k-Uniform complete hypergraph states stabilizers in terms of local operators
Gabriel M. Arantes, Vinícius Salem, Danilo Cius, Bárbara Amaral
TL;DR
This work addresses the challenge of expressing the inherently nonlocal stabilizers of $k$-uniform complete hypergraph states as linear combinations of local operators. By deriving a generalized CZ expansion and providing an explicit closed-form for the expansion coefficients $C_m$, the authors enable a local-operator perspective on hypergraph state stabilizers, demonstrated through the explicit example $|H^{4}_{3}\rangle$. They show that, while this approach yields a precise decomposition, the resulting sign structure (negative $C_m$ for $k>2$) prevents straightforward construction of Bell inequalities in the same spirit as for graph states. Despite this limitation, the local expansion offers a valuable tool for stabilizer-based analysis and potential applications in quantum error correction and self-testing, with future work exploring alternative Bell functionals and special cases where $C_0=0$.
Abstract
In this work, we present a novel method to express the stabilizer of a k-uniform complete hypergraph state as a linear combination of local operators. Quantum hypergraph states generalize graph states and exhibit properties that are not shared by their graph counterparts, most notably, their stabilizers are intrinsically nonlocal, as hyperedges can involve arbitrary subsets of vertices. Our formulation provides an explicit description of the stabilizers for k-uniform complete hypergraphs and may offer new insights for exploring these states within the stabilizer formalism. In particular, this approach could facilitate the construction of new Bell inequalities or find applications in quantum error correction.