Clauser-Horne-Shimony-Holt Research Articles

Study of quantum nonlocality by CHSH function and its extension in disordered fermions

Quantum nonlocality is an important concept in quantum physics. In this work, we study the quantum nonlocality in a fermion many-body system under quasi-periodic disorders. The Clauser-Horne-Shimony-Holt (CHSH) inequality is systematically investigated, which quantifies quantum nonlocality between two sites. We find particular behaviors of the quantifiers of quantum nonlocality around the extended and critical phase transitions in the system and further that the CHSH inequality is not broken in the globally averaged picture of the maximum value of the quantum nonlocality, but the violation probability of the CHSH inequality for two site pairs in the system becomes sufficiently finite in the critical phase and on a critical phase boundary. Further, we investigate an extension of the CHSH inequality, Mermin-Klyshko-Svetlichny (MKS) polynomials, which can characterize multipartite quantum nonlocality. We also find a similar behavior to the case of CHSH inequality. In particular, in the critical regime and on a transition point, the adjacent three-qubit MKS polynomial in a portion of the system exhibits a quantum nonlocal violation regime with a finite probability in the critical phase.

Space and Time Correlations in Quantum Histories

AbstractThe formalism of generalized quantum histories allows a symmetrical treatment of space and time correlations, by taking different traces of the same history density matrix. In this framework, the characterization of spatial and temporal entanglement is revisited. An operative protocol is presented, to map a history state into the ket of a static composite system. It is demonstrated, by examples, how the Leggett–Garg and the temporal Clauser‐Horne‐Shimony‐Holt (CHSH) inequalities can be violated in this approach.

SOS decomposition for general Bell inequalities in two qubits systems and its application to quantum randomness

Bell non-locality is closely related with device independent quantum randomness. In this paper, we present a kind of sum-of-squares (SOS) decomposition for general Bell inequalities in two qubits systems. By using the obtained SOS decomposition, we can then find the measurement operators associated with the maximal violation of considered Bell inequality. We also practice the SOS decomposition method by considering the (generalized) Clauser-Horne-Shimony-Holt (CHSH) Bell inequality, the Elegant Bell inequality, the Gisin inequality and the Chained Bell inequality as examples. The corresponding SOS decompositions and the measurement operators that cause the maximum violation values of these Bell inequalities are derived, which are consistent with previous results. We further discuss the device independent quantum randomness by using the SOS decompositions of Bell inequalities. We take the generalized CHSH inequality with the maximally entangled state and the Werner state that attaining the maximal violations as examples. Exact value or lower bound on the maximal guessing probability using the SOS decomposition are obtained. For Werner state, the lower bound can supply a much precise estimation of quantum randomness when p tends to 1.

The iteration formula of (n, 2, d) full-correlated multi-component Bell function and its applications

No matter for a scientific or technological reason, constructing Bell inequalities for multi-partite and high-dimensional systems is a significant task. The Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) inequality is expressed in the form of iteration formula and correlation functions, and is the generalization of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multi-partite cases. In the sense of detecting the nonlocality of the noisy maximally entangled states with the most white noises for the corresponding quantum system, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality, expressed in the joint probabilities form, is the high-dimensional analogue of the Clauser-Horne (CH) inequalities on joint probabilities. In the sense of detecting the nonlocality of the noisy Greenberger-Horne-Zeilinger(GHZ) states with the most white noises for the n-qudit system, it is a challenging task to construct the multi-partite analogue of the CGLMP inequality with iteration formula and correlation functions. In this paper, we generalize the multi-component correlation functions [Phys. Rev. A 71, 032 107 (2005)] for bipartite d-dimensional systems to n-partite d-dimensional ones, introduce the general full-correlated multi-component Bell function In,d , and construct the corresponding Bell inequality. By this way, we can reproduce both the CGLMP inequality and the Bell inequality in [Phys. Rev. A 71, 032 107 (2005)] for the case n = 2, and the MABK inequality for the case d = 2. Inspired by the iteration formula form of the MABK inequality, we prove that for prime d the general Bell function In,d can be reformulated by iterating two Bell functions In−1,d . As applications, for prime d, confined to the unbiased symmetric (d × 2)-port beam splitters and the noisy n-qudit GHZ states, we recover the most robust Bell inequalities for small n and d, such as for the (3, 2, 3), (4, 2, 3), (5, 2, 3), and (3, 2, 5) Bell scenarios, with the iteration formula and the most robust Bell inequalities for the (2, 2, d) scenario. This implies that the iteration formula is an efficient way of constructing the multi-partite analogues of the CGLMP inequality with correlation functions. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.

