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Criteria for Davies Irreducibility of Markovian Quantum Dynamics

Yikang Zhang, Thomas Barthel

Abstract

The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e., the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e., they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators $L_a$ and the operator $K=iH+\sum_a L^\dagger_aL_a$, involving the Hamiltonian $H$, is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.

Criteria for Davies Irreducibility of Markovian Quantum Dynamics

Abstract

The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e., the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e., they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators and the operator , involving the Hamiltonian , is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.
Paper Structure (10 sections, 16 theorems, 50 equations)

This paper contains 10 sections, 16 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Motivation: Evans theorem and irreducibility
  3. Davies irreducibility and Evans irreducibility
  4. A classical criterion for Davies irreducibility
  5. Algebraic criteria for Davies irreducibility
  6. Examples for Davies reducible and irreducible two-level systems
  7. Examples for Davies irreducible many-body systems I
  8. Examples for Davies irreducible many-body systems II
  9. Dissipative stabilization of pure states
  10. Discussion

Key Result

Theorem 2.1

For a Markovian system eq:QME, a projection operator $\hat{P}=\hat{P}^\dag=\hat{P}^2$ is conserved if and only if $\hat{P}$ is an element of the commutant $\{\hat{L}_\alpha, \hat{L}_\alpha^\dag,\hat{H}\}'$.

Theorems & Definitions (46)

  • Theorem 2.1: Evans theorem
  • proof
  • Example 2.1: Two-site ferromagnet
  • Theorem 2.2: Frigerio's first theorem
  • Definition 3.1: Evans irreducibility and Davies irreducibility
  • Proposition 3.1: Evans-reducibility in the Schrödinger picture
  • proof
  • Proposition 3.2: Implications of Davies and Evans reducibility
  • proof
  • Theorem 3.3: Davies irreducibility and unique steady states
  • ...and 36 more