Spectral Gap Bounds for Quantum Markov Semigroups via Correlation Decay

TL;DR

The paper introduces a canonical purified Hamiltonian to bridge dynamical relaxation rates of σ-reversible quantum Markov semigroups with static correlation decay in the invariant state σ. By relating the spectral gap of this purified Hamiltonian to a spatial mixing condition Δ_σ(A:C|D), the authors derive explicit, dimension-dependent gap bounds for Davies generators and for Gibbs states of local commuting Hamiltonians. They prove system-size independent gaps in 1D for general local Hamiltonians and in 2D for Kitaev quantum double models, with detailed bounds that quantify β-dependence and finite-range locality. The results yield practical implications for thermalization, mixing times, and stability of topological quantum memories at finite temperature, and they outline extensions to non-commuting models, Hopf-algebra generalizations, and potential log-Sobolev improvements.

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.

Figures (6)

• Figure 1: Possible decompositions of the torus into four subregions $\Lambda = ABCD$. In $(i)$ and $(ii)$ the subset $B$ is formed by the union of $B_{1}$ and $B_{2}$, whereas in $(i)$ the set $D$ is empty.
• Figure 2: The first picture represent a splitting of the ring as in Assumption \ref{['Assumption:1Dgap']}, whereas the other pictures correspond to the three possible choices for $D$ when considering $\Delta_{\sigma}(A:C|D)$.
• Figure 3: The square lattice on a torus (left) and a quantum spin system with spins located at the midpoints of the edges (right). The markings along the borders of both squares indicate the pairwise identification of edges, following a standard topological representation of the torus. A similar convention will be used for the cylinder, where only one pair of opposite edges is identified.
• Figure 4: Examples of rectangular regions within the square lattice $\Lambda_{N} \equiv {\mathcal{E}}_{N}$. The top row shows two examples of proper rectangles, while the bottom row displays two examples of cylinders. In each case, the left image highlights the edges within the region, and the right image highlights the corresponding spins. In the figures that follow, we will adopt the edge-based (left) representation.
• Figure 5: On the left, a selected subinterval $I$ of the ring, whose sites are identified with $[1,m]$. On the right, a partition of the $1D$ ring into four subintervals $\Lambda_{N}=A I_{1} C I_{2}$, where $I_{1}$ and $I_{2}$ shield $A$ from $C$. The endpoints of $A$ are marked as $l$ and $k$, and the endpoints of $C$ as $i$ and $j$.

Theorems & Definitions (98)

• Definition 1
• Proposition 2
• proof
• Definition 3
• Definition 4
• Theorem 5: Structure of QMS generators
• Definition 6
• Proposition 7
• Remark 8
• Proposition 9