Generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-locality inequalities in a star-network configuration and their optimal quantum violations

Abstract

Network Bell experiments reveal a form of nonlocality conceptually different from standard Bell nonlocality. Standard multiparty Bell experiments involve a single source shared by a set of observers. In contrast network Bell experiments feature multiple independent sources and each of them may distribute physical systems to a set of observers who perform randomly chosen measurements. The $n$-locality scenario in star-network configuration is an instance of network Bell experiment. Such a scenario involves $n$ number of edge observers (Alices) a central observer (Bob) and $n$ number of independent sources having no prior correlation. Each Alice shares an independent state with the central observer Bob. Usually in network Bell experiments one considers that each party measures only two observables. In this work we propose a nontrivial generalization of $n$-locality scenario in star-network configuration where each Alice performs some integer $m$ number of binary-outcome measurements and the central party Bob performs ${2}^{m\ensuremath{-}1}$ measurements. We derive a family of generalized $n$-locality inequalities for any arbitrary $m$. Using an elegant sum-of-squares approach we derive the optimal quantum violation of the aforementioned inequalities when each and every Alice measures $m$ number of mutually anticommuting observables. For $m=2$ and 3 one obtains the optimal quantum value with a two-qubit entangled state shared between each Alice and Bob. We further demonstrate that the optimal quantum violation of $n$-locality inequality for any arbitrary $m$ can be obtained when every Alice shares $\ensuremath{\lfloor}m/2\ensuremath{\rfloor}$ copies of two-qubit maximally entangled state with the central party Bob. We also argue that for $m&gt;3$ a single copy of a two-qubit-entangled state may not be enough to exhibit the violation of $n$-locality inequality but multiple copies of it can activate the quantum violation. We discuss the implications of our study and raise some open questions.

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