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arXiv:2212.14325 · Physical Review A

Sharing nonlocality in a quantum network by unbounded sequential observers

Shyam Sundar Mahato, A. K. Pan

Published Oct 24, 2022• 19 citations
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Abstract

Of late, there has been an upsurge of interest in studying the sequential sharing of various forms of quantum correlations, viz., nonlocality, preparation contextuality, coherence, and entanglement. In this work, we explore the sequential sharing of nonlocality in a quantum network. We first consider the simplest case of the two-input bilocality scenario that features two independent sources and three parties, including two edge parties and a central party. We demonstrate that, in the symmetric case when the sharing is considered for both the edge parties, the nonlocality can be shared by, at most, two sequential observers per edge party. However, in the asymmetric case, when the sharing across one edge party is considered, we show that, at most, six sequential observers can share the nonlocality in the network. We extend our investigation to the two-input $n$-local scenario in the star-network configuration that features an arbitrary $n$ number of edge parties and one central party. In the asymmetric case, we demonstrate that the network nonlocality can be shared by an unbounded number of sequential observers across one edge party for a suitably large value of $n$. Further, we generalize our study for an arbitrary $m$ input $n$-local scenario in the star-network configuration. We show that, even for an arbitrary $m$ input scenario, the nonlocality can be shared by an unbounded number of sequential observers. However, increasing the input $m$, one has to employ a larger number of edge parties $n$ than that of the two-input case to demonstrate the sharing of an unbounded number of sequential observers.

DOI
10.1103/physreva.106.042218

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Of late, there has been an upsurge of interest in studying the sequential sharing of various forms of quantum correlations, viz., nonlocality, preparation contextuality, coherence, and entanglement. In this work, we explore the sequential sharing of nonlocality in a quantum network. We first consider the simplest case of the two-input bilocality scenario that features two independent sources and three parties, including two edge parties and a central party. We demonstrate that, in the symmetric case when the sharing is considered for both the edge parties, the nonlocality can be shared by, at most, two sequential observers per edge party. However, in the asymmetric case, when the sharing across one edge party is considered, we show that, at most, six sequential observers can share the nonlocality in the network. We extend our investigation to the two-input nnn-local scenario in the star-network configuration that features an arbitrary nnn number of edge parties and one central party. In the asymmetric case, we demonstrate that the network nonlocality can be shared by an unbounded number of sequential observers across one edge party for a suitably large value of nnn. Further, we generalize our study for an arbitrary mmm input nnn-local scenario in the star-network configuration. We show that, even for an arbitrary mmm input scenario, the nonlocality can be shared by an unbounded number of sequential observers. However, increasing the input mmm, one has to employ a larger number of edge parties nnn than that of the two-input case to demonstrate the sharing of an unbounded number of sequential observers.
In an entanglement swapping scenario, if two sources distributing entangled states between three parties are independent, local correlations lead to a different kind of inequality than the standard Bell inequalities, known as network local models. A highly demanding task is to find out a way to involve many players nontrivially in a quantum network since measurements, in general, disturb the system. To this end, we consider here an alternative way of sharing network nonlocality when two observers initially share a maximally entangled, a nonmaximally entangled and a noisy entangled state. We report that by employing unsharp measurements performed by one of the observers, at most six pairs can sequentially demonstrate the violation of bilocal inequality while a maximum of two pairs of observers can exhibit bi-nonlocality when both the observers perform unsharp measurements. We also find the critical noise involved in unsharp measurements in each round to illustrate the bi-nonlocality for a fixed shared entangled state as a resource. We also connect the entanglement content of the shared state, quantified via von-Neumann entropy for pure states and entanglement of formation for Werner states with the maximum number of rounds showing violation of bilocal inequality. By considering elegant joint measurement at the node such that the entanglement content in the elements of the joint measurement by the third party gets reduced, we observe that the maximum number reduces to two sequential sharing of bi-nonlocality even for the maximally entangled state when the settings at each side are taken to be three and fixed.
Quantum network nonlocality sharing provides a unique perspective for constructing large-scale quantum networks and holds promise for numerous potential applications. In this paper we demonstrate the network nonlocality sharing via unsharp measurements in a two-forked nnn-layer tree-shaped network, which features ${2}^{n}\ensuremath{-}2$ independent sources and ${2}^{n}\ensuremath{-}1$ nodes. We investigate the $({2}^{n}\ensuremath{-}2)$-local scenarios via any mmm-sided sequential measurements, and conclude that arbitrarily many independent observers on each sequential side can share the non-$({2}^{n}\ensuremath{-}2)$-locality for a sufficiently large value of nnn. For a finite nnn, we present a method that enables directly obtaining the number of observers on each sequential side, who can share the non-$({2}^{n}\ensuremath{-}2)$-locality simultaneously. To interpret our results more intuitively, we discuss the simplest case of n=3n=3n=3, and find that at most 30, 6, 3, and 2 sequential observers per sequential side can simultaneously share the non-6-locality in mmm-sided cases for m=1,2,3,4m=1,2,3,4m=1,2,3,4, respectively.
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.
Based on the quantum violation of the bipartite Bell inequality, it has been established that the sharing of nonlocality can be demonstrated for at most two sequential observers at one end and at most one pair of observers at both ends. In this work we study the sharing of nonlocality and preparation contextuality based on a bipartite Bell inequality, involving arbitrary nnn measurements by one party and ${2}^{n\ensuremath{-}1}$ measurements by another party. Such a Bell inequality has two bounds, the local bound and the preparation noncontextual bound, which is smaller than the local bound. We show that while nonlocality can be shared only by the first pair of the sequential observers, the preparation contextuality can be shared by an arbitrary pair of independent sequential observers at both ends.
Quantum nonlocality is a significant feature in quantum information theory, prompting recent investigations into the potential reuse of postmeasurement states to uncover nonlocality among sequentially measuring observers. While prior studies primarily focused on bipartite or tripartite systems and observers with one chain, such as multiple Bobs with a single Alice or multiple Charlies with a single Alice and Bob, our work extends beyond this framework. We explore sequential nonlocality in systems comprising more parties and observer chains. Our findings reveal that in nnn-partite systems, regardless of whether it is a single-chain or double-chain scenario, there exist unbounded sequential observers capable of detecting nonlocality through violations of the Mermin inequality. In contrast to the conjecture that sequential Bell nonlocality cannot manifest with multiple Alices and Bobs in bipartite systems [i.e., the double-chain setting, see S. M. Cheng et al., Phys. Rev. A 104, L060201 (2021)], our results suggest that increasing the number of subsystems may enable more observer chains to detect nonlocality alongside single observers. Our study advances research on sequential nonlocality, providing valuable insights into its detection across diverse scenarios.
Sequential measurements present a novel framework for constructing large-scale quantum networks, particularly in quantum correlation sharing. In this paper, we design projective measurements and combination strategies to achieve nonlocality sharing in star-shaped networks by leveraging shared classical randomness. This network consists of nnn branches, each containing one or two sequential observers. By investigating two-sided nonlocality sharing within a 2-local network, we identify the optimal projective measurement combination strategy based on the double violation of the 2-locality inequality. Utilizing this optimal strategy, we demonstrate that four inequalities can be violated simultaneously in the 2-local scenario. Furthermore, in an nnn-local scenario involving mmm ($1\ensuremath{\le}m\ensuremath{\le}n$) branches with two sequential observers, our optimal strategy shows that non-nnn-locality sharing among all observers can be achieved through the simultaneous violation of 2mn{2}^{m} n2mn-locality inequality. Our findings provide a distinctive perspective on quantum correlation sharing in networks.