Mixing Time of Open Quantum Systems via Hypocoercivity

Abstract

Understanding the mixing of open quantum systems is a fundamental problem in physics and quantum information science. Existing approaches for estimating the mixing time often rely on the spectral gap estimation of the Lindbladian generator, which can be challenging to obtain in practice. We propose a novel theoretical framework to estimate the mixing time of open quantum systems that treats the Hamiltonian and dissipative part separately, thus circumventing the need for a priori estimation of the spectral gap of the full Lindbladian generator. This framework yields mixing time estimates for a class of quantum systems that are otherwise hard to analyze, even though it does not apply to arbitrary Lindbladians. The technique is based on the construction of an energy functional inspired by the hypocoercivity of (classical) kinetic theory.

Figures (2)

• Figure 1: The intuition behind hypocoercivity is depicted as follows: The blue curve represents the kernel of $\mathcal{D}$, that is, $\ker\mathcal{D}$, on which $\mathcal{D}$ has no effect. The Hamiltonian component acts as a mixing mechanism that pushes the system out of the kernel of $\mathcal{D}$, so that the dissipative component can damp the dynamics, guiding the system towards equilibrium.
• Figure 1: The spectral gap (computed numerically) and the lower bound $\lambda$ (computed using \ref{['eq:choice_of_params_C_lam']}) as functions of the dissipation strength $\gamma$ for the single-qutrit example (left) and the Heisenberg model (right) under dephasing noise as discussed below. The parameters are chosen to be $\omega=J_x=J_y=J_z=h=1$, and the Heisenberg model consists of $N=4$ sites.

Theorems & Definitions (13)

• Theorem 1: Main result
• Definition 2: Mixing time
• Corollary 3: Mixing time estimate
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6: Main result
• proof
• Corollary 7: Convergence in the Schrödinger picture