Non-Hermitian CHSH* Game with a Single Trapped-Ion Qubit

Abstract The Clauser–Horne–Shimony–Holt (CHSH) game provides a captivating illustration of the advantages of quantum strategies over classical ones. In a recent study, a variant of the CHSH game leveraging a single qubit system, referred to as the CHSH* game, has been identified. We demonstrate that this mapping relationship between these two games remains effective even for a non-unitary gate. Here we delve into the breach of Tsirelson’s bound in a non-Hermitian system, predicting changes in the upper and lower bounds of the player’s winning probability when employing quantum strategies in a single dissipative qubit system. We experimentally explore the impact of the CHSH* game on the player’s winning probability in a single trapped-ion dissipative system, demonstrating a violation of Tsirelson’s bound under the influence of parity-time ( P T ) symmetry. These results contribute to a deeper understanding of the influence of non-Hermitian systems on quantum games and the behavior of quantum systems under P T symmetry, which is crucial for designing more robust and efficient quantum protocols.

Search for Entanglement between Spatially Separated Living Systems: Experiment Design, Results, and Lessons Learned

Statistically significant violations of the Clauser–Horne–Shimony–Holt (CHSH) inequality are the “gold standard” test for quantum entanglement between spatially separated systems. Here, we report an experimental design that implements a CHSH test between bioelectric state variables for a human subject and bioelectric and/or biochemical state variables for cultured human cells in vitro. While we were unable to obtain evidence for entanglement with this design, observing only classical correlation, we report lessons learned and suggest possible avenues for future studies.

Quantum non-local correlation testing of the Werner state in non-Markovian environment

Research on whether quantum states retain quantum non-local correlation properties after evolving in non-Markovian environments has significant applications in the field of quantum information. This article presents the evolution results of the density matrix of quantum states over time in various non-Markovian environments. Specifically, we examine two types of non-Markovian phase damping environments: Random Telegraph (RT) noise and Ornstein-Uhlenbeck (OU) noise, as well as a nonMarkovian amplitude damping (AD) environment. By utilizing the Clauser-Horne-Shimony-Holt (CHSH) inequality, the quantum non-local correlation testing of the Werner state following its evolution in these nonMarkovian environments is studied. The results show significant differences in the quantum non-local correlation testing results of the Werner state after evolving in different non-Markovian environments. Notably, the Werner state displays information backflow in the RT noise environment and the AD environment, resulting in periodic oscillations in its quantum non-local correlation testing. This suggests that under certain conditions, the quantum state can transition from a state devoid of quantum non-local correlation back to a state possessing such correlation as evolution time progresses. The results also show that the Werner state exhibits information backflow phenomena in both RT noise and AD environments, leading to periodic oscillations in its quantum non-local correlation testing. Furthermore, these periods are inversely proportional to certain parameters, such as $\sqrt{\left(\frac{2 \gamma}{a}\right)^2-1}$ and $\sqrt{2 \frac{\Gamma}{\gamma}-\left(\frac{\Gamma}{\gamma}\right)^2}$. In contrast, in the OU noise environment, no information backflow is observed, leading to a decrease in the value of the quantum non-local correlation test with increasing evolution time. In most AD and OU noise environments, there exists a specific maximum evolution time $\gamma t_{\max }$ within which successful quantum non-local correlation testing can be conducted. This maximum evolution time $\gamma t_{\max }$ varies nonlinearly with increasing fidelity and inversely with increasing $\Gamma / \gamma$ parameters. In comparison, the maximum evolution time for successful quantum non-local correlation testing in the OU noise environment surpasses that in the AD environment under the same conditions, indicating that the AD environment exerts a more pronounced weakening effect on the quantum non-local correlation properties of the Werner state.

