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.
We study the relation between distillability of multipartite states and violation of Bell's inequality. We prove that there exist multipartite bound entangled states (i.e. non-separable, non-distillable states) that violate a multipartite Bell inequality. This implies that (i) violation of Bell's inequality is not a sufficient condition for distillability and (ii) some bound entangled states cannot be described by a local hidden variable model.
We construct ($d\times d$)-dimensional bound entangled states, which violate, for any $d>2$, a bipartite Bell inequality introduced in this paper. We conjecture that the proposed class of Bell inequalities acts as a dimension witness for bound entangled states: For any $d>2$ there exists a Bell inequality from this class that can be violated with bound entangled states only if their Hilbert space dimension is at least $d\times d$. Numerics supports this conjecture up to $d=8$.
We report Bell violations with classical light prepared in both entangled and non-entangled polarization-path, binary states. Our results show that violations of constraints such as the Bell-Clauser-Horn-Shimony-Holt inequality do not necessarily falsify local-realism. Correlations in the realm of classical statistical optics, which are not of the Bell type, may lead to Bell violations.
The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in Quantum theory is an intriguing one in this regard. Here, we make significant progress in answering this question, by showing that every tight Bell inequality is violated by 'Almost Quantum' correlations, a semi-definite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multi-o…
It is shown that the model introduced by Cabello et al. to criticize the Franson experiment suffers from the same weakness as the previous model introduced by Aerts et al. with the same purpose. It is also shown why we can assume with confidence that the Franson experiment does violate local realism.
As a consequence of having a positive partial transpose, bound entangled states lack many of the properties otherwise associated with entanglement. It is therefore interesting to identify properties that distinguish bound entangled states from separable states. In this paper, it is shown that some bound entangled states violate a non-symmetric class of local uncertainty relations (quant-ph/0212090). This result indicates that the asymmetry of non-classical correlations may be a characteristic feature of bound entanglement.
We show that a single Bell's inequality with two dichotomic observables for each observer, which is originated from Hardy's nonlocality proof without inequalities, is violated by all entangled pure states of a given number of particles, each of which may have a different number of energy levels. Thus Gisin's theorem is proved in its most general form from which it follows that for pure states Bell's nonlocality and quantum entanglement are equivalent.
First, we present a Bell type inequality for n qubits, assuming that m out of the n qubits are independent. Quantum mechanics violates this inequality by a ratio that increases exponentially with m. Hence an experiment on n qubits violating of this inequality sets a lower bound on the number m of entangled qubits. Next, we propose a definition of maximally entangled states of n qubits. For this purpose we study 5 different criteria. Four of these criteria are found compatible. For any number n of qubits, they determine an orthogonal basis consisting of maximally entangled states generalizing t…
It has been believed that statistical inequality such as Bell inequality should be modified once measurement independence (MI), the assumption that observers can freely choose measurement settings without changing the probability distribution of hidden variables, is relaxed. However, we show that there exists the possibility that Bell inequality is still valid even if MI is relaxed. MI is only a sufficient condition to derive Bell inequality when both determinism and setting independence, usually called as local realism, are satisfied. We thus propose a new condition necessary and sufficient f…
Maximally entangled states should maximally violate the Bell inequality. In this paper, it is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local unitary transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of …
A proof of Bell's theorem without inequalities for two maximally entangled particles is proposed using the technique of quantum teleportation. It follows Hardy's arguments for a non-maximally entangled state with the help of two auxiliary particles without correlation. The present proof can be tested by measurements with 100% probability.
In Kaszlikowski [Phys. Rev. Lett. {\bf 85}, 4418 (2000)], it has been shown numerically that the violation of local realism for two maximally entangled $N$-dimensional ($3 \leq N$) quantum objects is stronger than for two maximally entangled qubits and grows with $N$. In this paper we present the analytical proof of this fact for N=3.
