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arXiv:1709.04260 · Physical Review A

Quantifying Bell nonlocality with the trace distance

Samuraí Brito, Bárbara Amaral, Rafael Chaves

Published Feb 21, 2018• 54 citations
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Abstract

Measurements performed on distant parts of an entangled quantum state can generate correlations incompatible with classical theories respecting the assumption of local causality. This is the phenomenon known as quantum nonlocality that, apart from its fundamental role, can also be put to practical use in applications such as cryptography and distributed computing. Clearly, developing ways of quantifying nonlocality is an important primitive in this scenario. Here, we propose to quantify the nonlocality of a given probability distribution via its trace distance to the set of classical correlations. We show that this measure is a monotone under the free operations of a resource theory and, furthermore, that it can be computed efficiently with a linear program. We put our framework to use in a variety of relevant Bell scenarios also comparing the trace distance to other standard measures in the literature.

DOI
10.1103/physreva.97.022111

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Measurements performed on distant parts of an entangled quantum state can generate correlations incompatible with classical theories respecting the assumption of local causality. This is the phenomenon known as quantum nonlocality that, apart from its fundamental role, can also be put to practical use in applications such as cryptography and distributed computing. Clearly, developing ways of quantifying nonlocality is an important primitive in this scenario. Here, we propose to quantify the nonlocality of a given probability distribution via its trace distance to the set of classical correlations. We show that this measure is a monotone under the free operations of a resource theory and, furthermore, that it can be computed efficiently with a linear program. We put our framework to use in a variety of relevant Bell scenarios also comparing the trace distance to other standard measures in the literature.
Bell's theorem shows that local measurements on entangled states give rise to correlations incompatible with local-hidden-variable models. The degree of quantum nonlocality is not maximal though, as there are even more nonlocal theories beyond quantum theory still compatible with the nonsignaling principle. In spite of decades of research, we still have a very fragmented picture of the whole geometry of these different sets of correlations. Here we employ both analytical and numerical tools to ameliorate that. First, we identify two different classes of Bell scenarios where the nonsignaling correlations can behave very differently: In one case, the correlations are generically quantum and nonlocal while in the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignaling correlations, we compute the distribution of a nonlocality quantifier based on the trace distance to the local set. With that we conclude that the nonlocal correlations can show a concentration phenomenon: Their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed.
Understanding the quantitative relation between entanglement and Bell nonlocality is a long-standing open problem of fundamental and practical interest. Here, we tackle this problem in a general Bell scenario. {We observe that lying in the center of quantifying these properties are two minimal distances: one from a state to separable states (entanglement), and the other from a correlation to local correlations (nonlocality).} We find that these two distances can be related to each other -- the minimal correlation distance provides a lower bound for the minimal state distance, which allows us to derive nontrivial bounds on many entanglement measures with an arbitrary nonlocal correlation. Moreover, with the on-hand structural knowledge of entanglement and nonlocality in the (n,2,2)(n, 2, 2)(n,2,2) Bell scenario, we refine our estimate significantly.
Understanding the quantitative relation between entanglement and Bell nonlocality is a long-standing open problem of fundamental and practical interest. Here, we tackle this problem in a general Bell scenario. We observe that lying in the center of quantifying these properties are two minimal distances: one from a state to separable states (entanglement), and the other from a correlation to local correlations (nonlocality). We find that these two distances can be related to each other—the minimal correlation distance provides a lower bound for the minimal state distance, which allows us to derive nontrivial bounds on many entanglement measures with an arbitrary nonlocal correlation. Moreover, with the on-hand structural knowledge of entanglement and nonlocality in the ( n , 2, 2) Bell scenario, we refine our estimate significantly.
