We consider quantum systems composed of $N$ qubits, and the family of all Bell's correlation inequalities for two two-valued measurements per site. We show that if a $N$-qubit state $ρ$ violates any of these inequalities, then it is at least bipartite distillable. Indeed there exists a link between the amount of Bell's inequality violation and the degree of distillability. Thus, we strengthen the interpretation of Bell's inequalities as detectors of useful entanglement.
We study the relation between violation of Bell inequalities and distillability properties of quantum states. Recently, Dür has shown that there are some multiparticle bound entangled states, non-separable and non-distillable, that violate a Bell inequality. We prove that for all the states violating this inequality there exist at least one splitting of the parties into two groups such that some pure-state entanglement can be distilled, obtaining a connection between Bell inequalities and bipartite distillable entanglement.
The relation between Bell inequalities with two two-outcome measurements per site and distillability is analyzed in systems of an arbitrary number of quantum bits. We observe that the violation of any of these inequalities by a quantum state implies that pure-state entanglement can be distilled from it. The corresponding distillation protocol may require that some of the parties join into several groups. We show that there exists a link between the amount of the Bell inequality violation and the size of the groups they have to form for distillation. Thus, a strong violation is always sufficien…
A multipartite quantum state violates a Bell inequality asymptotically if, after jointly processing by general local operations an arbitrarily large number of copies of it, the result violates the inequality. In the bipartite case we show that asymptotic violation of the CHSH-inequality is equivalent to distillability. Hence, bound entangled states do not violate it. In the multipartite case we consider the complete set of full-correlation Bell inequalities with two dichotomic observables per site, also called WWZB-inequalities. We show that asymptotic violation of any of these inequalities by…
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.
Entanglement and nonlocality are both fundamental aspects of quantum theory, and play a prominent role in quantum information science. The exact relation between entanglement and nonlocality is however still poorly understood. Here we make progress in this direction by showing that, contrary to what previous work suggested, quantum nonlocality does not imply entanglement distillability. Specifically, we present analytically a 3-qubit entangled state that is separable along any bipartition. This implies that no bipartite entanglement can be distilled from this state, which is thus fully bound e…
Quantum entanglement plays a central role in many areas of physics, from quantum information science to many-body systems. In order to grasp the essence of this phenomenon, it is fundamental to understand how different manifestations of entanglement relate to each other. In 1999, Peres conjectured that Bell nonlocality is equivalent to distillability of entanglement. The intuition of Peres was that the non-classicality of an entangled state, as witnessed via Bell inequality violation, implies that pure entanglement can be distilled from this state, hence making it useful for most quantum infor…
We present a set of Bell inequalities which are sufficient and necessary for separability of general pure multipartite quantum states in arbitrary dimensions. The relations between Bell inequalities and distillability are also studied. We show that any quantum states that violate one of these Bell inequalities are distillable.
The information capacities and ``distillability'' of a quantum channel are studied in the presence of auxiliary resources. These include prior entanglement shared between the sender and receiver and free classical bits of forward and backward communication. Inequalities and trade-off curves are derived. In particular an alternative proof is given that in the absence of feedback and shared entanglement, forward classical communication does not increase the quantum capacity of a channel.
It is shown that even if the linear entropy of mixed two-qubit state is not smaller then 0.457, Bell - CHSH inequalities can be violated. This contradicts the result obtained in the paper of E. Santos [1].
We present a set of Bell inequalities for multiqubit quantum systems. These Bell inequalities are shown to be able to detect multiqubit entanglement better than previous Bell inequalities such as Werner-Wolf-Zukowski- Brukner ones. Computable formulas are presented for calculating the maximal violations of these Bell inequalities for any multiqubit states.
A technique, which we call homogenization, is applied to transform CH-type Bell inequalities, which contain lower order correlations, into CHSH-type Bell inequalities, which are defined for highest order correlation functions. A homogenization leads to inequalities involving more settings, that is a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: a homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example we obtain $3\times3\times3$ CHSH-type Bell inequalities b…
We discuss general Bell inequalities for bipartite and multipartite systems, emphasizing the connection with convex geometry on the mathematical side, and the communication aspects on the physical side. Known results on families of generalized Bell inequalities are summarized. We investigate maximal violations of Bell inequalities as well as states not violating (certain) Bell inequalities. Finally, we discuss the relation between Bell inequality violations and entanglement properties currently discussed in quantum information theory.
As shown by Pitowsky, the Bell inequalities are related to certain classes of probabilistic inequalities dealt with by George Boole, back in the 1850s. Here a short presentation of this relationship is given. Consequently, the Bell inequalities can be obtained without any assumptions of physical nature, and merely through mathematical argument.
We derive Bell inequalities for graph states by generalizing the approach proposed by Ardehali [Phys. Rev. A 46, 5375 (1992)] for Greenberger-Horne-Zeilinger (GHZ) states. Using this method, we demonstrate that Bell inequalities with nonstabilizer observables are often superior to the optimal GHZ-Mermin-type (or stabilizer-type) Bell inequalities.
We present a prescription for obtaining Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give examples of some families of such inequalities. The inequalities are violated by certain classes of states for which all standard Bell's inequalities with two measurement settings per observer are satisfied.
We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. The tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.
We present generic Bell inequalities for multipartite multi-dimensional systems. The inequalities that any local realistic theories must obey are violated by quantum mechanics for even-dimensional multipartite systems. A large set of variants are shown to naturally emerge from the generic Bell inequalities. We discuss particular variants of Bell inequalities, that are violated for all the systems including odd-dimensional systems.
