Clauser-Horne-Shimony-Holt Research Articles

Robust violation of a multipartite Bell inequality from the perspective of a single-system game

Recently, Fan et al. [Mod. Phys. Lett. A 36, 2150223 (2021)] presented a generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, to identify [Formula: see text]-qubit Greenberger–Horne–Zeilinger states. They showed an interesting phenomenon that the maximal violation of the generalized CHSH inequality is robust under some specific noises. In this work, we map the inequality to the CHSH game, and consequently to the CHSH* game in a single-qubit system. This mapping provides an explanation for the robust violations in [Formula: see text]-qubit systems. Namely, the robust violations, resulting from the degeneracy of the generalized CHSH operators correspond to the symmetry of the maximally entangled two-qubit states and the identity transformation in the single-qubit game. This explanation enables us to exactly demonstrate that the degeneracy is [Formula: see text].

Experimental simulation of a monogamy relation between quantum contextuality and nonlocality on an NMR quantum processor

The quantum no-disturbance principle imposes a fundamental monogamy relation on quantum contextuality and quantum nonlocality. We experimentally demonstrated the simulation of this monogamy relation on a ququart-qubit system using three NMR qubits. We constructed experimental tests of the violation of the Klyachko-Can-Binicioglu-Shumovski (KCBS) inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality. We measured the average CHSH and KCBS values at the single point where the quantum boundary meets the no-disturbance boundary, at which point the tradeoff inequality becomes an equality. The tradeoff relation is always satisfied, implying that the violation of either the KCBS or the CHSH inequality automatically forbids the violation of the other.

Generalising the Horodecki criterion to nonprojective qubit observables

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 two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing necessary and sufficient conditions for arbitrary 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 show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements. Finally, we use the CHSH inequality to obtain a simple necessary condition for the compatibility of two qubit observables.

Estimation of the CHSH Parameter Using HOM Interference

The Clauser–Horne–Shimony–Holt (CHSH) experiment is an essential test of nonlocality in quantum mechanics and can be used to validate the principle of entanglement. In addition to verifying entanglement, the measurable CHSH parameter can also be used to gauge the quality of the entanglement present in a system. The measurement of Hong–Ou–Mandel (HOM) interference is another important fundamental experiment in quantum optics that measures the indistinguishability of a pair of photons. In this article, we demonstrate how the results of a HOM interference experiment, a relatively simple experiment, can be used to obtain an estimate for the value of the CHSH <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$S$</tex-math></inline-formula> parameter, which is a more complicated measurement. We experimentally demonstrate that the HOM interference technique is capable of providing an estimate of the value of the CHSH parameter that is within one standard deviation of measurement error when spectral impairments are present. We expect that this technique will aid in the calibration of quantum optical systems.

Quantum nonlocal test of “X” state based on geometric interpretation of CHSH inequality

Quantum nonlocal correlation is one of the important features that distinguish the quantum theory from classical theory. As a typical quantum mixed state, the study of quantum nonlocal correlation based on the “X” state is of great importance for the verification of the correctness of quantum theory and the application of quantum information theory. In this work, with the traditional Clauser-Horne-Shimony-Holt (CHSH) inequality testing for quantum nonlocal correlations, we propose a strategy for testing the quantum nonlocal correlations based on the geometric interpretation of the “X” state. By using the geometric interpretation of the “X” state, which is described by the transform of Bloch sphere, it is possible to investigate the optimal selection of measurement settings. The maximum value of CHSH inequalities can also obtained from the physical images. Finally, the range of parameters for a successful quantum nonlocal correlation testing based on the CHSH inequality for the “X” state is studied. The results show that when &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$f = 1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the “X” state will be reduced to a normal pure entangled state, and the quantum nonlocal correlation testing results are in full agreement with the traditional ones. This result proves the correctness of the geometric interpretation strategy proposed in this work. When&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$f \lt 1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, only some of the “X” states can be used for e successfully testing the quantum nonlocal correlations. It is also found that the range of fidelity values that can successfully test the quantum nonlocal correlations will be further increased by increasing the values of &lt;i&gt;r&lt;/i&gt;. In particular, when &lt;i&gt;r&lt;/i&gt; = 1, the range of fidelity value will reach a largest one (e.g. &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$f \gt 0.781$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20220445_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;). The results in this work can provide the reference for experimentally testing the quantum nonlocal correlation by using the “X” state.

Violating Bell inequality using weak coherent states

We present an experimental investigation of two-photon interference using a continuous-wave laser. We demonstrate the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality using the phase-randomized weak coherent states from a continuous-wave laser. Our implementation serves as an approach to reveal the quantum nature of a state that is considered to be a classical state.

