In this paper, we characterize the maximal violation of Ardehali's inequality of $n$ qubits by showing that GHZ's states and the states obtained from them by local unitary transformations are the unique states that maximally violate the Ardehali's inequalities. This concludes that Ardehali's inequalities can be used to characterize maximally entangled states of $n$ qubits, as the same as Mermin's and Bell-Klyshko's inequalities.
Any pure entangled state of two particles violates a Bell inequality for two-particle correlation functions (Gisin's theorem). We show that there exist pure entangled N>2 qubit states that do not violate any Bell inequality for N particle correlation functions for experiments involving two dichotomic observables per local measuring station. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description.
Entanglement is often verified by a violation of an inequality like a Bell inequality or an entanglement witness. Considerable effort has been devoted to the optimization of such inequalities in order to obtain a high violation. We demonstrate theoretically and experimentally that such an optimization does not necessarily lead to a better entanglement test, if the statistical error is taken into account. Theoretically, we show for different error models that reducing the violation of an inequality can improve the significance. Experimentally, we observe this phenomenon in a four-photon experim…
A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of $\sqrt 2$, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection…
A new interpretation offers a consistent conceptual basis for nonrelativistic quantum mechanics. The violation of Bell's inequality is explained by maintaining realism, inductive inference and Einstein separability.
We consider the problem of certifying binary observables based on a Bell inequality violation alone, a task known as self-testing of measurements. We introduce a family of commutation-based measures, which encode all the distinct arrangements of two projective observables on a qubit. These quantities by construction take into account the usual limitations of self-testing and since they are "weighted" by the (reduced) state, they automatically deal with rank-deficient reduced density matrices. We show that these measures can be estimated from the observed Bell violation in several scenarios and…
We consider violation of CHSH inequality for states before and after entanglement swapping. We present a pair of initial states which do not violate CHSH inequality however the final state violates CHSH inequality for some results of Bell measurements performed in order to swap entanglement.
A new interpretation offers a consistent conceptual basis for nonrelativistic quantum mechanics. The Einstein-Podolsky-Rosen (EPR) paradox is solved and the violation of Bell's inequality is explained by maintaining realism, inductive inference and Einstein separability.
We investigate a classical statistical model and show that Mermin's version of a Bell inequality is violated. We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for measurement processes for quantum systems.
Bell's inequality has been derived several times from quite different basic assumptions, which imply different conclusions. This resulted into widespread confusion regarding the exact implications of the experimental violations of the inequality. In this article, the structures of Bell's and of Peres' derivations are analyzed, and the title question is explicitly answered.
The consistency test and error estimation for the data concerning recently reported violation of Bell inequality for Josephson phase qubits are presented in details. It is pointed out that the deviation of the Bell signal from the classical limit might still be challenged.
We show that a recent observation by Yan leads to a method to experimentally test whether a higher-than-quantum violation of the Clauser-Horne-Shimony-Holt Bell inequality is possible (assuming that the sum of probabilities of pairwise exclusive propositions cannot exceed 1). The test requires reaching the maximum quantum violation of a noncontextuality inequality involving sequences of three compatible measurements on a five-dimensional quantum system.
An inequality is deduced from Einstein's locality and a supplementary assumption. This inequality defines an experiment which can actually be performed with present technology to test local realism. Quantum mechanics violate this inequality a factor of 1.5. In contrast, quantum mechanics violates previous inequalities (for example, Clauser-Horne-Shimony-Holt inequality of 1969, Freedman-Clauser inequality of 1972, Clauser-Horne inequality of 1974) by a factor of $\sqrt 2$. Thus the magnitude of violation of the inequality derived in this paper is approximately $20.7%$ larger than the magnitude…
We consider violations of a Clauser-Horne-Shimony-Holt-type steering inequality for quantum states of systems of indistinguishable particles restricted by a particle-number-superselection rule. We check for violations in non-interacting Bose-Einstein condensate and N00N states, by using two copies of the states for bypassing the superselection rule. The superselection rule prevents the states from maximally violating the steering inequality. But the steering inequality violations are higher than Bell inequality violations for the same states. This in particular implies, in certain cases, that …
We present new bell inequalities for arbitrary dimensional bipartite quantum systems. The maximal violation of the inequalities is computed. The Bell inequality is capable of detecting quantum entanglement of both pure and mixed quantum states more effectively.