The worst violation of Bell's inequality for $n$ qbits is of size $2^{\frac{n-1}{2}}$ and it is obtained by a specific operator acting on a specific state. We show, to the contrary, that for a vast majority of Bell operators the worst violation is bounded by $O((n\log n)^{1/2})$, below experimental detection. With respect to the extremal operators, introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)], we show that a large majority of them have a norm bounded by $O(n^{1/2})$.
The spectral decomposition of all (n,2,2) Bell operators ($2^{2^n}$ in number, $n \ge 2$), as introduced by Werner and Wolf, is done. Its implications on the characterization of Bell operators as probes of entanglement are considered in detail.
We present a local-hidden-variable model for positive-operator-valued measurements (an LHVPOV model) on a class of entangled generalized Werner states, thus demonstrating that such measurements do not always violate a Bell-type inequality. We also show that, in general, if the state $ρ'$ can be obtained from $ρ$ with certainty by local quantum operations without classical communication then an LHVPOV model for the state $ρ$ implies the existence of such a model for $ρ'$.
We investigate how relativity influences the spin correlation of entangled fermions measured by moving detectors. In particular, we show that the Clauser-Horne-Shimony-Holt Bell inequality is not violated by quantum mechanics when the left and right spin detectors move fast enough.
The nature of quantum correlations in networks featuring independent sources of entanglement remains poorly understood. Here, focusing on the simplest network of entanglement swapping, we start a systematic characterization of the set of quantum states leading to violation of the so-called "bilocality" inequality. First, we show that all possible pairs of entangled pure states can violate the inequality. Next, we derive a general criterion for violation for arbitrary pairs of mixed two-qubit states. Notably, this reveals a strong connection between the CHSH Bell inequality and the bilocality i…
Polarization-entangled photon pair generation based on two scalar scattering processes of the vector four photon scattering has been demonstrated experimentally in high nonlinear microstructure fiber with birefringence. By controlling the pump polarization state, polarization-entangled Bell states can be realized. It is provides a simple way to realize efficient and compact fiber based polarization-entangled photon pair sources.
A silicon quantum photonic circuit was proposed and demonstrated as an integrated quantum light source for telecom band polarization entangled Bell state generation and dynamical manipulation. Biphoton states were firstly generated in four silicon waveguides by spontaneous four wave mixing. They were transformed to polarization entangled Bell states through on-chip quantum interference and quantum superposition, and then coupled to optical fibers. The property of polarization entanglement in generated photon pairs was demonstrated by two-photon interferences under two non-orthogonal polarizati…
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.
Contrary to the usual assumption that the experimental preparation of pure entangled states can be described by mixed states due to white noise, a more realistic description for polarization-entangled states produced by parametric down-conversion is that they are mixed states due to decoherence in a preferred polarization basis. This distinction between white and colored noise is crucial when we look for maximal violations of Bell's inequalities for two-qubit and two-qutrit entangled states. We find that violations of Bell's inequalities with realistic noise for polarization-entangled photons …
Kar's recent proof showing that a maximally entangled state of two spin-1/2 particles gives the largest violation of a Bell inequality is extended to N spin-1/2 particles (with N greater than or equal to 3). In particular, it is shown that all the states yielding a direct contradiction with the assumption of local realism do generally consist of a superposition of maximally entangled states.
We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed and lead to a violation of a Clauser-Horne Bell inequality for any finite photon number N. We present also an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such states. We test other Bell-type inequalities. Some are violated by a constant amount for any N.
We investigate two-particle phase-space distributions in classical mechanics characterized by a well-defined value of the total angular momentum. We construct phase-space averages of observables related to the projection of the particles' angular momenta along axes with different orientations. It is shown that for certain observables, the correlation function violates Bell's inequality. The key to the violation resides in choosing observables impeding the realization of the counterfactual event that plays a prominent role in the derivation of the inequalities. This situation can have statistic…
In this Letter we show that in relativistic regime maximally entangled state of two spin-1/2 particles not only gives maximal violation of the Bell-CHSH inequality but also gives the largest violation attainable for any pairs of four spin observables that are noncommuting for both systems. Also we extend our results to three spin-1/2 particles. We obtain the largest eigenvalue of Bell operator and show that this value is equal to expectation value of Bell operator on GHZ state.