Hidden nonmacrorealism: Reviving the Leggett-Garg inequality with stochastic operations

The Leggett-Garg inequality (LGI) distinguishes nonmacrorealistic channels from macrorealistic ones by constraining the experimental outcomes of the underlying system. In this work, we propose a class of channels which, initially, cannot violate the LGI (in the form of the temporal Bell inequality) but can violate it after the application of stochastic pre- and post- operations (SPPOs). As a proof-of-principle experiment, we demonstrate the stochastic pre- and post- operations in an amplitude-damping channel with photonic qubits. We denote the above phenomenon as hidden nonmacrorealistic channels. We also discuss the relationship between this hidden nonmacrorealistic channels (in terms of the temporal Clauser-Horne-Shimony-Holt (CHSH) inequality) and the strongly nonlocality-breaking channel, which breaks the hidden spatial CHSH nonlocality for arbitrary states. In general, if the channel satisfies hidden nonmacrorealism, it is not a strongly CHSH nonlocality-breaking channel.

Bell correlations in a split two-mode-squeezed Bose-Einstein condensate

We propose and analyze a protocol for observing a violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using two spatially separated Bose-Einstein condensates (BECs). To prepare the Bell-correlated state, spin-changing collisions are used to first prepare a two-mode squeezed BEC. This is then split into two BECs by controlling the spatial wavefunction, e.g., by modifying the trapping potential. Finally, spin-changing collisions are also exploited locally, to compensate local squeezing terms. The correlators appearing in the inequality are evaluated using three different approaches. In the first approach, correlators are estimated using normalized expectation values of number operators, in a similar way to evaluating continuous-variable Bell inequalities. An improvement to this approach is developed using the sign-binning of total spin measurements, which allows for the construction of two-outcome measurements and violations of the CHSH inequality without auxiliary assumptions. Finally, we show a third approach where maximal violations of the CH inequality can be obtained by assigning zero values to local vacua outcomes under a no-enhancement assumption. The effect of loss and imperfect detection efficiency is investigated and the observed violations are found to be robust to noise.

Investigation of the Aharonov-Bohm and Aharonov-Casher topological phases for quantum entangled states

Aharonov-Bohm (AB) and Aharonov-Casher (AC) effects are treated fully relativistically in 2+1 dimensions. The influences of the relevant geometric and topological phases on an entangled spin-1/2 system are studied. It is shown that for the AC case the correlation function of the Clauser-Horne-Shimony-Holt (CHSH) inequality for certain choices of the spin measurements depends on the AC phase explicitly.

Greenberger–Horne–Zeilinger states: Their identifications and robust violations

The [Formula: see text]-qubit Greenberger–Horne–Zeilinger (GHZ) states are the maximally entangled states of [Formula: see text] qubits, which have had many important applications in quantum information processing, such as quantum key distribution and quantum secret sharing. Thus how to distinguish the GHZ states from other quantum states becomes a significant problem. In this work, by presenting a family of the generalized Clauser–Horne–Shimony–Holt (CHSH) inequality, we show that the [Formula: see text]-qubit GHZ states can be indeed identified by the maximal violations of the generalized CHSH inequality under some specific measurement settings. The generalized CHSH inequality is simple and contains only four correlation functions for any [Formula: see text]-qubit system, thus has the merit of facilitating experimental verification. Furthermore, we present a quantum phenomenon of robust violations of the generalized CHSH inequality in which the maximal violation of Bell’s inequality can be robust under some specific noises adding to the [Formula: see text]-qubit GHZ states.

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.

Characterizing nonlocality of pure symmetric three-qubit states

We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality. We make use of the elegant parametrization in the canonical form of these states, proposed by Meill and Meyer (Phys. Rev. A, 96, 062310 (2017)) based on Majorana geometric representation. The reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state do not violate the CHSH inequality and hence they are CHSH-local. However, when Alice and Bob perform a CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. We have also shown that two different families of three-qubit pure symmetric states, consisting of two and three distinct spinors (qubits) respectively, can be distinguished based on the strength of violation in the conditional CHSH nonlocality test. Furthermore, we identify six of the 46 classes of tight Bell inequalities in the three-party, two-setting, two-outcome i.e., (3,2,2) scenario (Phys. Rev. A 94, 062121 (2016)). Among the two inequivalent families of three-qubit pure symmetric states, only the states belonging to three distinct spinor class show maximum violations of these six tight Bell inequalities.

Device-independent quantum key distribution with asymmetric CHSH inequalities

The simplest device-independent quantum key distribution protocol is based on the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality and allows two users, Alice and Bob, to generate a secret key if they observe sufficiently strong correlations. There is, however, a mismatch between the protocol, in which only one of Alice's measurements is used to generate the key, and the CHSH expression, which is symmetric with respect to Alice's two measurements. We therefore investigate the impact of using an extended family of Bell expressions where we give different weights to Alice's measurements. Using this family of asymmetric Bell expressions improves the robustness of the key distribution protocol for certain experimentally-relevant correlations. As an example, the tolerable error rate improves from 7.15% to about 7.42% for the depolarising channel. Adding random noise to Alice's key before the postprocessing pushes the threshold further to more than 8.34%. The main technical result of our work is a tight bound on the von Neumann entropy of one of Alice's measurement outcomes conditioned on a quantum eavesdropper for the family of asymmetric CHSH expressions we consider and allowing for an arbitrary amount of noise preprocessing.

