

We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.
Paper Structure (6 sections, 18 theorems, 204 equations, 1 table)

This paper contains 6 sections, 18 theorems, 204 equations, 1 table.

Table of Contents

  1. Introduction
  2. Convergence in trace distance for quantum Markov semigroups
  3. Fermi Ornstein-Uhlenbeck semigroup
  4. Adjoint action of the Casimir element as a Lindblad generator
  5. Bose Ornstein-Uhlenbeck semigroup
  6. Gap of the Lindblad generator for simple Lie algebras

Key Result

Lemma 2.1

Let $(P_t)_{t\ge 0}$ be a QMS on ${\mathcal{A}}$. Given $n\in{\mathbb N}$, $0\le t_1\le \cdots \le t_n$, and $a_1,\cdots, a_n \in {\mathcal{A}}$, set Then $(P_t)_{t\ge 0}$ is reversible with respect to $\sigma$ if and only if for each $n\in{\mathbb N}$, $0\le t_1\le \cdots \le t_n\le T$, and $a_1,\cdots, a_n \in {\mathcal{A}}$,

Theorems & Definitions (32)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 22 more