Equivalence of Genuine Multipartite Entanglement and Nonlocality of Nearly Symmetric Multiqubit Pure States
Jakub Wójcik, Wojciech Bruzda, Ignacy Stachura, Remigiusz Augusiak
TL;DR
The work tackles whether pure genuine multipartite entangled states must exhibit genuine multipartite nonlocality, and proves this equivalence for a class of nearly symmetric $n$-qubit states by leveraging a recently proposed GMNL Bell inequality together with Hardy's paradox and a canonical/state-decomposition analysis. The authors consider states in $\mathbb{C}^2 \otimes \mathrm{Sym}((\mathbb{C}^2)^{\otimes(n-1)})$ with a Dicke-state decomposition and show that, after suitable symmetric measurements on $n-1$ parties, the remaining two-qubit state can be driven into a Hardy-type configuration that necessarily violates the GMNL Bell inequality. A key technical step is showing that a polynomial in $\tan\alpha$ with coefficients $C_m$ can only vanish for discrete $\alpha$ unless the state is biseparable (i.e., $h_k = \lambda h_k'$), which contradicts GME; hence GMNL follows for these states. Overall, the result strengthens the case for an equivalence between $GME$ and $GMNL$ in multipartite quantum theory and points to a path for a general proof by extending symmetry assumptions.
Abstract
Whether every pure genuinely multipartite entangled (GME) state necessarily exhibits genuine multipartite nonlocality (GMNL) remains an open question. By combining a recently proposed Bell inequality [I. Stachura \textit{et al.}, \href{https://iopscience.iop.org/article/10.1088/1367-2630/ad7753}{New J. Phys. \textbf{26}, 093029 (2024)}] with Hardy's paradox and the canonical decomposition of pure states, we analytically demonstrate that all highly symmetric, genuinely entangled multipartite qubit states exhibit genuine multipartite nonlocality, thereby supporting Gisin's conjecture in the multipartite setting. This result constitutes a step toward a general proof of the conjectured equivalence between GME and GMNL in quantum theory.