The relationship between the degree of violation of Bell-CHSH inequality and measurement uncertainty in two classes of spin squeezing mechanisms

In solid-state systems, spin squeezing states have acquired much interest in quantum information processing since they are not only essential from a basic physics standpoint but also have many applications, notably reducing noise in interferometry. This paper addresses the study of quantum nonlocality as examined by the violation of the Bell and Clauser–Horne–Shimony–Holt (CHSH) inequality in two classes of spin squeezing states and how this relates to the quantum-memory-assisted entropic uncertainty relation (QMA–EUR). The spin squeezing states in the two classes are systematically generated by two twisting nonlinear interactions. These separately describe two spin-[Formula: see text] interaction mechanisms that undergo one-axis twisting (OAT) and two-axis countertwisting (TAC). We investigate each spin squeezing mechanism in a thermal bath ([Formula: see text]) with a transverse magnetic field. Our main results indicate that the degree of violation of the Bell-CHSH inequality and the entropic uncertainty relation (EUR) is typically anticorrelated in both mechanisms. We demonstrate that the spin squeezing states in the OAT mechanism violate the Bell-CHSH inequality and thus exhibit Bell nonlocality only at low temperatures ([Formula: see text]), whereas in the TAC mechanism, increasing the strength of the spin squeezing interaction leads to a violation of the Bell-CHSH inequality even at finite temperatures. We have also shown that there is always an expansion of the EUR and a decrease in the degree of violation of the Bell-CHSH inequality in the high-temperature regime. In addition, the transverse magnetic field exhibits different behavior under the considered temperature regimes in the two classes of spin squeezing states. Other results will also be discussed.

Exploring quantum correlations of two-qubit Heisenberg chain model influenced by magnetic dipole–dipole, magnetic field, and a symmetric cross interaction

The concepts of concurrence, 3-steerability, and Clauser–Horne–Shimony–Holt (CHSH) inequality are employed to investigate the environmental impacts on the quantum correlations of the two-spin-1/2 Heisenberg XYZ chain model. In particular, the effects of a homogeneous magnetic field, symmetric cross interaction, and dipole–dipole interaction on the entanglement, degree of steerability, and non-locality are discussed. Results show that the entanglement and steering phenomena are bounded by the non-locality for both positive and negative values of dipole–dipole coupling. It has also been observed that higher symmetric cross-interaction strengthens the quantum correlations, whereas the homogeneous magnetic field weakens the quantumness of the system. The findings indicate that a magnetic field normal to the magnetic dipole yields better quantum correlations than when they are parallel.

Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles

In this paper, we investigate the critical efficiency of detectors to observe Bell nonlocality using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles, such as hyperentangled states, and the product of Pauli measurements. It is known that in a Clauser-Horne-Shimony-Holt (CHSH) Bell test the symmetric detection efficiency of $82.84\%$ can be tolerated for the two-qubit maximally entangled state. We beat this enigmatic threshold by entangling two particles with multiple degrees of freedom. The obtained upper bounds of the symmetric detection efficiency thresholds are $80.86\%$, $73.99\%$ and $69.29\%$ for two, three and four copies of the two-qubit maximally entangled state, respectively. The number of measurements and outcomes in the respective cases are 4, 8 and 16. To find the improved thresholds, we use large-scale convex optimization tools, which allows us to significantly go beyond state-of-the-art results. The proof is exact up to three copies, while for four copies it is due to reliable numerical computations. Specifically, we used linear programming to obtain the two-copy threshold and the corresponding Bell inequality, and convex optimization based on Gilbert's algorithm for three and four copies of the two-qubit state. We show analytically that the symmetric detection efficiency threshold decays exponentially with the number of copies of the two-qubit state. Our techniques can also be applied to more general Bell nonlocality scenarios with more than two parties.