We study a recently proposed Einstein-Podolsky-Rosen steering inequality [arXiv- 1412.8178 (2014)]. Analogous to Clauser-Horne-Shimony-Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., 2 parties, 2 measurements per party and 2 outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article, using an equivalence between measurement incompatibility (non joint measurability) and steering, we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation am…
Mermin's inequality is the generalization of the Bell-CHSH inequality for three qubit states. The violation of the Mermin inequality guarantees the fact that there exists quantum non-locality either between two or three qubits in a three qubit system. In the absence of an analytical result to this effect, in order to check for the violation of Mermin's inequality one has to perform a numerical optimization procedure for even three qubit pure states. Here we derive an analytical formula for the maximum value of the expectation of the Mermin operator in terms of eigenvalues of symmetric matrices…
We prove that for every Bell's inequality and for a broad class of protocols, there always exists a multi-party communication complexity problem, for which the protocol assisted by states which violate the inequality is more efficient than any classical protocol. Moreover, for that advantage Bell's inequality violation is a necessary and sufficient criterion. Thus, violation of Bell's inequalities has a significance beyond that of a non-optimal-witness of non-separability.
We have observed a violation of the Cauchy-Schwarz inequality in the macroscopic regime by more than 8 standard deviations. The violation has been obtained while filtering out only the low frequency noise of the quantum-correlated beams that results from the technical noise of the laser used to generate them. We use bright intensity-difference squeezed beams produced by four-wave mixing as the source of the correlated fields. We also demonstrate that squeezing does not necessarily imply a violation of the Cauchy-Schwarz inequality.
The standard Bell inequality experiments test for violation of local realism by repeatedly making local measurements on individual copies of an entangled quantum state. Here we investigate the possibility of increasing the violation of a Bell inequality by making collective measurements. We show that nonlocality of bipartite pure entangled states, quantified by their maximal violation of the Bell-Clauser-Horne inequality, can always be enhanced by collective measurements, even without communication between the parties. For mixed states we also show that collective measurements can increase the…
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)].
We report the measurement of a Bell inequality violation with a single atom and a single photon prepared in a probabilistic entangled state. This is the first demonstration of such a violation with particles of different species. The entanglement characterization of this hybrid system may also be useful in quantum information applications.
In contrast to Bell's inequalities which test the correlations between multiple spatially separated systems, the Leggett-Garg inequalities test the temporal correlations between measurements of a single system. We experimentally demonstrate the violation of the Leggett-Garg inequality in a classical optical system using only the polarization degree-of-freedom of a laser beam. Our results show maximal violations of the Leggett-Garg inequality.
Correlations for the Bell gedankenexperiment are constructed using probabilities given by quantum mechanics, and nonlocal information. They satisfy Bell's inequality and exhibit spatial non stationarity in angle. Correlations for three successive local spin measurements on one particle are computed as well. These correlations also exhibit non stationarity, and satisfy the Bell inequality. In both cases, the mistaken assumption that the underlying process is wide-sense-stationary in angle results in violation of Bell's inequality. These results directly challenge the wide-spread belief that vio…
We identify what ideal correlated photon number states are to required to maximize the discrepancy between local realism and quantum mechanics when a quadrature homodyne phase measurement is used. Various Bell inequality tests are considered.
By calculating entanglement measures and quantum violation of Bell-type inequality, we reveal the relationship between entanglement measure and the amount of quantum violation for a family of four-qubit entangled states. It has been demonstrated that the Bell-type inequality is completely violated by these four-qubit entangled states. The plot of entanglement measure as a function of the expectation value of Bell operator shows that entanglement measure first decreases and then increases smoothly with increasing quantum violation.
