Spectral Gap Bounds for Quantum Markov Semigroups via Correlation Decay

TL;DR

This paper establishes a connection between the spectral gap of a canonical Hamiltonian derived from a quantum state and the correlation decay properties of that state, with applications to quantum Markov semigroups and specific models.

Contribution

It introduces a canonical purified Hamiltonian framework to relate spectral gaps to correlation decay, applicable to Gibbs states and quantum double models.

Findings

Spectral gap bounds are derived from correlation decay conditions.

The approach applies to finite-range 1D models and Kitaev's quantum double models.

The method provides lower bounds for the spectral gap of quantum Markov semigroups.

Abstract

Starting from an arbitrary full-rank state of a lattice quantum spin system, we define a "canonical purified Hamiltonian" and characterize its spectral gap in terms of a spatial mixing condition (or correlation decay) of the state. When the state considered is a Gibbs state of a local, commuting Hamiltonian at positive temperature, we show that the spectral gap of the canonical purified Hamiltonian provides a lower bound to the spectral gap of a large class of reversible generators of quantum Markov semigroup, including local and ergodic Davies generators. As an application of our construction, we show that the mixing condition is always satisfied for any finite-range 1D model, as well as by Kitaev's quantum double models.