Universal coarse geometry of spin systems

TL;DR

This paper introduces a universal coarse geometry framework for quantum spin systems, linking state correlations and dynamics to intrinsic geometric structures that are stable under perturbations.

Contribution

It defines a canonical coarse structure for spin states and dynamics, establishing their stability and dependence on geometric decay of correlations in quantum systems.

Findings

Coarse structures characterize decay of correlations in spin states.

Stability of coarse structures under quasi-local perturbations.

Compatibility between dynamical and spatial coarse structures.

Abstract

The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associated coarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state $\phi$ on an (abstract) spin system with an infinite collection of sites $X$, we define a universal coarse structure $\mathcal{E}_{\phi}$ on the set $X$ with the property that a state has decay of correlations with respect to a coarse structure $\mathcal{E}$ on $X$ if and only if $\mathcal{E}_{\phi}\subseteq \mathcal{E}$. We show that under mild assumptions, the coarsely connected completion $(\mathcal{E}_{\phi})_{con}$ is stable under quasi-local perturbations of the state $\phi$. We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated to a quantum circuit $\alpha$ and the coarse structure of the state $\psi\circ \alpha$ where $\psi$ is any product state.