Multi-component correlation functions are developed by utilizing d-outcome measurements. Based on the multi-component correlation functions, we propose a Bell inequality for bipartite d-dimensional systems. Violation of the Bell inequality for continuous variable (CV) systems is investigated. The violation of the original Einstein-Podolsky-Rosen state can exceed the Cirel'son bound, the maximal violation is 2.96981. For finite value of squeezing parameter, violation strength of CV states increases with dimension d. Numerical results show that the violation strength of CV states with finite squ…
I study the dynamics of the violation of the CHSH inequality for two qubits interacting with a common zero-temperature non-Markovian environment. I demonstrate sudden violation of the inequality for two qubits initially prepared in a factorized state. Due to the strong coupling between the qubits and the reservoir, the dynamics is characterized by numerous sharp revivals. Furthermore I focus on a more realistic physical system in which the spontaneous emission for the qubits is taken into account. When including spontaneous emission even for small decay parameters, revivals in the violation ar…
We find that Bell's inequality can be significantly violated (up to Tsirelson's bound) with two-mode entangled coherent states using only homodyne measurements. This requires Kerr nonlinear interactions for local operations on the entangled coherent states. Our example is a demonstration of Bell-inequality violations using classical measurements. We conclude that entangled coherent states with coherent amplitudes as small as 0.842 are sufficient to produce such violations.
Processes with an indefinite causal structure may violate a causal inequality, which quantifies quantum correlations that arise from a lack of causal order. In this paper, we show that when the inequalities are analysed with a Gaussian-localised field theoretic definition of particles and labs, the causal indeterminacy of the fields themselves allows a causal inequality to be violated within the causal structure of Minkowski spacetime. We quantify the violation of the inequality and determine the optimal ordering of observers.
The correct laws of thermodynamics, either classical or quantum without probabilities of statistical quantum mechanics are valid for all systems, large or small, including systems of one spin or one particle only. So no violations can occur in nanoscale electric circuits.
In quantum information theory, entropic inequalities act as the necessary and sufficient conditions to noncontextuality. Here, we first experimentally demonstrate the violation of the entropic noncontextual inequality in a four-level quantum system, by using the five-qubit IBM quantum computer. The experimental result disproves the existence of a local realist model.
Using joint Shannon entropy, we propose an inequality for a four-level system, which is satisfied in any non-contextual realist hidden variable model. We show that this entropic inequality is violated by quantum mechanics for a range of entangled, as well as product states. Our inequality can be experimentally tested using the existing technology.
We investigate the behavior of the maximal violations of the CHSH inequality and V$\grave{e}$rtesi's inequality under the local filtering operations. An analytical method has been presented for general two-qubit systems to compute the maximal violation of the CHSH inequality and the lower bound of the maximal violation of V$\acute{e}$rtesi's inequality over the local filtering operations. We show by examples that there exist quantum states whose non-locality can be revealed after local filtering operation by the V$\acute{e}$rtesi's inequality instead of the CHSH inequality.
We show that the maximum quantum violation of the Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality is exactly the maximum value satisfying the following principle: The sum of probabilities of pairwise exclusive events cannot exceed 1. We call this principle "global exclusivity," since its power shows up when it is applied to global events resulting from enlarged scenarios in which the events in the inequality are considered jointly with other events. We identify scenarios in which this principle singles out quantum contextuality, and show that a recent proof excluding nonlocal boxes follows…
Bell-inequality violation and entanglement, measured by Wootters' concurrence and negativity, of two qubits initially in Werner or Werner-like states coupled to thermal reservoirs are analyzed within the master equation approach. It is shown how this simple decoherence process leads to generation of states manifesting the relativity of two-qubit entanglement measures.
