Small sets of genuinely nonlocal Greenberger-Horne-Zeilinger states in multipartite systems
Zong-Xing Xiong, Yongli Zhang, Mao-Sheng Li, Lvzhou Li · Physical Review A · 2024
A set of orthogonal multipartite quantum states are called (distinguishability-based) genuinely nonlocal if they are locally indistinguishable across any bipartition of the subsystems. In this work, we consider the problem of constructing small genuinely nonlocal sets consisting of generalized Greenberger-Horne-Zeilinger (GHZ) states in multipartite systems. For system ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}N}$ where $N$ is large, using the language of group theory, we show that a tiny proportion $\mathrm{\ensuremath{\Theta}}(1/\sqrt{{2}^{N}})$ of the states among the $N$-qubit GHZ basis suffice to exhibit genuine nonlocality. Similar arguments also hold for the canonical generalized GHZ bases in systems ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, wherever $d$ is even and $N$ is large. What is more, moving to the condition that any fixed $N$ is given, we show that $d+1$ genuinely nonlocal generalized GHZ states exist in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, provided the local dimension $d$ is sufficiently large. As an additional merit, within and beyond an asymptotic sense, the latter result also indicates some evident limitations of the ``trivial othogonality-preserving local measurements'' (TOPLM) technique that has been utilized frequently for detecting genuine nonlocality.