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.
We have performed a Bell-type test for energy-time entangled qutrits. A method of inferring the Bell violation in terms of an associated interference visibility is derived. Using this scheme we obtained a Bell value of $2.784 \pm 0.023$, representing a violation of $34 σ$ above the limit for local variables. The scheme has been developed for use at telecom wavelengths and using proven long distance quantum communication architecture to optimize the utility of this high dimensional entanglement resource.
The robustness of Bell's inequality (in CHSH form) violation by entangled state in the simultaneous presence of colored and white noise in the system is considered. A twophoton polarization state is modeled by twoparameter density matrix. Setting parameter values one can vary the relative fraction of pure entangled Bell's state as well as white and colored noise fractions. Bell's operator-parameter dependence analysis is made. Computational results are compared with experimental data [quant-ph/0511265] and with results computed using a oneparameter density matrix [doi: 10.1103/PhysRevA.72.0521…
An efficient and economical scheme is proposed for the perfect quantum teleportation of n-qubit non-maximally entangled state of generalized Bell-type. A Bell state is used as the quantum channel in the proposed scheme. It is also shown that the controlled teleportation of this n-qubit state can be achieved by using a GHZ state or a GHZ-like state as quantum channel. The proposed schemes are economical because for the perfect and controlled teleportation of n-qubit non-maximally entangled state of generalized Bell-type we only need a Bell state and a tripartite entangled state respectively. It…
Bell inequalities have traditionally been used to demonstrate that quantum theory is nonlocal, in the sense that there exist correlations generated from composite quantum states that cannot be explained by means of local hidden variables. With the advent of device-independent quantum information protocols, Bell inequalities have gained an additional role as certificates of relevant quantum properties. In this work we consider the problem of designing Bell inequalities that are tailored to detect maximally entangled states. We introduce a class of Bell inequalities valid for an arbitrary number…
We explore the consequences of denying the "emptiness of paths not taken," EPNT, a premise of Bernstein, Greenberger, Horne, and Zeilinger in their paper titled, Bell theorem without inequalities. Carrying out the negation of EPNT leads to the concept of a "shadow stream." Streams are essentially particle implementations of the path amplitudes in Feynman's formulation of quantum mechanics, resulting in a simple and consistent extension of the standard postulates for quantum mechanics. Following up on the negation of EPNT using shadow streams, we analyze the experimental outcomes in a standard …
Braid theories are applied to quantum computation processes, where to each crossing in the Braid diagram a unitary Yang-Baxter operator R is associated, representing either a Braiding matrix or a universal quantum gate. By operating with Braid operators on the computational basis of n-QUBITS states, orthonormal entangled states are obtained, referred here as general Bell states. The 3-QUBITS Bell states are explicitly developed and the present methods are generalized to any n-QUBITS system. The quantum properties of the general Bell states are analyzed and these properties are related to concu…
Any Bell test consists of a sequence of measurements on a quantum state in space-like separated regions. Thus, a state is better than others for a Bell test when, for the optimal measurements and the same number of trials, the probability of existence of a local model for the observed outcomes is smaller. The maximization over states and measurements defines the optimal nonlocality proof. Numerical results show that the required optimal state does not have to be maximally entangled.
Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein|Podolsky|Rosen (EPR) spin correlation is calculated and shown to coincide with the quantum mechanical one for the 1/2-spin particles.