We consider maximal violations of the Leggett-Garg inequality, obtained by maximising over all possible measurement operators, in relation to non-unitary aspects of the system dynamics. We model the action of an environment on a qubit in terms of generic quantum channels and relate the maximal value of the Leggett-Garg correlator to the channel parameters. We focus on unital channels, and hence on decoherence. In certain important cases, exact relations between the channel parameters and maximal violations can be found. Moreover, we demonstrate the existence of distinct thresholds for the chan…
In the case of bipartite two qubits systems, we derive the analytical expression of bound of Bell operator for any given pure state. Our result not only manifest some properties of Bell inequality, for example which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also give the explicit values of maximal violation for any pure state. Finally we point out that for two qubits systems there is no mixed state which can produce maximal violation of Bell inequality.
We report the experimental observations of Bell Inequality Violations (BIV) in entangled photons causally separated by a rotating mirror. A Foucault mirror gating geometry is used to causally isolate the entangled photon source and detectors. We report an observed BIV of CHSH-$S = 2.30 \pm 0.07 > 2.00$. This result rules out theories that explain correlations with traveling communication between source and detectors.
Entangled coherent states for multiple bosonic modes, also referred to as multimode cat states, not only are of fundamental interest, but also have practical applications. The nonclassical correlation among these modes is well characterized by the violation of the Mermin-Klyshko inequality. We here study Mermin-Klyshko inequality violations for such multi-mode entangled states with rotated quantum-number parity operators. Our results show that the Mermin-Klyshko signal obtained with these operators can approach the maximal value even when the average quantum number in each mode is only 1, and …
We numerically demonstrate that "mode-entangled states" based on the transverse modes of classical optical fields in multimode waveguides violate Bell's inequality. Numerically simulating the correlation measurement scheme of Bell's inequality, we obtain the normalized correlation functions of the intensity fluctuations for the two entangled classical fields. By using the correlation functions, the maximum violations of Bell's inequality are obtained. This implies that the two classical fields in the mode-entangled states, although spatially separated, present a nonlocal correlation.
The correlations between two qubits belonging to a three-qubit system can violate the Clauser-Horne-Shimony-Holt-Bell inequality beyond Cirel'son's bound [A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)]. We experimentally demonstrate such a violation by 7 standard deviations by using a three-photon polarization-entangled Greenberger-Horne-Zeilinger state produced by Type-II spontaneous parametric down-conversion. In addition, using part of our results, we obtain a violation of the Mermin inequality by 39 standard deviations.
We demonstrate the first experimental violation of a spin-1 Bell inequality. The spin-1 inequality is a calculation based on the Clauser, Horne, Shimony and Holt formalism. For entangled spin-1 particles the maximum quantum mechanical prediction is 2.552 as opposed to a maximum of 2, predicted using local hidden variables. We obtained an experimental value of 2.27 $\pm 0.02$ using the four-photon state generated by pulsed, type-II, stimulated parametric down-conversion. This is a violation of the spin-1 Bell inequality by more than 13 standard deviations.
Violation of a Bell inequality guarantees the existence of quantum correlations in a quantum state. A pure bipartite quantum state, having nonvanishing quantum correlation, always violates a Bell inequality. Such correspondence is absent for multipartite pure quantum states. For a shared multipartite quantum state, we establish a connection between the monogamy of Bell inequality violation and genuine multi-site entanglement as well as monogamy-based multiparty quantum correlation measures. We find that generalized Greenberger-Horne-Zeilinger states and another single-parameter family states w…
Bell's inequality violation (BIQV) for correlations of polarization is studied for a {\it product} state of two two-mode squeezed vacuum (TMSV) states. The violation allowed is shown to attain its maximal limit for all values of the squeezing parameter, $ζ$. We show via an explicit example that a state whose entanglement is not maximal allow maximal BIQV. The Wigner function of the state is non negative and the average value of either polarization is nil.