Interplay of nonlocality and incompatibility breaking qubit channels

Incompatibility and nonlocality are not only of foundational interest but also act as important resources for quantum information theory. In CHSH (Clauser--Horne--Shimony--Holt) scenario, the incompatiblity of a pair of observables is known to be equivalent to Bell nonlocality. Here, we investigate these notions in the context of qubit channels. The Bell-CHSH inequality has a greater perspective -- compared to any genuine tri-partite nonlocality scenario -- while determining about the interplay between nonlocality breaking qubit channels and incompatibility breaking qubit channels. In Bell CHSH scenario, we prove that if the conjugate of a channel is incompatibility breaking, then the channel is itself nonlocality breaking and the converse also holds provided the channel is unital. However, this equivalence is not straightforwardly generalized to multi-partite systems, due to the absence of an equivalence relation between incompatiblity and nonlocality in the multi-partite scenario. We investigate this relation in tripartite scenario by considering some well known states like GHZ and W states and using the notion of Mermin and Svetlichny nonlocality. By subjecting the parties in question to unital qubit channels, we identify the range of state and channel parameters for which incompatiblity coexists with nonlocality. Further, we identify the set of unital qubit channels that is Mermin/Svetlichny nonlocality breaking \emph{irrespective} of the input state.

Synchronous Observation of Bell Nonlocality and State-Dependent Contextuality.

Bell nonlocality and Kochen-Specker contextuality are two remarkable nonclassical features of quantum theory, related to strong correlations between outcomes of measurements performed on quantum systems. Both phenomena can be witnessed by the violation of certain inequalities, the simplest and most important of which are the Clauser-Horne-Shimony-Holt (CHSH) and the Klyachko-Can-Binicioğlu-Shumovski (KCBS), for Bell nonlocality and Kochen-Specker contextuality, respectively. It has been shown that, using the most common interpretation of Bell scenarios, quantum systems cannot violate both inequalities concomitantly, thus suggesting a monogamous relation between the two phenomena. In this Letter, we show that the joint consideration of the CHSH and KCBS inequalities naturally calls for the so-called generalized Bell scenarios, which, contrary to the previous results, allows for joint violation of them. In fact, this result is not a special feature of such inequalities: We provide very strong evidence that there is no monogamy between nonlocality and contextuality in any scenario where both phenomena can be observed. We also implement a photonic experiment to test the synchronous violation of both CHSH and KCBS inequalities. Our results agree with the theoretical predictions, thereby providing experimental proof of the coexistence of Bell nonlocality and contextuality in the simplest scenario, and lead to novel possibilities where both concepts could be jointly employed for quantum information processing protocols.

Cosmic primordial density fluctuations and Bell inequalities

The temperature measurements, $T$, of the perturbed cosmic microwave background, performed by the cosmic background explorer satellite (COBE), are considered. A dichotomist function, $f=\ifmmode\pm\else\textpm\fi{}1$, is defined such that $f=+1$ if $\ensuremath{\delta}T>0$ and $f=\ensuremath{-}1$ if $\ensuremath{\delta}T<0$, where $\ensuremath{\delta}T$ stands for the fluctuation of $T$. Then, it is assumed that behind the appearance of these fluctuations there is local realism. Under this assumption, some specific Clauser-Horne-Shimony-Holt (CHSH) inequalities are proved for these fluctuation temperatures measured by COBE in the different sky directions. The calculation of these inequalities from the actual temperature measurements shows that these inequalities are not violated. This result cannot be anticipated by calculating the commutators of the cosmic density quantum operators. This must be remarked here since, in the case of a system of two entangled spin $\frac{1}{2}$ particles, its CHSH inequalities violation can be inferred from the nonvanishing value of the corresponding spin measurement commutators. The above nonviolation of the observed cosmic CHSH inequalities is compatible with the existence of local realism behind the cosmic measurement results. Nevertheless, assuming again local realism, some new cosmic CHSH inequalities can be derived for the case of the WMAP measurements whose accuracy is better than the one of the above considered COBE measurements. More specifically, in the WMAP case, some significant cross correlations between the temperature and polarization maps are detected, and the new cosmic CHSH inequalities are the ones built with these cross correlations. Now, the occasional violation of these CHSH inequalities would mean the failure of the assumed local realism in accordance with the quantum origin of the primordial temperature and polarization fluctuations in the framework of standard inflation.

