Bell-Clauser-Horne-Shimony-Holt Research Articles

CHSH Bell tests for optical hybrid entanglement

Optical hybrid entanglement can be created between two qubits, one encoded in a single photon and another one in coherent states with opposite phases. It opens the path to a variety of quantum technologies, such as heterogeneous quantum networks, merging continuous- and discrete-variable encoding, and enabling the transport and interconversion of information. However, reliable characterization of the non-local nature of this quantum state is limited so far to full quantum state tomography. Here, we perform a thorough study of Clauser–Horne–Shimony–Holt Bell inequality tests, enabling practical verification of quantum nonlocality for optical hybrid entanglement. We show that a practical violation of this inequality is possible with simple photon number on/off measurements if detection efficiencies stay above 82%. Another approach, based on photon-number parity measurements, requires 94% efficiency but works well in the limit of higher photon populations. Both tests use no postselection of the measurement outcomes and they are free of the fair-sampling hypothesis. Our proposal paves the way to performing loophole-free tests using feasible experimental tasks such as coherent state interference and photon counting.

Remarks on the Bell-Clauser-Horne-Shimony-Holt inequality

We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.

Quantum sets of the multicolored-graph approach to contextuality

The CHSH inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter (CSW) came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the CSW approach, one may think that the theta body of the CHSH graph, is equal to the quantum set of the CHSH Bell inequality, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while CSW approach only demands the exclusivity structure of a non-contextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like CHSH, the colored-graph approach was introduced. Does it make any difference to think about CHSH as a Bell scenario or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph and the NC CHSH inequality by a simple graph, which is the shadow of the colored graph. In general, we have that the theta body of the colored graph is a subset of the theta body of its shadow graph in the same way that the Lov\`asz number, which corresponds to the quantum bound, of the simple graph is greater than or equal to the Lov\`asz number of the colored graph. In the case of the CHSH inequality, we have that the Lov\`asz numbers are equal. Does this accident also hold for the corresponding quantum sets? Is it true that the theta bodies are equal, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negativ. This implies that there are quantum correlations which can not be obtained under Bell restrictions.

L$\ddot {u}$der Rule, Von Neumann Rule and Cirelson’s Bound of Bell CHSH Inequality

In [PRL, 113, 050401 (2014)] the authors have shown that instead of L$\ddot{u}$der rule, if degeneracy breaking von Neumann projection rule is adopted for state reduction, the quantum value of three-time Leggett-Garg inequality can exceed it's L$\ddot{u}$ders bound. Such violation of L$\ddot{u}$ders bound may even approach algebraic maximum of the inequality in the asymptotic limit of system size. They also claim that for Clauser-Horne-Shimony-Holt (CHSH) inequality such violation of L$\ddot{u}$ders bound (known as Cirelson's bound) cannot be obtained even when the measurement is performed sequentially first by Alice followed by Bob. In this paper, we have shown that if von Neumann projection rule is used, quantum bound of CHSH inequality exceeds it's Cirelson's bound and may also reach its algebraic maximum four. This thus provide a strong objection regarding the viability of von Neumann rule as a valid state reduction rule. Further, we pointed out that the violation of Cirelson's bound occurs due to the injection of additional quantum non-locality by the act of implementing von Neumann measurement rule.

Entanglement swapping with autonomous polarization-entangled photon pairs from a warm atomic ensemble.

Entanglement swapping forms a key concept in the realization of scalable quantum networks and large-scale quantum communication. For the practical implementation of entanglement swapping, completely autonomous entanglement sources and a joint Bell-state measurement (BSM) between two independent photons are essential. Here, we experimentally demonstrate entanglement swapping between two independent polarization-entangled photon-pair sources obtained via spontaneous four-wave mixing (SFWM) in a Doppler-broadened atomic ensemble of ${^{87}}{\rm Rb}$87Rb atoms. From the joint BSM, we estimate the ${\rm S}$S parameter in the Clauser-Horne-Shimony-Holt (CHSH) form of Bell's inequality and confirm the violation of the Bell-CHSH inequality by ${\rm S}={2.32} \pm {0.07}$S=2.32±0.07 with 4.5 standard deviations. We believe that this work is an important step toward realizing practical scalable quantum networks.

Relaxed Bell inequalities with arbitrary measurement dependence for each observer

Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local "hidden variables" that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as "measurement independence" (also referred to as "freedom of choice" or "settings independence"). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters $M_1$ and $M_2$, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\hat{M}_1$ and $\hat{M}_2$, deriving a modified Bell inequality for each description. These "relaxed" inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.