Different Bell Inequalities as Probes to Detect Quantum Phase Transitions

We introduce a method which based on Bell inequalities, to study quantum phase transitions. By using the non-linear programming, we compare two different kinds of Bell inequalities, the original Bell inequality and Clauser-Horne-Shimony-Holt (CHSH) inequality. And we find that the original Bell inequality is more accurate in detecting the Bell non-locality. By defining the maximal violation of Bell inequalities, we calculate two kinds of transitions, the one is magnetic transition in the spin- $\frac {1}{2}$ XX model and the other is topological transition in the Kitaev honeycomb model. The critical points are detected successfully. Compared with traditional methods, our method requires no prior knowledge of order parameters and it is base-free.

Sharing quantum nonlocality and genuine nonlocality with independent observables

Recently the authors in [Phys. Rev. Lett. 125, 090401 (2020)] considered the following scenario: Alice and Bob each have half of a pair of entangled qubit state. Bob measures his half and then passes his part to a second Bob who measures again and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality with the single Alice. By taking the maximally entangled pure two-qubit state as an example, it has been constructively proved that arbitrarily many independent Bobs can share the nonlocality with the single Alice. Here we demonstrate that arbitrarily many independent observers can share the nonlocality of a single arbitrary dimensional bipartite entangled but not necessary two-qubit entangled state. Further, taking the generalized GHZ states as an example, we show that at most two Charlies can share the genuine nonlocality of a single generalized GHZ state with an Alice and a Bob.

Sagnac-type entangled photon source using only conventional polarization optics

We designed and implemented a novel combination of a Sagnac-interferometer with a Mach–Zehnder interferometer for a source of polarization-entangled photons. The new versatile configuration does not require multi-wavelength polarization optics, yet it performs with a good polarization quality and phase-stability over a wide wavelength range. We demonstrate the interferometer using only standard commercial optics to experimentally realize the pulsed generation of polarization-entangled photon-pairs at wavelengths of 764 nm and 1221 nm via type-I spontaneous four-wave mixing in a polarization-maintaining fiber. Polarization entanglement was verified by a polarization-correlation measurement with a visibility of 95.5% from raw coincidence counts and the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality with S = 2.70 ± 0.04. The long-term phase-stability was characterized by an Allan deviation of 8° over an integration time of about 1 h with no active phase-stabilization.

Post-Measurement Adjustment of the Coincidence Window in Quantum Optics Experiments

We report on an electronic coincidence detection circuit for quantum photonic applications implemented on a field-programmable gate array (FPGA), which records each the time separation between detection events coming from single-photon detectors. We achieve a coincidence window as narrow as 500 ps with a series of optimizations on a readily-available and affordable FPGA development board. Our implementation allows real-time visualization of coincidence measurements for multiple coincidence window widths simultaneously. To demonstrate the advantage of our high-resolution visualization, we certified the generation of polarized entangled photons by collecting data from multiple coincidence windows with minimal accidental counts, obtaining a violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality by more than 338 standard deviations. Our results have shown the applicability of our electronic design in the field of quantum information.

What Do Bell-Tests Prove? A Detailed Critique of Clauser-Horne-Shimony-Holt Including Counterexamples

Many future directions of scientific endeavors depend on quantum theory and the precise interpretation and significance of the entanglement of quantum-particles. This interpretation depends in turn on the physical meaning of so called Bell-tests that are mostly performed using entangled photons and randomly switched polarizers to measure their polarization at distant locations. This paper presents a detailed critique of the well known theory of Bell tests given by Clauser-Horne-Shimony-Holt (CHSH). It is demonstrated that several important steps of the CHSH derivations contain serious inaccuracies of the underlying physics and probability theory and even a calculus error. As a consequence, the Bell-CHSH theory cannot be used to demonstrate extreme and opposite interpretations of entanglement such as super-luminal influences or alternatively super-determinism that cast aspersions on Einstein’s concepts of locality and separability.

Bell inequalities for entangled qubits: quantitative tests of quantum character and nonlocality on quantum computers.

This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.

Various formulations of inequivalent Leggett–Garg inequalities

In two-party, two-input and two-output measurement scenario only relevant Bell’s inequality is the Clauser–Horne–Shimony–Holt (CHSH) form. They also provide the necessary and sufficient conditions (NSCs) for local realism. Any other form, such as, Clauser–Horne and Wigner forms reduce to the CHSH one. The standard Leggett–Garg inequalities, proposed for testing incompatibility between macrorealism and quantum theory, are often considered to be the temporal analog of CHSH inequalities. However, they do not provide the NSCs for macrorealism. There is thus scope of formulating new macrorealist inequalities inequivalent and stronger than the standard Leggett–Garg inequalities. In this paper, we propose two different classes of macrorealistic inequalities. A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory, and the other class of inequalities are inequivalent to the all the other formulations of Leggett–Garg inequalities both in macrorealist model and in quantum theory. The latter class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett–Garg inequalities fails to do so. We extend the formulations of inequivalent Leggett–Garg inequalities to the four-time and two-time measurement scenarios. Further, we provide a brief discussion about the alternative formulation of macrorealism known as the no-signaling in time (NSIT) conditions.