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arXiv:2112.15450 · Physical Review A

Generalized n -locality inequalities in a star-network configuration and their optimal quantum violations

Sneha Munshi, Rahul Kumar, A. K. Pan

Published Oct 21, 2021• 28 citations
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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 $m>3$ 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.

DOI
10.1103/physreva.104.042217

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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 nnn-locality scenario in star-network configuration is an instance of network Bell experiment. Such a scenario involves nnn number of edge observers (Alices) a central observer (Bob) and nnn 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 nnn-locality scenario in star-network configuration where each Alice performs some integer mmm number of binary-outcome measurements and the central party Bob performs 2m\ensuremath−1{2}^{m\ensuremath{-}1}2m\ensuremath−1 measurements. We derive a family of generalized nnn-locality inequalities for any arbitrary mmm. Using an elegant sum-of-squares approach we derive the optimal quantum violation of the aforementioned inequalities when each and every Alice measures mmm number of mutually anticommuting observables. For m=2m=2m=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 nnn-locality inequality for any arbitrary mmm can be obtained when every Alice shares \ensuremath⌊m/2\ensuremath⌋\ensuremath{\lfloor}m/2\ensuremath{\rfloor}\ensuremath⌊m/2\ensuremath⌋ copies of two-qubit maximally entangled state with the central party Bob. We also argue that for m>3m>3m>3 a single copy of a two-qubit-entangled state may not be enough to exhibit the violation of nnn-locality inequality but multiple copies of it can activate the quantum violation. We discuss the implications of our study and raise some open questions.
The multipartite quantum networks feature multiple independent sources, in contrast to the conventional multipartite Bell experiment involving a single source. Despite the initial independence of resources, the multiple observers in the network can suitably choose measurements on their local subsystems and generate a form of quantum nonlocality across the network. So far, network nonlocality has been explored when each source produces a two-qubit entangled state. In this work we demonstrate the network nonlocality when each party performs a black-box measurement, and the dimension of the system remains unspecified. In an interesting work, by considering each source produces two-qubit entangled states in the conventional bilocal scenario, Gisin et al. [Phys. Rev. A 96, 020304 (2017)] demonstrated a correspondence between the violations of bipartite Clauser-Horne-Shimony-Halt inequality and the bilocality inequality. We introduce a variant of the sum-of-squares approach to reproduce their results without assuming the dimension of the system. We then generalize the argument for network nonlocality in star-network topology. Furthermore, we propose a set of nnn-locality inequalities in a star-network configuration where each of the nnn parties performs an arbitrary number of dichotomic measurements and demonstrates the above correspondence between the quantum violations of the nnn-locality inequalities and the chained Bell inequalities. A similar correspondence is demonstrated based on a recently formulated family of nnn-locality inequalities whose optimal quantum violation cannot be obtained when each source emits a two-qubit entangled state and requires multiple copies of two-qubit entangled states. Throughout this paper, each party in the network performs black-box measurements, and the dimension of the system remains unspecified.
Abstract Bell experiment in a network gives rise to a form of quantum nonlocality which is conceptually different from traditional multipartite Bell nonlocality. In this work, the star‐network configuration involving arbitrary n independent sources and parties, including n edge parties and one central party is considered. Each of the n edge parties shares a physical system with the central party. Each edge party receives number of inputs, and the central party receives an arbitrary m number of inputs. The conditional dependence on the inputs of each edge party is imposed so that the local probabilities satisfy a set of constraints. A family of generalized n‐locality inequalities is proposed in the arbitrary input scenario by imposing the set of constraints on inputs. The optimal quantum violation of the inequalities is derived by using an elegant sum‐of‐squares approach without specifying the dimension of the quantum system. Notably, the optimal quantum value is achieved only when the set of linear constraints on inputs is satisfied, which, in turn, self‐tests the observables required for each edge party. It shows that while conditional dependence on inputs significantly reduces the n‐local bound of the inequalities, the optimal quantum violation remains invariant. It argues that this implies a more robust test of network non‐locality, which can be revealed for smaller visibility parameters of the corresponding state. Further, the network nonlocality is characterized and examine its correspondence with suitably derived standard Bell nonlocality.
This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in a generalized star network with an (n,m,k)(n,m,k)(n,m,k) configuration, comprising nnn independent branches, mmm sequential Alices per branch, and kkk measurement settings per party. On each branch, the intermediate Alices implement optimal weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary measurement settings and weak-measurement strengths. The network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in (2,2,6)(2,2,6)(2,2,6) and (2,2,465)(2,2,465)(2,2,465) cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.
Abstract Multipartite nonlocality in a network is conceptually different from standard multipartite Bell nonlocality. In recent times, network nonlocality has been studied for various topologies. A linear‐chain topology of the network is considered and the quantum nonlocality (the non‐n‐locality) is demonstrated. Such a network scenario involves n number of independent sources and parties, two edge parties (Alice and Charlie), and central parties (Bobs). It is commonly assumed that each party receives only two inputs. In this work, a generalized scenario where the edge parties receive an arbitrary n number of inputs (equals to a number of independent sources) is considered and each of the central parties receives two inputs. A family of generalized n‐locality inequalities for a linear‐chain network for arbitrary n is derived and the optimal quantum violation of the inequalities is demonstrated. An elegant sum‐of‐squares approach enabling the derivation of the optimal quantum violation of aforesaid inequalities without assuming the dimension of the system is introduced. It is shown that the optimal quantum violation requires the observables of edge parties to be mutually anticommuting.
The concept of bilocality was introduced to study the correlations which arise in an entanglement-swapping scenario, where one has two sources which can naturally be taken to be independent. This additional constraint leads to stricter requirements than simply imposing locality, in the form of bilocality inequalities. In this work we consider a natural generalization of the bilocality scenario; namely, the star network consisting of a single central party surrounded by nnn edge parties, each of which shares an independent source with the center. We derive inequalities which are satisfied by all local correlations in this scenario, for the cases when the central party performs (i) two dichotomic measurements (ii) a single Bell-state measurement. We demonstrate quantum violations of these inequalities and study both the robustness to noise and to losses.
Abstract This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in generalized star networks with an ( n , m , k ) configuration, comprising n independent branches, m sequential Alices per branch, and k measurement settings per party. On each branch, the intermediate Alices implement weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary two-qubit source states, measurement settings and weak-measurement strengths. Network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in the ( 2 , 2 , 6 ) and ( 2 , 2 , 465 ) cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.