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arXiv:1901.08287 · Physical Review Letters

Limits on Correlations in Networks for Quantum and No-Signaling Resources

Marc-Olivier Renou, Yuyi Wang, Sadra Boreiri, Salman Beigi, Nicolas Gisin, Nicolas Brunner

Published Aug 15, 2019• 72 citations
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Abstract

A quantum network consists of independent sources distributing entangled states to distant nodes which can then perform entangled measurements, thus establishing correlations across the entire network. But how strong can these correlations be? Here we address this question, by deriving bounds on possible quantum correlations in a given network. These bounds are nonlinear inequalities that depend only on the topology of the network. We discuss in detail the notably challenging case of the triangle network. Moreover, we conjecture that our bounds hold in general no-signaling theories. In particular, we prove that our inequalities for the triangle network hold when the sources are arbitrary no-signaling boxes which can be wired together. Finally, we discuss an application of our results for the device-independent characterization of the topology of a quantum network.

DOI
10.1103/physrevlett.123.070403

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A quantum network consists of independent sources distributing entangled states to distant nodes which can then perform entangled measurements, thus establishing correlations across the entire network. But how strong can these correlations be? Here we address this question, by deriving bounds on possible quantum correlations in a given network. These bounds are nonlinear inequalities that depend only on the topology of the network. We discuss in detail the notably challenging case of the triangle network. Moreover, we conjecture that our bounds hold in general no-signaling theories. In particular, we prove that our inequalities for the triangle network hold when the sources are arbitrary no-signaling boxes which can be wired together. Finally, we discuss an application of our results for the device-independent characterization of the topology of a quantum network.
The possibility of Bell inequality violations in quantum theory had a profound impact on our understanding of the correlations that can be shared by distant parties. Generalizing the concept of Bell nonlocality to networks leads to novel forms of correlations, the characterization of which is, however, challenging. Here, we investigate constraints on correlations in networks under the natural assumptions of no-signaling and independence of the sources. We consider the triangle network with binary outputs, and derive strong constraints on correlations even though the parties receive no input, i.e., each party performs a fixed measurement. We show that some of these constraints are tight, by constructing explicit local models (i.e. where sources distribute classical variables) that can saturate them. However, we also observe that other constraints can apparently not be saturated by local models, which opens the possibility of having nonlocal (but non-signaling) correlations in the triangle network with binary outputs.
Quantum networks play a major role in long-distance communication, quantum cryptography, clock synchronization, and distributed quantum computing. Generally, these protocols involve many independent sources sharing entanglement among distant parties that, upon measuring their systems, generate correlations across the network. The question of which correlations a given quantum network can give rise to remains almost uncharted. Here we show that constraints on the observable covariances, previously derived for the classical case, also hold for quantum networks. The network topology yields tests that can be cast as semidefinite programs, thus allowing for the efficient characterization of the correlations in a wide class of quantum networks, as well as systematic derivations of device-independent and experimentally testable witnesses. We obtain such semidefinite tests for fixed measurement settings, as well as parties that independently choose among collections of measurement settings. The applicability of the method is demonstrated for various networks, and compared with previous approaches.
Entangled quantum networks provide great flexibilities and scalabilities for quantum information processing or quantum Internet. Most of results are focused on the nonlocalities of quantum networks. Our goal in this work is to explore new characterizations of any networks with theory-independent configurations. We firstly prove the configuration inequality for any network using the fractional independent set of the associated graph. These inequalities can be built with polynomial-time complexity. The new result allows featuring correlations of any classical network depending only on its network topology. Similar inequalities hold for all entangled quantum networks with any local measurements. This shows an inherent feature of quantum networks under local unitary operations. It is then applied for verifying almost all multipartite entangled pure states with linear complexity, and witnessing quantum network topology without assumption of inputs. The configuration theory is further extended for any no-signalling networks. These results may be interesting in entanglement theory, quantum information processing, and quantum networks.
The exploration of the concept of nonlocality beyond standard Bell scenarios in quantum network architectures unveils fundamentally new forms of correlations that hold a strong potential for future applications of quantum communication networks [1]. It is necessary to adapt theoretical advances to realistic configurations to materialize this potential. Here we consider a quantum triangle network, illustrated in Fig. 1, for which it was shown in theory that, remarkably, quantum nonlocality without inputs can be demonstrated for sources with an arbitrarily small level of independence [2]. We realize experimentally such correlated sources,ρAB,ρAC\rho_{AB}, \rho_{AC}ρAB​,ρAC​, andρBC\rho_{BC}ρBC​, by carefully engineering the output state of a single AlGaAs multiplexed entangled-photon source [3], exploiting energy-matched channels cut in its broad spectrum. This simulated triangle network is then used to violate a Bell-like inequality that we derive to capture the effect of noise in the correlations present in our system [4]. We also rigorously validate our findings by analyzing the mutual information between the generated states. Our results allow us to deepen our understanding of network nonlocality while also pushing its practical relevance for quantum communication networks.
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In the above-mentioned reference, the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios has yet to be provided. The present work characterizes correlations arising due to fixed local measurements in a triangle network under a source independence assumption. Precisely speaking, a set of criteria is framed in the form of Bell-type inequalities, each of which is necessarily satisfied by trilocal correlations. Possible quantum violation of at least one criterion from the set is analyzed, which in turn points out the utility of the set of criteria to detect nonlocality (nontrilocality) in quantum triangle networks. Interestingly, measurement on a local product state basis turns out to be sufficient to generate nontrilocal correlations in some quantum networks. Noise tolerance of the detection criteria is discussed followed by a generalization of the framework for demonstrating correlations in any nnn-sided polygon where nnn is finite.
The problem of characterizing classical and quantum correlations in networks is considered. Contrary to the usual Bell scenario, where distant observers share a physical system emitted by one common source, a network features several independent sources, each distributing a physical system to a subset of observers. In the quantum setting, the observers can perform joint measurements on initially independent systems, which may lead to strong correlations across the whole network. In this work, we introduce a technique to systematically map a Bell inequality to a family of Bell-type inequalities bounding classical correlations on networks in a star-configuration. Also, we show that whenever a given Bell inequality can be violated by some entangled state ρ , then all the corresponding network inequalities can be violated by considering many copies of ρ distributed in the star network. The relevance of these ideas is illustrated by applying our method to a specific multi-setting Bell inequality. We derive the corresponding network inequalities, and study their quantum violations.