The so-called quasi-Bell entangled coherent states in a thermal environment are studied. In the analysis, we assume thermal noise affects only one of the two modes of each state. First the matrix representation of the density operators of the quasi-Bell entangled coherent states in a thermal environment is derived. Secondly we investigate the entanglement property of one of the quasi-Bell entangled coherent states with thermal noise. At that time a lower bound of the entanglement of formation for the state is computed. Thirdly the minimax discrimination problem for two cases of the binary set …
Network Bell experiments give rise to a form of quantum nonlocality that conceptually goes beyond Bell's theorem. We investigate here the simplest network, known as the bilocality scenario. We depart from the typical use of the Bell State Measurement in the network central node and instead introduce a family of symmetric iso-entangled measurement bases that generalise the so-called Elegant Joint Measurement. This leads us to report noise-tolerant quantum correlations that elude bilocal variable models. Inspired by these quantum correlations, we introduce network Bell inequalities for the biloc…
We study GHZ-type and W-type three-mode entangled coherent states. Both the types of entangled coherent states violate Mermin's version of the Bell inequality with threshold photon detection (i.e., without photon counting). Such an experiment can be performed using linear optics elements and threshold detectors with significant Bell violations for GHZ-type entangled coherent states. However, to demonstrate Bell-type inequality violations for W-type entangled coherent states, additional nonlinear interactions are needed. We also propose an optical scheme to generate W-type entangled coherent st…
It is shown that the correlations between two qubits selected from a trio prepared in a W state violate the Clauser-Horne-Shimony-Holt inequality more than the correlations between two qubits in any quantum state. Such a violation beyond Cirel'son's bound is smaller than the one achieved by two qubits selected from a trio in a Greenberger-Horne-Zeilinger state [A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)]. However, it has the advantage that all local observers can know from their own measurements whether their qubits belongs or not to the selected pair.
We define a family of binary outcome $n$-party $m\leq n$ settings per party Bell inequalities whose members require the least detection efficiency for their violation among all known inequalities of the same type. This gives upper bounds for the minimum value of the critical efficiency --- below which no violation is possible --- achievable for such inequalities. For $m=2$, our family reduces to the one given by Larsson and Semitecolos in 2001. For $m>2$, a gap remains between these bounds and the best lower bounds. The violating state near the threshold efficiency always approaches a product …
We review and compare Bell-state measurement and quantum teleportation schemes using linear optics with three different types of resources, i.e., two-photon pairs, entangled coherent states and hybrid entangled states. Remarkably, perfect teleportation with linear optics is possible in principle based on a hybrid approach that combines two-photon pairs and entangled coherent states. It turns out that hybrid approach allows to perform Bell-state measurement and local Pauli operations in a deterministic way, while it requires photon number resolving measurements.
Bipartite Bell inequalities can be simultaneously violated by two different pairs of observers when weak measurements and signaling is employed. Here we experimentally demonstrate the violation of two simultaneous CHSH inequalities by exploiting a two-photon polarization maximally entangled state. Our results demonstrate that large double violation is experimentally achievable. Our demonstration may have impact for Quantum Key Distribution or certification of Quantum Random Number generators based on weak measurements.
Long-distance Bell-type experiments are presented. The different experimental challenges and their solutions in order to maintain the strong quantum correlations between energy-time entangled photons over more than 10 km are reported and the results analyzed from the point of view of tests of fundamental physics as well as from the more applied side of quantum communication, specially quantum key distribution. Tests using more than one analyzer on each side are also presented.
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…
Previous work on Bell's inequality realised in the laboratory has used entangled photons. Here we describe how entangled atoms can violate Bell's inequality, and how these violations can be measured with a very high detection efficiency. We first discuss a simple scheme based on two-level atoms inside a cavity to prepare the entangled state. We then discuss a scheme using four-level atoms, which requires a parameter regime much easier to access experimentally using current technology. As opposed to other schemes, our proposal relies on the presence of finite decay rates and its implementation …
Entangled pure-states, Werner-states and generalized mixed-states of any structure, spanning a 2x2 Hilbert space are created by a novel high-brilliance universal source of polarization-entangled photon pairs. The violation of a Bell inequality has been tested for the first time with a pure-state, indeed a conceptually relevant ideal condition, and with Werner-states. The generalized ''maximally entangled mixed states'' (MEMS) were also synthetized for the first time and their exotic properties investigated by means of a quantum tomographic technique.
Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative …
Non-classical states of light field have been exploited to provide marvellous results in quantum information science. Effectiveness of nonclassical states depends on whether physical parameter as signal is continuous or digital. Here we present an investigation on the potential of quasi Bell state of entangled coherent states in quantum reading of the classical digital memory which was pioneered by Pirandola. This is a typical example of discrete quantum discrimination. We show that the quasi Bell state gives the error free performance in the quantum reading that cannot be obtained by any clas…
All the states of N qubits can be classified into N-1 entanglement classes from 2-entangled to N-entangled (fully entangled) states. Each class of entangled states is characterized by an entanglement index that depends on the partition of N. The larger the entanglement index of an state, the more entangled or the less separable is the state in the sense that a larger maximal violation of Bell's inequality is attainable for this class of state.
We suggest to test the premise of ``macroscopic local realism'' which is sufficient to derive Bell inequalities when measurements of photon number are only accurate to an uncertainty of order $n$ photons, where $n$ is macroscopic. Macroscopic local realism is only sufficient to imply, in the context of the original Einstein-Podolsky-Rosen argument, fuzzy ``elements of reality'' which have a macroscopic indeterminacy. We show therefore how the violation of local realism in the presence of macroscopic uncertainties implies the failure of ``macroscopic local realism''. Quantum states violating th…
We use polarization-entangled photon pairs to demonstrate quantum nonlocality in an experiment suitable for advanced undergraduates. The photons are produced by spontaneous parametric downconversion using a violet diode laser and two nonlinear crystals. The polarization state of the photons is tunable. Using an entangled state analogous to that described in the Einstein-Podolsky-Rosen ``paradox,'' we demonstrate strong polarization correlations of the entanged photons. Bell's idea of a hidden variable theory is presented by way of an example and compared to the quantum prediction. A test of th…
Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Though Bell nearly showed a violation using the CHSH inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he doesn't give a quantitative estimation of the maximum achievable violation for the wavefunction he considers. In this article, we show how his strategy can be reimagined using the transv…
We experimentally study the violation of the CGLMP inequality for entangled 2-qubit and 2-qutrit states with different degrees of entanglement using numerically optimized measurement settings. The qudits are encoded and manipulated in the frequency spectrum of broadband energy-time entangled photons by taking into account a spatial light modulator. The latter allows to discretize the spectrum into bins. By controlling each frequency bin individually, the generation of maximally and non-maximally entangled qutrits is verified through quantum state tomography.
Bell inequalities and nonlocality have been widely studied in one-dimensional quantum systems. As a kind of quantum correlation, it is expected that bipartite nonlocaity should be present in quantum systems, just as bipartite entanglement does. Surprisingly, for various models, two-qubit states do not violate Bell inequalities, i.e., they are local. Recently, it is realized that the results are related to the monogamy trade-off obeyed by bipartite Bell correlations, thus it is believed that for general translation invariant systems, two-qubit states should not violate the Bell inequality(Olive…
We have developed a scheme to generate, control, transmit and measure entangled photonic qutrits (two photons each of dimension d = 3). A Bell test of this source has previously been reported elsewhere [1], therefore, here we focus on how the control of the system is realized. Motivated by these results, we outline how the scheme can be used for two specific quantum protocols, namely key distribution and coin tossing and discuss some of their advantages and disadvantages.
This note is a reply to M. Navascués' claim that "all entangled states violate Leggett's crypto-nonlocality" [arXiv:1303.5124v2]. I argue that such a conclusion can only be reached if one introduces additional assumptions that further restrict Leggett's notion of "crypto-nonlocality". If a contrario one sticks only to Leggett's original axioms, there exist entangled states whose correlations are always compatible with Leggett's crypto-nonlocality---which is thus a genuinely different concept from quantum separability. I clarify in this note the relation between these two notions, together also…
We experimentally demonstrate that violations of Bell's inequalities for two-photon polarization-entangled states with colored noise are extremely robust, whereas this is not the case for states with white noise. Controlling the amount of noise by using the timing compensation scheme introduced by Kim et al. [Phys. Rev. A 67, 010301(R) (2003)], we have observed violations even for states with very high noise, in excellent agrement with the predictions of Cabello et al. [Phys. Rev. A 72, 052112 (2005)].
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.