Quantum entanglement with Freedman's inequality

The assumption of local realism imposes constraints, such as Bell inequalities, on quantities obtained from measurements. In recent years, various tests of local realism have gained popularity in undergraduate laboratories, giving students the exciting opportunity to experimentally contradict this philosophical assumption. The standard test of the CHSH (Clauser–Horne–Shimony–Holt) Bell inequality requires 16 measurements, whereas a test of Freedman's inequality requires only three measurements. The calculations required to test Freedman's inequality are correspondingly simpler and the theory is less abstract. We suggest that students may benefit from testing Freedman's inequality before proceeding to the CHSH inequality and other more complicated experiments. Our measured data violated Freedman's inequality by more than six standard deviations.

Correlations, Nonlocality and Usefulness of an Efficient Class of Two-Qubit Mixed Entangled States

Abstract We establish an analytical relation between the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality and weak measurement strengths under noisy conditions. We show that the analytical results obtained in this article are of utmost importance for proposing a new class of two-qubit mixed states for quantum information processing. Our analysis further shows that the states proposed here are better resources for quantum information in comparison to other two-qubit mixed entangled states.

Measurement incompatibility does not give rise to Bell violation in general

In the case of a pair of two-outcome measurements, incompatibility is equivalent to Bell nonlocality. Indeed, any pair of incompatible two-outcome measurements can violate the Clauser–Horne–Shimony–Holt Bell inequality, which has been proven by Wolf et al (2009 Phys. Rev. Lett. 103 230402). In the case of more than two measurements the equivalence between incompatibility and Bell nonlocality is still an open problem, though partial results have recently been obtained. Here we show that the equivalence breaks for a special choice of three measurements. In particular, we present a set of three incompatible two-outcome measurements, such that if Alice measures this set, independent of the set of measurements chosen by Bob and the state shared by them, the resulting statistics cannot violate any Bell inequality. On the other hand, complementing the above result, we exhibit a set of N measurements for any that is -wise compatible, nevertheless it gives rise to Bell violation.

Effects of measurement dependence on generalized Clauser-Horne-Shimony-Holt Bell test in the single-run and multiple-run scenarios

Bell tests, as primitive tools to detect nonlocality in bipartite systems, rely on an assumption, i.e., measurement independence. Since ensuring measurement independence in a practical Bell test is difficult, it is crucial to explore the effects of relaxing this assumption. Recently, in the simplest Clauser-Horne-Shimony-Holt Bell (CHSH-Bell) test, Koh et al. [Phys. Rev. Lett. 109, 160404 (2012)] built the relation among measurement dependence, guessing probability, and the maximum value of CHSH-Bell correlation function that an adversary (Eve) can fake. As well, Pope et al. [Phys. Rev. A 88, 032110 (2013)] and Yuan et al. [Phys. Rev. A. 91, 032111 (2015)] settled the same problem in the multiple-run scenario with the general input distribution and the factorizable one, respectively. However, pertinent results in the generalized CHSH-Bell test are still missing. Here, we study this problem and establish the relation among measurement dependence, guessing probability, and the maximum value of the generalized CHSH-Bell correlation function that Eve can fake. Furthermore, we also consider the multiple-run scenario and show the relations in both input distributions. Interestingly, compared with the simplest CHSH-Bell test, we find that it is more difficult for Eve to fake a violation in the generalized CHSH-Bell test in some special cases. We expect that our conclusions will serve as a reference for quantum information processing tasks such as quantum key distribution, randomness expansion, and other tasks in the device-independent framework.

Self-testing protocols based on the chained Bell inequalities

Self-testing is a device-independent technique based on non-local correlations whose aim is to certify the effective uniqueness of the quantum state and measurements needed to produce these correlations. It is known that the maximal violation of some Bell inequalities suffices for this purpose. However, most of the existing self-testing protocols for two devices exploit the well-known Clauser–Horne–Shimony–Holt Bell inequality or modifications of it, and always with two measurements per party. Here, we generalize the previous results by demonstrating that one can construct self-testing protocols based on the chained Bell inequalities, defined for two devices implementing an arbitrary number of two-output measurements. On the one hand, this proves that the quantum state and measurements leading to the maximal violation of the chained Bell inequality are unique. On the other hand, in the limit of a large number of measurements, our approach allows one to self-test the entire plane of measurements spanned by the Pauli matrices X and Z. Our results also imply that the chained Bell inequalities can be used to certify two bits of perfect randomness.

Noise robustness of the incompatibility of quantum measurements

The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH) Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noise-based quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson's bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.

Contrasting classical probability concepts with quantum mechanical behavior in the undergraduate laboratory

In this paper, we report an experimental activity suitable for undergraduate students that makes them question their ideas about locality and realism. The experiment here reported complements previous educational approaches to Bell inequalities, since the usually called S function, that quantifies correlations, is mapped by measuring it for different detection angles. The students themselves worked in a pre-aligned setup that allowed them to test and violate the Clauser–Horne–Shimony–Holt–Bell inequality for two distant photons entangled in polarization.

