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arXiv:2408.10489 · Physical Review Research

Interplay of quantum resources in nonlocality tests

Hai-Hao Dong, Yuwei Zhu, Su-Yi Cheng, Xingjian Zhang, Cheng-Long Li, Ying-Zhao Li, Hao Li, Lixing You, Xiongfeng Ma, Qiang Zhang, Jian-Wei Pan

Published Jun 5, 2025• 1 citation
arXivHTMLPDF

Abstract

Nonlocality, evidenced by the violation of Bell inequalities, not only signifies entanglement but also highlights measurement incompatibility in quantum systems. Utilizing the tilted Clauser-Horne-Shimony-Holt (CHSH) Bell inequality, our high-efficiency optical setup achieves a loophole-free violation of 2.0132 . This result provides a device-independent lower bound on entanglement, quantified as the entanglement of formation at 0.0159 . Moreover, by tuning the parameters of the tilted CHSH inequality, we enhance the estimation of measurement incompatibility, which is quantified by an effective overlap of 4.3883 × 10 − 5 . To explore the intricate interplay among nonlocality, entanglement, and measurement incompatibility, we generate mixed states, allowing for flexible modulation of entanglement via fast switching among the four Bell states using Pockels cells, achieving a fidelity above 99.10 % . Intriguingly, our results reveal a counterintuitive relationship where increasing incompatibility initially boosts nonlocality but eventually leads to its reduction. Typically, maximal nonlocality does not coincide with maximal incompatibility. This experimental study sheds light on the optimal management of quantum resources for Bell-inequality-based quantum information processing.

DOI
10.1103/physrevresearch.7.l022055

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Nonlocality, evidenced by the violation of Bell inequalities, not only signifies entanglement but also highlights measurement incompatibility in quantum systems. Utilizing the tilted Clauser-Horne-Shimony-Holt (CHSH) Bell inequality, our high-efficiency optical setup achieves a loophole-free violation of 2.0132 . This result provides a device-independent lower bound on entanglement, quantified as the entanglement of formation at 0.0159 . Moreover, by tuning the parameters of the tilted CHSH inequality, we enhance the estimation of measurement incompatibility, which is quantified by an effective overlap of 4.3883 × 10 − 5 . To explore the intricate interplay among nonlocality, entanglement, and measurement incompatibility, we generate mixed states, allowing for flexible modulation of entanglement via fast switching among the four Bell states using Pockels cells, achieving a fidelity above 99.10 % . Intriguingly, our results reveal a counterintuitive relationship where increasing incompatibility initially boosts nonlocality but eventually leads to its reduction. Typically, maximal nonlocality does not coincide with maximal incompatibility. This experimental study sheds light on the optimal management of quantum resources for Bell-inequality-based quantum information processing.
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.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.
Abstract The degree of experimentally attainable nonlocality, as gauged by the amount of loophole-free violation of Bell inequalities, remains severely limited due to inefficient detectors. We address an experimentally motivated question: Which quantum strategies attain the maximal loophole-free nonlocality in the presence of inefficient detectors? For any Bell inequality and any specification of detection efficiencies, the optimal strategies are those that maximally violate a tilted version of the Bell inequality in ideal conditions. In the simplest scenario, we demonstrate that the quantum strategies that maximally violate the tilted versions of Clauser-Horne-Shimony-Holt inequality are unique up to local isometries. However, self-testing via the standard sum of squares decomposition method turns out to be analytically intractable since even high levels of the Navascués--Pironio--Acín hierarchy are insufficient to saturate the maximum quantum violation of these inequalities. Instead, we utilize a novel Jordan's lemma-based proof technique to obtain robust analytical self-testing statements for the entire family of tilted-Bell inequalities. These results allow us to unveil intriguing aspects of the effect of inefficient detectors and the complexity of characterizing the set of quantum correlations, in the simplest Bell scenario.
Bell nonlocality is the experimentally testable incompatibility between quantum correlations and explanations based on local hidden variables. Its significance now extends beyond the foundations of quantum mechanics: a Bell-inequality violation can certify entanglement, randomness, measurement incompatibility, and cryptographic security without requiring a detailed model of the devices producing the observed data. This article develops the mathematical structure of Bell experiments and explains its role in device-independent quantum information. Conditional probability distributions are classified into local, quantum, and no-signalling sets; the Clauser-Horne-Shimony-Holt inequality is derived, and its classical, quantum, and algebraic limits are distinguished. Tsirelson’s bound is obtained from the CHSH operator, while Popescu-Rohrlich correlations demonstrate that no-signalling alone does not identify the quantum set. The relation between entanglement and nonlocality is examined through local entangled states, hidden nonlocality, and activation. Nonlocal games, semidefinite relaxations, self-testing, certified randomness, and device-independent quantum key distribution are treated as operational consequences of Bell violation. Particular attention is given to entropy accumulation, finite-data security, loophole-free experiments, and the limitations of device independence. Bell nonlocality emerges as both a foundational separation between classical and quantum causal models and a quantitative resource for cryptographic and informational tasks.
Abstract The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.
Nonlocality is one of the most important resources for quantum information protocols.The observation of nonlocal correlations in a Bell experiment is the result of appropriately chosen measurements and quantum states.We quantify the minimal purity to achieve a certain Bell value for any Bell operator.Since purity is the most fundamental resource of a quantum state, this enables us also to quantify the necessary coherence, discord, and entanglement for a given violation of two-qubit correlation inequalities.Our results shine a light on the Clauser-Horne-Shimony-Holt inequality by showing that for a fixed Bell violation an increase in the measurement resources does not always lead to a decrease of the minimal state resources.