Abstract One of the most striking features of quantum theory is that it allows spatially separated observers to share correlations that resist local-causal (classical) explanations, a phenomenon referred to as Bell nonlocality[1, 2]. Besides their foundational relevance, the nonlocal correlations enable distant observers to accomplish classically inconceivable information processing and cryptographic feats such as unconditionally secure device-independent key distribution schemes[3-5]. However, the distances over which nonlocal correlations can be realized in state-of-the-art Bell experiments remain severely limited owing to the high threshold efficiencies of the detectors and the fragility of the nonlocal correlations to experimental noise[6, 7]. Instead of looking for quantum strategies with marginally lower threshold requirements, we exploit the properties of loophole-free nonlocal correlations, which are experimentally attainable today, albeit at short distances, to extend them over arbitrarily large distances. Specifically, we consider Bell experiments wherein the spatially separated parties randomly choose the location of their measurement devices in addition to their measurement settings. We demonstrate that when devices close to the entanglement source are perfect and witness extremal loophole-free nonlocal correlations, such correlations can be extended to devices placed arbitrarily far from the source, with almost zero detection efficiency and visibility. To accommodate imperfections close to the source, we demonstrate a specific analytical tradeoff: the higher the loophole-free nonlocality close to the source, the lower the threshold requirements away from the source. We utilize this analytical tradeoff paired with optimal quantum strategies to estimate the critical requirements of a measurement device placed away from the source in the simplest non-trivial scenario and formulate a versatile numerical method applicable to generic network scenarios. Our results demonstrate that the properties of already feasible short-distance loophole-free nonlocal correlations can be exploited to extend them to longer distances, enabling device-independent information processing and cryptography fuelled by long-distance loophole-free nonlocal correlations to become near-term commonplace technologies.
Although entanglement is necessary for observing nonlocality in a Bell experiment, there are entangled states which can never be used to demonstrate nonlocal correlations. In a seminal paper [PRL 108, 200401 (2012)] F. extended the standard Bell experiment by allowing Alice and Bob to be asked quantum, instead of classical, questions. This gives rise to a broader notion of nonlocality, one which can be observed for every entangled state. In this work we introduce and study a resource theory of this type of nonlocality which we term Buscemi nonlocality. We propose a geometric quantifier measuring the ability of a given state and local measurements to produce nonlocal correlations and establish its operational significance. In particular, we show that any distributed measurement which can demonstrate nonlocal correlations provides strictly better performance than any distributed measurement which does not use entanglement in the task of distributed state discrimination. We also show that the maximal amount of nonlocality that can be generated using a given state is precisely equal to its entanglement content. Finally, we prove a quantitative relationship between: nonlocality, the ability to perform nonclassical teleportation, and entanglement. Using this relationship we propose new discrimination tasks for which nonclassical teleportation and entanglement lead to an advantage over their classical counterparts.
Quantum theory admits correlations between distant parties that are stronger than those permitted by classical physics. Such correlations, which are termed nonlocal, have been studied extensively since their discovery by John Bell in 1964. This thesis makes progress on various aspects of quantum nonlocality both from a foundational and practical perspective. We begin by studying a foundational question on the relationship between nonlocality and quantum entanglement. We show that the strongest form of nonlocality, also called pseudo-telepathy or full nonlocality, can be demonstrated by non-maximally entangled states and prove that all pure entangled states can demonstrate full nonlocality in the many-copy scenario. We also provide non-trivial examples of fully nonlocal mixed states. We then turn to the study of another phenomenon in quantum theory which is essential in almost all demonstrations of non-classicality—measurement incompatibility. We generalise the notion of measurement incompatibility and show that our notion has strong operational relevance: it captures the randomness in the outcomes of an untrusted measurement device against a classical adversary. We use our formalism to derive new bounds on the experimental losses and noise that can be tolerated in quantum communication protocols before they become insecure. We then begin a systematic study of routed Bell experiments, a variant of standard Bell experiments that was introduced to demonstrate nonlocality over long distances in the presence of losses and noise. We formalize the notion of genuine long-range quantum correlations in routed Bell experiments and develop numerical and analytical tools to characterize them. We show that routed Bell experiments allow for the demonstration of long-range quantum correlations at arbitrarily large distances using only entangled qubit states. We then prove the security of fully device-independent (DI) quantum key distribution (QKD) protocols based on routed Bell tests and quantify the resulting improvement in key rate and distance compared to standard DI QKD protocols.