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Abstract We consider the problem of building non-invertible quantum symmetries (as characterized by actions of unitary fusion categories) on noncommutative tori. We introduce a general method to construct actions of fusion categories on inductive limit C*-algberas using finite dimenionsal data, and then apply it to obtain AT-actions of arbitrary Haagerup-Izumi categories on noncommutative 2-tori, of the even part of the$$E_{8}$$ $E_8$subfactor on a noncommutative 3-torus, and of$$\text {PSU}(2)_{15}$$ $\text{PSU}{\left(2\right)}_{15}$on a noncommutative 4-torus. 

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 associatedcoarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state$$\phi $$ $\varphi$on an (abstract) spin system with an infinite collection of sitesX, we define a universal coarse structure$$\mathcal {E}_{\phi }$$ $E_{\varphi }$on the setXwith the property that a state has decay of correlations with respect to a coarse structure$$\mathcal {E}$$ $E$onXif and only if$$\mathcal {E}_{\phi }\subseteq \mathcal {E}$$ $E_{\varphi }\subseteq E$. We show that under mild assumptions, the coarsely connected completion$$(\mathcal {E}_{\phi })_{con}$$ ${\left(E_{\varphi }\right)}_{\mathrm{con}}$is stable under quasi-local perturbations of the state$$\phi $$ $\varphi$. 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$$\alpha $$ $\alpha$and the coarse structure of the state$$\psi \circ \alpha $$ $\psi \circ \alpha$where$$\psi $$ $\psi$is any product state. 

Abstract We show that the Levin-Wen model of a unitary fusion category$${\mathcal {C}}$$ $C$is a gauge theory with gauge symmetry given by the tube algebra$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$. In particular, we define a model corresponding to a$${\text {Tube}}({\mathcal {C}})$$ $\text{Tube}\left(C\right)$symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case$${\mathcal {C}}=\textsf{Hilb}(G,\omega )$$ $C=\mathrm{Hilb}(G,\omega )$, we show how our procedure reduces to the twisted gauging of a trivalG-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter. 

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a latticeL\subseteq \mathbb{R}^{n}satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebrasAof operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to\mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)), containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry\mathcal{D}, we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center\mathcal{Z}(\mathcal{D}). We use this to show that, for the double spin-flip action\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}, the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy ofS_{3}; hence, it is non-abelian, in contrast to the case with no symmetry.