Universal coarse geometry of spin systems

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 φ ϕ on an (abstract) spin system with an infinite collection of sites X , we define a universal coarse structure Eφ E ϕ on the set X with the property that a state has decay of correlations with respect to a coarse structure E E on X if and only if Eφ⊆ E E ϕ ⊆ E . We show that under mild assumptions, the coarsely connected completion (Eφ)con ( E ϕ ) con is stable under quasi-local perturbations of the state φ ϕ . 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 with a quantum circuit α α and the coarse structure of the state ψ ∘ α ψ ∘ α where ψ ψ is any product state.