Abstract Bell nonlocality refers to the nonlocal correlation strength among quantum systems that cannot be duplicated using local hidden variables, and uncertainty relations set the lower bound on the sum or product of variances for incompatible observables within a quantum system. In this work, the study of linking these two fundamental quantum features in terms of statistical characteristics within Bell tests is initiated. Two statistics determine the state‐dependent quantum upper bounds of the Bell values: the expectation values of local observables, and the joint uncertainty that is the product of local uncertainty. Specifically, it is showed that the joint measurement uncertainty determines the upper bounds of covariant Bell inequalities in the quantum region, with more joint measurement uncertainty required for more covariant correlation strengths. Conversely, the covariant Bell value defines the lower bound of joint uncertainty. In addition, given the maximally entangled states, Tsirelson's bounds of Bell inequalities can be achieved when the joint uncertainty reaches the maximum.
Abstract Bell nonlocality refers to the nonlocal correlation strength among quantum systems that cannot be duplicated using local hidden variables, and uncertainty relations set the lower bound on the sum or product of variances for incompatible observables within a quantum system. In this work, the study of linking these two fundamental quantum features in terms of statistical characteristics within Bell tests is initiated. Two statistics determine the state‐dependent quantum upper bounds of the Bell values: the expectation values of local observables, and the joint uncertainty that is the product of local uncertainty. Specifically, it is showed that the joint measurement uncertainty determines the upper bounds of covariant Bell inequalities in the quantum region, with more joint measurement uncertainty required for more covariant correlation strengths. Conversely, the covariant Bell value defines the lower bound of joint uncertainty. In addition, given the maximally entangled states, Tsirelson's bounds of Bell inequalities can be achieved when the joint uncertainty reaches the maximum.
Abstract Bell’s inequalities are defined by sums of correlations involving non-commuting observables in each of the two systems. Violations of Bell’s inequalities are only possible because the precision of any joint measurement of these observables will be limited by quantum mechanical uncertainty relations. In this paper we explore the relation between the local measurement uncertainties and the magnitude of the correlations by preparing polarization entangled photon pairs and performing joint measurements of non-commuting polarization components at different uncertainty trade-offs. The change in measurement visibility reveals the existence of a non-trivial balance between the measurement uncertainties where the probabilities of a specific pair of measurement outcomes approaches zero because of the particular combination of enhancement and suppression of the experimentally observed correlations. The occurrence of these high-contrast results shows that the quantum correlations between the photons are close to their maximal value, confirming that the Cirel’son bound of Bell’s inequality violations is defined by the minimal uncertainties that limit the precision of joint measurements.
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.
Bell inequalities are mathematical constructs that demarcate the boundary between quantum and classical physics.A new class of multiplicative Bell inequalities originating from a volume maximization game (based on products of correlators within bipartite systems) has been recently proposed.For these new Bell parameters, it is relatively easy to find the classical and quantum, i.e., Tsirelson, limits.Here, we experimentally test the Tsirelson bounds of these inequalities using polarization-entangled photons for a different number of measurements (n), each party can perform.For n = 2, 3, 4, we report the experimental violation of local hidden variable theories.In addition, we experimentally compare the results with the parameters obtained from a fully deterministic strategy, and observe the conjectured nature of the ratio.Finally, utilizing the principle of "relativistic independence" encapsulating the locality of uncertainty relations, we theoretically derive and experimentally test new and richer quantum bounds for both the multiplicative and the additive Bell parameters for n = 2. Our findings strengthen the deep correspondence between local and nonlocal correlations, and pave the way for both theoretical (e.g., better understanding of nonlocal correlations) and practical (e.g., Bell tests and quantum technologies with inefficient detectors) applications.
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.
Quantum nonlocality and the uncertainty principle are two fundamental cornerstones of quantum theory, reflecting totally different aspects that distinguish it from classical theory. However, Oppenheim and Wehner introduced fine-grained uncertainty relations (FGURs) to account for the degree of quantum nonlocality. In the simplest Bell scenario, they showed that the quantum nonlocality measured by the maximal violation of some Bell inequality was determined by the FGUR. Along this line of research, we first derive explicit FGURs with general weights about probability distributions of two general projective measurements on local systems. Then we split those joint probabilities of Bell inequalities into a single-party conditional probability and the remaining joint probabilities so that the weighted FGURs can be applied straightforwardly to obtain the maximal violation in various scenarios ranging from bipartite, tripartite, to a complete set of multipartite Bell inequalities. Furthermore, in the bipartite scenario, the exact correlation boundary can be shown to be solely determined by our weighted local FGURs.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.