We present a generalized quantum scattering circuit which can be used to perform non-invasive quantum measurements, and implement it on NMR qubits. Such a measurement is a key requirement for testing temporal non-contextual inequalities. We use this circuit to experimentally demonstrate the violation of the Peres-Mermin inequality (which is the temporal analog of a Klyachko-Can- Binicioglu-Shumovsky (KCBS) inequality), on a three-qubit NMR quantum information processor. Further, we experimentally demonstrate the violation of a transformed Bell-type inequality (the spatial equivalent of the tem…
Here we demonstrate, for the first time, violation of Bell's inequality using a triggered quantum dot photon-pair source without post-selection. Furthermore, the fidelity to the expected Bell state can be increased above 90% using temporal gating to reject photons emitted at times when collection of uncorrelated light is more probable. A direct measurement of a CHSH Bell inequality is made showing a clear violation, highlighting that a quantum dot entangled photon source is suitable for communication exploiting non-local quantum correlations.
The best upper bound for the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of $\pm 1$ valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendieck's constant ($\mathcal{K}_{G}\approx 1.73\pm0.06$) as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four $\pm 1$ valued observables on both sites of a correlation experiment and show that the quantum violation approaching 1.58. Moreover, we estimate the maximal quantum violation of …
Einstein's locality is invoked to derive a correlation inequality. In the case of ideal experiments, this inequality is equivalent to Bell's original inequality of 1965 which, as is well known, is violated by a maximum factor of 1.5. The crucial point is that even in the case of real experiments where polarizers and detectors are non-ideal, the present inequality is violated by a factor of 1.5, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of $\sqrt 2$. The larger magnitude of violation can be o…
We present a much simplified version of the CGLMP inequality for the 2 x 2 x d Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of d the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2 x 2 x inf sce…
We study the violations of Bell inequality for thermal states of qubits in a multi-qubit Heisenberg model as a function of temperature and external magnetic fields. Unlike the behaviors of the entanglement the violation can not be obtained by increasing the temperature or the magnetic field. The threshold temperatures of the violation are found be less than that of the entanglement. We also consider a realistic cavity-QED model which is a special case of the mutli-qubit Heisenberg model.
Optical beams with topological singularities have a Schmidt decomposition. Hence, they display features typically associated with bipartite quantum systems; in particular, these classical beams can exhibit entanglement. This classical entanglement can be quantified by a Bell inequality formulated in terms of Wigner functions. We experimentally demonstrate the violation of this inequality for Laguerre-Gauss (LG) beams and confirm that the violation increases with increasing orbital angular momentum. Our measurements yield negativity of the Wigner function at the origin for $\LG_{10}$ beams, whe…
For two qubits belonging to Alice and Bob, we derive an approach to setup the bound of Bell operator in the condition that Alice and Bob continue to perform local vertical measurements. For pure states we find that if the entanglement of the two qubits is less than 0.2644 (measured with von Neumann entropy) the violation of the Bell inequality will never be realized, and only when the entanglement is equal to 1 the maximal violation ($2\sqrt{2}$) can occur. For specific form of mixed states, we prove that the bound of the Bell inequality depends on the concurrence. Only when the concurrence is…
The model of a quantum-optical device for a conditional preparation of entangled states from input mixed states is presented. It is demonstrated that even thermal or pseudo-thermal radiation can be entangled in such a way, that Bell-inequalities are violated.
In this paper we present the results of numerical calculations about the minimal value of detection efficiency for violating the Clauser - Horne inequality for qutrits. Our results show how the use of non-maximally entangled states largely improves this limit respect to maximally entangled ones. A stronger resistance to noise is also found.
Violation of a Bell-like inequality for a spin-energy entangled neutron state has been confirmed in a polarimetric experiment. The proposed inequality, in Clauser-Horne-Shimony-Holt (CHSH) formalism, relies on correlations between the spin and energy degree of freedom in a single-neutron system. The entangled states are generated utilizing a suitable combination of two radio-frequency fields in a neutron polarimeter setup. The correlation function S is determined to be 2.333+/-0.005, which violates the Bell-like CHSH inequality by more than 66 standard deviations.
A macroscopic quantum model of a two-level system (the analogue of a half-spin particle) is described. The model is employed for simulating not only the system under study, but the measurement process as well. Single- and two-particle state models of a quantum system are constructed. The Einstein-Podolsky-Rosen paradox and Bell's inequality are discussed within the framework of the model.
