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Subspace Stabilization Analysis for Non-Markovian Open Quantum Systems

Shikun Zhang, Kun Liu, Daoyi Dong, Xiaoxue Feng, Feng Pan

TL;DR

The paper addresses stabilizing quantum information in non-Markovian open systems by deriving necessary and sufficient conditions for subspace invariance under a time-convolution master equation with memory kernel $\gamma$, and, under a commutativity condition, sufficient conditions for subspace attractivity via a double-integral Lyapunov functional. The approach combines algebraic block-decomposition criteria for invariance with Lyapunov analysis of integro-differential dynamics to establish asymptotic convergence to a target subspace, validated by a three-level numerical example. The main contributions extend subspace stabilization theory from Markovian (Lindblad) to non-Markovian regimes, providing practical invariance tests and an attractivity criterion rooted in $L^1$ kernel properties and Barbalat-type arguments. This work enables robust initialization and decoherence-free operation for quantum information processing devices where memory effects are significant. The results lay groundwork for broader non-Markovian quantum control, with future directions including additional models and deeper dynamical-property studies.

Abstract

Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian H, coupling operator L, and memory kernel function γ, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.

Subspace Stabilization Analysis for Non-Markovian Open Quantum Systems

TL;DR

The paper addresses stabilizing quantum information in non-Markovian open systems by deriving necessary and sufficient conditions for subspace invariance under a time-convolution master equation with memory kernel , and, under a commutativity condition, sufficient conditions for subspace attractivity via a double-integral Lyapunov functional. The approach combines algebraic block-decomposition criteria for invariance with Lyapunov analysis of integro-differential dynamics to establish asymptotic convergence to a target subspace, validated by a three-level numerical example. The main contributions extend subspace stabilization theory from Markovian (Lindblad) to non-Markovian regimes, providing practical invariance tests and an attractivity criterion rooted in kernel properties and Barbalat-type arguments. This work enables robust initialization and decoherence-free operation for quantum information processing devices where memory effects are significant. The results lay groundwork for broader non-Markovian quantum control, with future directions including additional models and deeper dynamical-property studies.

Abstract

Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian H, coupling operator L, and memory kernel function γ, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.

Paper Structure

This paper contains 6 sections, 4 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-Markovian System Model
  3. Subspace Invariance
  4. Subspace Attractivity
  5. Numerical Example and Simulation
  6. Conclusion and Future Work

Key Result

Theorem 1

The following conditions (i),(ii),(iii) are necessary and sufficient for $\mathcal{H}_S$ to be an invariant subspace. (i) (ii) (iii) Denote by $\rho_S(t;\rho_S^0)$ the trajectory, with initial value $\rho_S^0$, which satisfies the following integro-differential equation. where Then, $\forall \rho_S^0 \in \mathcal{D}(\mathcal{H}_S)$,

Figures (1)

  • Figure 1: Four different initial values for $\text{tr}(\rho_R)$: 1, 0.75, 0.5, 0.25 are chosen. Simulation results show that $\text{tr}(\rho_R)$ vanishes as time elapses, demonstrating subspace attractivity.

Theorems & Definitions (11)

  • Definition 1: Subspace Invariance
  • Theorem 1: Subspace Invariance
  • proof
  • Corollary 1
  • proof
  • Definition 2: Subspace Attractivity
  • Lemma 1
  • proof
  • Theorem 2: Subspace Attractivity
  • proof
  • ...and 1 more