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.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.

arXiv:2312.02588Dec 5, 20230 citations

Extending loophole-free nonlocal correlations to arbitrarily large distances

Anubhav Chaturvedi, Giuseppe Viola, Marcin Pawłowski

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.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.

arXiv:2211.14231npj Quantum InformationJan 11, 202412 citations

Quantifying Bell nonlocality of a pure two-qudit state via its entanglement

Elena R. Loubenets, Sergey Kuznetsov, Louis Hanotel

Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.

arXiv:2303.16794Journal of Physics A Mathematical and TheoreticalJan 5, 20246 citations

Interplay among entanglement, measurement incompatibility, and nonlocality

Yuwei Zhu, Xingjian Zhang, Xiongfeng Ma

Abstract Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.Abstract Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.

arXiv:2303.08407Quantum Science and TechnologyJun 21, 20247 citations

Exact Identification of Quantum Correlations in the Minimal Bell Scenario

Liang-Liang Sun, Jie Wang, Sixia Yu

Bell nonlocality places the long-standing philosophical disputes over the nature of quantum theory on a mathematically rigorous, experimentally testable footing, positioning quantum mechanics between local realism and the most general no-signaling theories. Yet the precise quantum boundary has remained elusive even in the minimal Bell scenario, where the extreme points generating the set are known. Here we characterize this boundary through a computable necessary and sufficient criterion: a correlation is quantum if and only if it can be decomposed into an extreme quantum correlation and a classical correlation. The criterion follows from our proof that every Bell functional exhibiting quantum advantage admits a unique quantum maximizer. We further classify the finitely many maximal-face structures containing each nonlocal exposed point according to their local deterministic vertices. By sharply distinguishing physically realizable correlations from nonphysical ones, these results resolve a long-standing open problem and provide a new foundation for identifying the principles underlying quantum theory and determining the ultimate information-theoretic limits of quantum correlations.Bell nonlocality places the long-standing philosophical disputes over the nature of quantum theory on a mathematically rigorous, experimentally testable footing, positioning quantum mechanics between local realism and the most general no-signaling theories. Yet the precise quantum boundary has remained elusive even in the minimal Bell scenario, where the extreme points generating the set are known. Here we characterize this boundary through a computable necessary and sufficient criterion: a correlation is quantum if and only if it can be decomposed into an extreme quantum correlation and a classical correlation. The criterion follows from our proof that every Bell functional exhibiting quantum advantage admits a unique quantum maximizer. We further classify the finitely many maximal-face structures containing each nonlocal exposed point according to their local deterministic vertices. By sharply distinguishing physically realizable correlations from nonphysical ones, these results resolve a long-standing open problem and provide a new foundation for identifying the principles underlying quantum theory and determining the ultimate information-theoretic limits of quantum correlations.

quant-pharXiv:2610.34364Oct 27, 20260 citations

Inequality relation between entanglement and Bell nonlocality for arbitrary two-qubit states

Xiao‐Gang Fan, Zhiyong Ding, Fei Ming +3 more

Entanglement and Bell nonlocality are used to describe quantum inseparabilities. Bell-nonlocal states form a strict subset of entangled states. A natural question arises concerning how much territory Bell nonlocality occupies entanglement for a general two-qubit entangled state. In this work, we investigate the relation between entanglement and Bell nonlocality by using lots of randomly generated two-qubit states, and give out a constraint inequality relation between the two quantum resources. For studying the upper or lower boundary of the inequality relation, we discover maximally (minimally) nonlocal entangled states, which maximize (minimize) the value of the Bell nonlocality for a given value of the entanglement. Futhermore, we consider a special kind of mixed state transformed by performing an arbitrary unitary operation on werner state. It is found that the special mixed state's entanglement and Bell nonlocality are related to ones of a pure state transformed by the unitary operation performed on the Bell state.Entanglement and Bell nonlocality are used to describe quantum inseparabilities. Bell-nonlocal states form a strict subset of entangled states. A natural question arises concerning how much territory Bell nonlocality occupies entanglement for a general two-qubit entangled state. In this work, we investigate the relation between entanglement and Bell nonlocality by using lots of randomly generated two-qubit states, and give out a constraint inequality relation between the two quantum resources. For studying the upper or lower boundary of the inequality relation, we discover maximally (minimally) nonlocal entangled states, which maximize (minimize) the value of the Bell nonlocality for a given value of the entanglement. Futhermore, we consider a special kind of mixed state transformed by performing an arbitrary unitary operation on werner state. It is found that the special mixed state's entanglement and Bell nonlocality are related to ones of a pure state transformed by the unitary operation performed on the Bell state.

arXiv:1909.00346Sep 1, 20191 citation

(n, 2, 2)

Bell scenario, we refine our estimate significantly.