Categorical Quantum Volume Operator

TL;DR

This paper generalizes the quantum volume operator to curved discrete geometries using fusion categories, extending its applicability to anyon models and recovering classical limits in specific cases.

Contribution

It introduces a novel construction of the quantum volume operator for ribbon and spherical fusion categories, broadening the scope beyond traditional $ ext{SU}(2)$ frameworks.

Findings

Volume operator for ribbon fusion categories computed.

Hermitian volume operator obtained for unitary categories.

Standard $ ext{SU}(2)$ volume operator recovered as a limit.

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 $k\rightarrow\infty$.