Universal coarse geometry of spin systems

• URL: http://arxiv.org/abs/2411.07912v1
• Date: Tue, 12 Nov 2024 16:39:33 GMT
• Title: Universal coarse geometry of spin systems
• Authors: Ali Elokl, Corey Jones,
• Abstract summary: We show 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. We establish a basic compatibility between the dynamical coarse structure associated to a quantum circuit $alpha$ and the coarse structure of the state $psicirc alpha$ where $psi$ is any product state.
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• Abstract: The prospect of realizating 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 intrinic 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 the state only depend on the coarse structure of the 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.

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