Bell-inequality tests using asymmetric entangled coherent states in asymmetric lossy environments

We study an asymmetric form of two-mode entangled coherent state (ECS), where the two local amplitudes have different values, for testing the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We find that the asymmetric ECSs have obvious advantages over the symmetric form of ECSs in testing the Bell-CHSH inequality. We further study an asymmetric strategy in distributing an ECS over a lossy environment and find that such a scheme can significantly increase violation of the inequality.

Tailored two-photon correlation and fair-sampling: a cautionary tale

We demonstrate an experimental test of the Clauser–Horne– Shimony–Holt (CHSH) Bell inequality which seemingly exhibits correlations beyond the limits imposed by quantum mechanics. Inspired by the idea of Fourier synthesis, we design analysers that measure specific superpositions of orbital angular momentum (OAM) states, such that when one analyser is rotated with respect to the other, the resulting coincidence curves are similar to a square-wave. Calculating the CHSH Bell parameter, S, from these curves result to values beyond the Tsirelson bound of . We obtain S = 3.99 ± 0.02, implying almost perfect nonlocal Popescu–Rohrlich correlations. The ‘super-quantum’ values of S is only possible in our experiment because our experiment, subtly, does not comply with fair-sampling. The way our Bell test fails fair-sampling is not immediately obvious and requires knowledge of the states being measured. Our experiment highlights the caution needed in Bell-type experiments based on measurements within high-dimensional state spaces such as that of OAM, especially in the advent of device-independent quantum protocols.

Correlation and nonlocality measures as indicators of quantum phase transitions in several critical systems

We have investigated the quantum phase transitions in the ground states of several critical systems, including transverse field Ising and XY models as well as XY with multiple spin interactions, XXZ and the collective system Lipkin–Meshkov–Glick models, by using different quantumness measures, such as entanglement of formation, quantum discord, as well as its classical counterpart, measurement-induced disturbance and the Clauser–Horne–Shimony–Holt–Bell function. Measurement-induced disturbance is found to detect the first and second order phase transitions present in these critical systems, while, surprisingly, it is found to fail to signal the infinite-order phase transition present in the XXZ model. Remarkably, the Clauser–Horne–Shimony–Holt–Bell function is found to detect all the phase transitions, even when quantum and classical correlations are zero for the relevant ground state.

On Possible Violation of the Clauser–Horne–Shimony–Holt Bell Inequality in a Classical Context

It has been shown that there is a small possibility to experimentally violate the CHSH Bell inequality in a 'classical' context. The probability of such a violation has been estimated in the framework of a classical probabilistic model in the language of a random-walk representation.

Complementary contributions of indeterminism and signaling to quantum correlations

Simple quantitative measures of indeterminism and signaling, $I$ and $S$, are defined for models of statistical correlations. It is shown that any such model satisfies a generalized Bell-type inequality, with tight upper bound $B(I,S)$. This upper bound explicitly quantifies the complementary contributions required from indeterminism and signaling, for modeling any given violation of the standard Bell--Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. For example, all models of the maximum quantum violation must either assign no more than $80%$ probability of occurrence to some underlying event, and/or allow a nonlocal change of at least $60%$ in an underlying marginal probability of one observer in response to a change in measurement setting by a distant observer. The results yield a corresponding complementarity relation between the numbers of local random bits and nonlocal signaling bits required to model a given violation. A stronger relation is conjectured for simulations of singlet states. Signaling appears to be a useful resource only if a ``gap'' condition is satisfied, corresponding to being able to nonlocally flip some underlying marginal probability $p$ to its complementary value $1\ensuremath{-}p$.

A Method for Systematic Construction of Bell-Like Inequalities and a Proposal of a New Type of Test

The Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, which is proven in the context of the local hidden variable theory, has been used as a test to reveal failure of the hidden variable theory and to prove validity of the quantum theory. We note that violation of the BCHSH inequality is caused by noncommutativity of quantum observables and find a systematic method for constructing generalizations of the BCHSH inequality. This method is applied for inventing a new quantity which is defined in a two-qubit system and satisfies a new type of inequality. This provides a fair test of the quantum theory. Remaining problems are also discussed.

Experimental Bell-inequality violation without the postselection loophole

We report on an experimental violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality using energy-time entangled photons. The experiment is not free of the locality and detection loopholes, but is the first violation of the Bell-CHSH inequality using energy-time entangled photons which is free of the postselection loophole described by Aerts et al. [Phys. Rev. Lett. 83, 2872 (1999)].