Abstract The quantum bit error rate (QBER) is a central operational parameter in quantum communication, quantifying both channel noise and the potential presence of an eavesdropper. Here we establish that, for any two-qubit state, the QBER measured across two complementary bases serves as an operational witness of nonlocality and negative conditional entropy. By invoking the Horodecki criterion for violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality, we prove that an average QBER below approximately suffices to certify nonlocality. Moreover, applying entropic uncertainty relations, we show that QBER values below guarantee negative conditional entropy. We illustrate the framework with Werner and Gisin states and demonstrate its operational significance in the BBM92 QKD protocol, where parameter estimation simultaneously certifies nonlocality and negative conditional entropy. Our results therefore establish a direct connection between practical QKD parameter estimation and the certification of Bell nonlocality and negative conditional entropy.
Abstract The quantum bit error rate (QBER) is a central operational parameter in quantum communication, quantifying both channel noise and the potential presence of an eavesdropper. Here we establish that, for any two-qubit state, the QBER measured across two complementary bases serves as an operational witness of nonlocality and negative conditional entropy. By invoking the Horodecki criterion for violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality, we prove that an average QBER below approximately suffices to certify nonlocality. Moreover, applying entropic uncertainty relations, we show that QBER values below guarantee negative conditional entropy. We illustrate the framework with Werner and Gisin states and demonstrate its operational significance in the BBM92 QKD protocol, where parameter estimation simultaneously certifies nonlocality and negative conditional entropy. Our results therefore establish a direct connection between practical QKD parameter estimation and the certification of Bell nonlocality and negative conditional entropy.
We propose and experimentally implement an efficient procedure based on entanglement swapping to determine the Bell nonlocality measure of Horodecki et al. [Phys. Lett. A 200, 340 (1995)] and the fully entangled fraction of Bennett et al. [Phys. Rev. A 54, 3824 (1996)] of an arbitrary two-qubit polarization-encoded state. The nonlocality measure corresponds to the amount of the violation of the Clauser-Horne-Shimony-Holt (CHSH) optimized over all measurement settings. By using simultaneously two copies of a given state, we measure directly only six parameters. This is an experimental determination of these quantities without quantum state tomography or continuous monitoring of all measurement bases in the usual CHSH inequality tests. We analyze how well the measured degrees of Bell nonlocality and other entanglement witnesses (including the fully entangled fraction and a nonlinear entropic witness) of an arbitrary two-qubit state can estimate its entanglement. In particular, we measure these witnesses and estimate the negativity of various two-qubit Werner states. Our approach could especially be useful for quantum communication protocols based on entanglement swapping.
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.
Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of nonlocal correlations. The uncertainty principle states that there exist pairs of incompatible measurements on quantum systems such that their outcomes cannot both be predicted. On the other hand, nonlocal correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in the presence of entanglement. Here, we show that these two fundamental quantum effects are quantitatively related. Namely, we provide an entropic uncertainty relation for the outcomes of two binary measurements, where the lower bound on the uncertainty is quantified in terms of the maximum Clauser-Horne-Shimony-Holt value that can be achieved with these measurements. We discuss applications of this uncertainty relation in quantum cryptography, in particular, to certify quantum sources using untrusted devices.
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.
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.
The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising general two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing generalised necessary and sufficient conditions for noisy qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also give a strong upper bound for the CHSH parameter in the general case, and show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements.