Related papers: Categorical Quantum Volume Operator

Categorical Quantum Volume Operator

URL: http://arxiv.org/abs/2406.02111v3
Date: Thu, 29 Aug 2024 02:21:25 GMT
Title: Categorical Quantum Volume Operator
Authors: Alexander Hahn, Sebastian Murk, Sukhbinder Singh, Gavin K. Brennen,
Abstract summary: We show a generalization of the quantum volume operator quantifying the volume in curved three-dimensional discrete geometries. In both cases, we obtain a volume operator that is Hermitian, provided that the input category is unitary. As an illustrative example, we consider the case of $mathrmSU(2)_k$ and show that the standard $mathrmSU(2)$ volume operator is recovered in the limit $krightarrowinfty$
Score: 41.94295877935867
License: http://creativecommons.org/licenses/by/4.0/
Abstract: We present a generalization of the quantum volume operator quantifying the volume in curved three-dimensional discrete geometries. In its standard form, the quantum volume operator is constructed from tetrahedra whose faces are endowed with irreducible representations of $\mathrm{SU}(2)$. Here, we show two equivalent constructions that allow general objects in fusion categories as degrees of freedom. First, we compute the volume operator for ribbon fusion categories. This includes the important class of modular tensor categories (such as quantum doubles), which are the building blocks of anyon models. Second, we further generalize the volume operator to spherical fusion categories by relaxing the categorical analog of the closure constraint (known as tetrahedral symmetry). In both cases, we obtain a volume operator that is Hermitian, provided that the input category is unitary. As an illustrative example, we consider the case of $\mathrm{SU}(2)_k$ and show that the standard $\mathrm{SU}(2)$ volume operator is recovered in the limit $k\rightarrow\infty$.

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