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arXiv:2112.15451 · Physical Review A

Characterizing nonlocal correlations through various n -locality inequalities in a quantum network

Sneha Munshi, A. K. Pan

Published Mar 30, 2022• 23 citations
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Abstract

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 $n$-locality inequalities in a star-network configuration where each of the $n$ parties performs an arbitrary number of dichotomic measurements and demonstrates the above correspondence between the quantum violations of the $n$-locality inequalities and the chained Bell inequalities. A similar correspondence is demonstrated based on a recently formulated family of $n$-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.

DOI
10.1103/physreva.105.032216

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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 ${2}^{m\ensuremath{-}1}$ 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 $\ensuremath{\lfloor}m/2\ensuremath{\rfloor}$ 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.
Source-independent quantum networks are considered as a natural generalization to the Bell scenario where we investigate the nonlocal properties of quantum states distributed and measured in a network. Considering the simplest network of entanglement swapping, recently Gisin et al. [Phys. Rev. A 96, 020304(R) (2017)] and Andreoli [New. J. Phys. 19, 113020 (2017)] independently provided a systematic characterization of the set of quantum states leading to violation of the so-called bilocality inequality. In this work, we consider the complexities in quantum networks with an arbitrary number of parties distributed in chain-shaped and star-shaped networks. We derive the maximal violation of the ``nnn-local'' inequality that can be achieved by arbitrary two-qubit states for such chain- and star-shaped networks. This would further provide a deeper understanding of quantum correlations in complex structures.
The nonlocality revealed in a multiparty multisource network Bell experiment is conceptually different than the standard multiparty Bell nonlocality involving a single common source. Here, by introducing variants of asymmetric bilocal as well as trilocal network scenarios, we go beyond the typical bilocal network scenario where both the edge parties have an equal number of measurement settings. We first introduce an asymmetric bilocal network where one of the edge parties (say, Alice) receives ${2}^{n\ensuremath{-}1}$ inputs and the other edge party (say, Charlie) receives nnn inputs. We derive two variants of asymmetric bilocality inequalities and demonstrate their optimal quantum violations. Further, we explore two types of asymmetric trilocal scenarios: (i) when two edge parties receive ${2}^{n\ensuremath{-}1}$ inputs each and the other edge party receives nnn inputs, and (ii) when one edge party receives ${2}^{n\ensuremath{-}1}$ inputs and the other two edge parties have nnn inputs each. We use an elegant sum-of-squares technique that enables us to evaluate the quantum optimal values of the proposed network inequalities without assuming the dimension of the systems for both the asymmetric bilocal as well as the trilocal scenarios. Further, we demonstrate the robustness of the quantum violations of the proposed inequalities in the presence of white noise.
The nonlocal correlations of noisy quantum systems are important for both understanding nature and developing quantum technology. We consider the correlations of star and chain quantum networks in which noisy entanglement sources are measured by nonsignaling parties. We derive the necessary and sufficient conditions for when a broad family of star and chain nnn-locality inequalities achieve their maximal quantum violation assuming that each party's dichotomic observables are separable across qudit systems. When pairs of local dichotomic observables are considered on qubit systems, we derive maximal nnn-local violations that are larger and more robust to noise than the maximal nnn-locality violations reported previously. To obtain these larger values, we consider observables that we have not found in previous studies. Thus, we gain insights into self-testing entanglement sources and measurements in star and chain networks.
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.
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.