Remarks on the Clauser-Horne-Shimony-Holt inequality in relativistic quantum field theory

We present an investigation of the Clauser-Horne-Shimony-Holt (CHSH) inequality within a relativistic quantum field theory model built up with a pair of free massive scalar fields $({\ensuremath{\varphi}}_{A},{\ensuremath{\varphi}}_{B})$ where, as is customary, the indices $(A,B)$ refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A CHSH-type correlator is constructed and evaluated in the Fock vacuum by means of canonical quantization. Although the observed violation of the CHSH inequality turns out to be rather small as compared to Tsirelson's bound of quantum mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as the Higgs models, for which local Becchi-Rouet-Stora-Tyutin (BRST) invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to BRST invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant BRST environment.

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.

A matter-wave Rarity–Tapster interferometer to demonstrate non-locality

We present an experimentally viable approach to demonstrate quantum non-locality in a matter-wave system via a Rarity–Tapster interferometer using two s-wave scattering halos generated by colliding helium Bose–Einstein condensates. The theoretical basis for this method is discussed, and its suitability is experimentally quantified. As a proof of concept, we demonstrate an interferometric visibility of V=0.42(9), corresponding to a maximum CSHS-Bell parameter of S=1.1(1), for the Clauser–Horne–Shimony–Holt (CHSH) version of the Bell inequality, between atoms separated by sim 4 correlation lengths. This constitutes a significant step toward a demonstration of a Bell inequality violation for motional degrees of freedom of massive particles and possible measurements of quantum effects in a gravitationally sensitive system.

Tight Analytic Bound on the Trade-Off between Device-Independent Randomness and Nonlocality.

Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given nonlocal correlations is a key quantity of interest. Here, we derive tight analytic bounds on the maximum certifiable randomness as a function of the nonlocality as expressed using the Clauser-Horne-Shimony-Holt (CHSH) value. We show that for every CHSH value greater than the local value (2) and up to 3sqrt[3]/2≈2.598 there exist quantum correlations with that CHSH value that certify a maximal two bits of global randomness. Beyond this CHSH value the maximum certifiable randomness drops. We give a second family of Bell inequalities for CHSH values above 3sqrt[3]/2, and show that they certify the maximum possible randomness for the given CHSH value. Our work hence provides an achievable upper bound on the amount of randomness that can be certified for any CHSH value. We illustrate the robustness of our results, and how they could be used to improve randomness generation rates in practice, using a Werner state noise model.

Quantum steering as resource of quantum teleportation

Quantum steering, which lies between Bell nonlocality and entanglement, is a characteristic quantum correlation. Its application in quantum information tasks is worth pursuing. In this paper, we investigate the application of quantum steering as resource of quantum teleportation. It is found that two-qubit states violating the three-setting linear steering inequality are always useful for teleportation. Some steerable states which obey the Bell--Clauser-Horne-Shimony-Holt (CHSH) inequality could be used for teleportation. Two-qubit states that are optimal for quantum teleportation for a fixed amount of steering are present. The optimal states achieve the maximal average fidelity and also exhibit zero fidelity deviation.

Detecting entanglement between modes of light

We consider a subgroup of unitary transformations on a mode of light induced by a Mach-Zehnder Interferometer and an algebra of observables describing a photon-number detector preceded by the interferometer. We explore the uncertainty principles between such observables and their usefulness in performing a Bell-like experiment to show the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality under the physical assumption that the detector distinguishes between only zero photons and a nonzero number of photons. We show the local settings of the interferometer which lead to a maximal violation of the CHSH inequality.

Entanglement witnessing by arbitrarily many independent observers recycling a local quantum shared state

We investigate the scenario where an observer, Alice, shares a two-qubit state with an arbitrary number of observers, Bobs, via sequentially and independently recycling the qubit in possession of the first Bob. It is known that there exist entangled states which can be used to have an arbitrarily long sequence of Bobs who can violate the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. We show that there exist entangled states that do not violate the Bell inequality and whose entanglement can be detected by an arbitrary number of Bobs by suitably choosing the entanglement witness operator and the unsharp measurement settings by the Bobs. This proves that the set of states that can be used to witness entanglement sequentially is larger than those that can witness sequential violation of local realism. There exist, therefore, two-party quantum correlations that are Bell "classical", but whose entanglement "nonclassicality" can be witnessed sequentially and independently by an arbitrarily large number of observers at one end of the shared state with the single observer at the other end.