All maximally entangled two-qubit states violate local realism with the same probability under uniformly random projective measurements, yet they need not behave identically in randomized Bell tests. We show that when measurement settings are exchanged between the parties in sequential Bell experiments, permutation symmetry of the shared state determines the statistical relation between the two realizations. Permutationally invariant states yield identical nonlocality outcomes in both experiments, whereas asymmetric states can violate local realism in one realization but not in the other. This distinction leads to two operational consequences. First, it enables the detection of correlations between the measurement choices of Alice and Bob through the joint violation statistics. Second, in Bell tests with finite measurement pools, asymmetric maximally entangled states can significantly increase the probability of observing nonlocality without requiring additional resources. Our results identify permutation asymmetry as a useful feature in randomized Bell experiments and highlight a new role of symmetry in quantum nonlocality.
All maximally entangled two-qubit states violate local realism with the same probability under uniformly random projective measurements, yet they need not behave identically in randomized Bell tests. We show that when measurement settings are exchanged between the parties in sequential Bell experiments, permutation symmetry of the shared state determines the statistical relation between the two realizations. Permutationally invariant states yield identical nonlocality outcomes in both experiments, whereas asymmetric states can violate local realism in one realization but not in the other. This distinction leads to two operational consequences. First, it enables the detection of correlations between the measurement choices of Alice and Bob through the joint violation statistics. Second, in Bell tests with finite measurement pools, asymmetric maximally entangled states can significantly increase the probability of observing nonlocality without requiring additional resources. Our results identify permutation asymmetry as a useful feature in randomized Bell experiments and highlight a new role of symmetry in quantum nonlocality.
While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality. In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.
It is already known that one can always find a set of measurements on any two-qubit entangled state that will lead to a violation of the CHSH inequality. We provide an explicit state in terms of the angle between Alice's choice of measurements and the angle between Bob's choice of measurements, such that the CHSH inequality is always violated provided Alice's or Bob's choice of inputs are not collinear. We prove that inequalities with a corresponding Bell operator written as a linear combination of tensor products of Pauli matrices, excluding the identity, will generate the most nonlocal correlations using maximally entangled states in our experiment. From this result and a proposition from Horodecki et. al., we are able to construct the state that generates these optimal correlations. To achieve this state in a lab, one party must rotate their qubit using the orthogonal operation we provide and also rotate their Bloch sphere such that all their measurements lie in the same plane. We provide a comprehensive study of how Bell inequalities change when experiments introduce error via imperfect detection efficiency. The original cases of perfect efficiency are covered first and then a more realistic approach, when inefficient detectors are used, will follow. It is shown that less entanglement is needed to demonstrate more nonlocality in some Clasure-Horne-Shimony-Holt (CHSH) experiments when detector inefficiency is introduced. An example of this is shown for any given specific set of measurements in the CHSH Bell experiment. This occurs when one party has a detector of efficiency for each choice of input and the other party makes projective measurements. The efficiency can be pushed down to fifty percent while still violating the CHSH inequality, and for the experimental set-up illustrated, there is more nonlocality with less entanglement. Furthermore, it is shown that if the first party has an imperfect detector for only one choice of inputs rather than two, the efficiency can be brought down arbitrarily close to zero percent while still violating the CHSH inequality. Historically, nonlocality and entanglement were viewed as two equivalent resources, but recently this equality has come under question; these results further support this fundamental difference. Further more, we introduce Mermin's game in the case of relaxed conditions. The original constraints were that when the detectors in separate labs of a two-qubit experiment are in the same setting, then the results should be the same. We require that the outcomes are the same at least part of the time, given by some epsilon variable. Initially, one could find a maximum violation of one-fourth by allowing to parties to share the singlet state and have measurement settings one-hundred and twenty degrees apart from one another. By allowing some epsilon error in the perfect correlations regime, one can find a maximum violation of minus one plus the square root of two using the singlet state and measurement inputs that achieve Tsirelson's bound for the CHSH experiment. The reason is that we show Mermin's inequality is technically the CHSH inequality "in disguise", but with using constraints the CHSH experiment does not use. We derive Mermin's inequality under new conditions and give the projective measurements needed to violate maximally.
Nonlocality is the most characteristic feature of quantum mechanics. John Bell, in his seminal 1964 work, proved that local-realism imposes a bound on the correlations among the measurement statistics of distant observers. Surpassing this bound rules out local-realistic description of microscopic phenomena, establishing the presence of nonlocal correlation. To manifest nonlocality, it requires, in the simplest scenario, two measurements performed randomly by each of two distant observers. In this work, we propose a novel framework where three measurements, two on Alice's side and one on Bob's side, suffice to reveal quantum nonlocality and hence does not require all-out randomness in measurement choice. Our method relies on a very naive operational task in quantum information theory, namely, the minimal error state discrimination. As a practical implication this method constitutes an economical entanglement detection scheme, which uses a less number of entangled states compared to all such existing schemes. Moreover, the method applies to class of generalized probability theories containing quantum theory as a special example.
Abstract In the standard Bell experiment, two parties perform local projective measurements on a shared pair of entangled qubits to generate nonlocal correlations. However, these measurements completely destroy the entanglement, rendering the post-measurement state unable for subsequent use. For a long time, it was believed that only unsharp measurements can be used to share quantum correlations. Remarkably, recent research has shown that classical randomness assisted projective measurements are sufficient for sharing nonlocality (Steffinlongo and Tavakoli 2022 Phys. Rev. Lett. 129 230402). Here, by stochastically combining no more than two different projective measurement strategies, we report an experimental observation of double Clauser–Horne–Shimony–Holt inequality violations with two measurements in a sequence made on each pair of maximally and partially entangled polarization photons. Our results reveal that the double violation achieved by partially entangled states can be 11 standard deviations larger than that achieved by maximally entangled ones. Our scheme eliminates the requirement for entanglement assistance in previous unsharp-measurement-based sharing schemes, making it experimentally easier. Our work provides possibilities for sharing other types of quantum correlations in various physical systems with projective measurements.
Abstract Imperfect detection efficiency remains one of the major obstacles in achieving loophole-free Bell tests over long distances. At the same time, the challenge of establishing a common reference frame for measurements becomes more pronounced as the separation between parties increases. In this work, we tackle both of these issues by examining the impact of limited detection efficiency on the probability of Bell inequality violation with Haar random measurement settings. We derive analytical lower bounds on the violation probability for a two-qubit maximally entangled state, which is tight for correlation inequalities and perfect detection efficiencies. We further investigate it numerically for more qubits and settings using two detection efficiency models and an original method based on the linear programming. Beyond that, we show that the so-called typicality of Bell inequality violation, i.e. almost certain violation of local realism with sufficiently many particles or random measurement directions, holds even if the detection efficiency is limited. Our findings reveal that increasing the number of measurement settings can compensate for efficiency losses above the critical threshold. Additionally, we determine critical detection efficiencies for both two-party and three-party scenarios. For two parties, we recover previously established results, while for three parties, we derive a symmetric critical efficiency of η crit = 2 / 3 for the W and GHZ states within the binning model. In cases involving the no-detection outcome, we present a modified inequality using a pair of orthogonal observables for each party with η crit ≈ 0.7208 , which is notably less than 0.75 for the GHZ state. These results offer deeper insights into the limitations and possibilities for certifying non-locality in the presence of limited detection efficiency, shedding light on its robustness in practical Bell tests.
The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from d=2 up to d=10. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension d grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviors. Furthermore, in the latter approach the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison to maximally entangled states, so no anomaly is observed. Finally, the nonlocality of states with nonmaximal Schmidt rank is investigated.