Multipartite Bell-type inequalities are derived for general systems. They involve up to eight observables with arbitrary spectra on each site. These inequalities are closely related to the algebras of quaternions and octonions.
We overview series of multiqubit Bell's inequalities which apply to correlation functions. We present conditions that quantum states must satisfy to violate such inequalities.
We derive entropic Bell inequalities from considering entropy Venn diagrams. These entropic inequalities, akin to the Braunstein-Caves inequalities, are violated for a quantum mechanical Einstein-Podolsky-Rosen pair, which implies that the conditional entropies of Bell variables must be negative in this case. This suggests that the satisfaction of entropic Bell inequalities is equivalent to the non-negativity of conditional entropies as a necessary condition for separability.
Some temporal Bell inequalities are deduced under the assumption of realism and perfect correlation. No locality condition is needed. When the system is macroscopic, the perfect correlation assumption substitutes the noninvasive measurability hypothesis advanteousgely. The new inequalities are violated quantically. This violation is clearly more severe than the similar violation in the case of ordinary Bell inequalities. Some microscopic and macroscopic situations in which these inequalities could be tested are considered.
Since John S. Bell demonstrated the interest of studying linear combinations of probabilities in relation with the EPR paradox in 1964, Bell inequalities have lead to numerous developments. Unfortunately, the description of Bell inequalities is subject to several degeneracies, which make any exchange of information about them unnecessarily hard. Here, we analyze these degeneracies and propose a decomposition for Bell-like inequalities based on a set of reference expressions which is not affected by them. These reference expressions set a common ground for comparing Bell inequalities. We provid…
We present an analysis of the structure of Bell inequalities, mainly for the case of N qubits with two observables each. We show that these inequalities are related to Hadamard matrices and define Bell polynomials (in one variable) as an additional tool. With these aids we raise several conditions the coefficients of Bell inequalities must satisfy, and recursively generate the whole set of inequalities starting from N=1. Moreover, we prove some characteristic features of this set, such as that most of the inequalities contain all expectation values under consideration. Finally, we show how the…
Finding all Bell inequalities for a given number of parties, measurement settings, and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-…
Bell inequalities for position measurements are derived using the bits of the binary expansion of position-measurement results. Violations of these inequalities are obtained from the output state of the Non-degenerate Optical Parametric Amplifier.
We propose a variation of Bell inequalities for continuous variables that employs the Wigner function and Weyl symbols of operators in phase space. We present examples of Bell inequality violation which beat Cirel'son's bound.
The connection between quantum optical nonclassicality and the violation of Bell's inequalities is explored. Bell type inequalities for the electromagnetic field are formulated for general states(arbitrary number or photons, pure or mixed) of quantised radiation and their violation is connected to other nonclassical properties of the field. Classical states are shown to obey these inequalities and for the family of centered Gaussian states the direct connection between violation of Bell-type inequalities and squeezing is established.
The derivation of Bell inequalities in terms of quantum statistical (thermodynamic) entropies is considered. Inequalities of the Wigner form are derived but shown to be extremely limiting in their applicability due to the nature of the density matrices involved. This also helps to identify a limitation in the Cerf-Adami inequalities.
Bounds on the norm of quantum operators associated with classical Bell-type inequalities can be derived from their maximal eigenvalues. This quantitative method enables detailed predictions of the maximal violations of Bell-type inequalities.
A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequ…
Quantum nonlocality, one of the most important features of quantum mechanics, is normally connected in experiments with the violation of Bell-Clauser-Horne (Bell-CH) inequalities. We propose effective methods for the rearrangement and linear inequality to prove a large variety of Bell-CH inequalities. We also derive a set of Bell-CH inequalities by using these methods which can be violated in some quantum entangled states.
We present a set of Bell inequalities that gives rise to a finer classification of the entanglement for tripartite systems. These inequalities distinguish three possible bi-separable entanglements for three-qubit states. The three Bell operators we employed constitute an external sphere of the separable cube.
I review the relation of the Bell inequalities - characteristic of (classical) probabilities defined on Boolean logics - with noncontextual and local hidden variables theories of quantum mechanics and with quantum information.
A derivation of the full set of Bell inequalities involving correlation functions, for two parties, with binary observables, and three possible local settings. The procedure can be extended straightforwardly to multiparty correlations.
Which is the simplest logical structure for which there is quantum nonlocality? We show that there are only three bipartite Bell inequalities with quantum violation associated with the simplest graph of relationships of exclusivity with a quantum-classical gap. These are the most elementary logical Bell inequalities. We show that the quantum violation of some well-known Bell inequalities is related to them. We test the three Bell inequalities with pairs of polarization-entangled photons and report violations in good agreement with the quantum predictions. Unlike other experiments testing nonco…
Relations between teleportation, Bell's inequalities and inseparability are investigated. It is shown that any mixed two spin-$1\over2$ state which violates the Bell-CHSH inequality is useful for teleportation. The result is extended to any Bell's inequalities constructed of the expectation values of products of spin operators. It is also shown that there exist inseparable states which are not useful for teleportation within the standard scheme.
A method for construction of the multipartite Clauser-Horne-Shimony-Holt (CHSH) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. A unified framework to establish all kinds of CHSH-type Bell inequalities by increasing step by step the number of observers is given. As an application, compact Bell inequalities, for eight observers, involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus is experimentally friendly in mul…
We derive tight Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give a necessary and sufficient condition for a general quantum state to violate the new inequalities. The inequalities are violated by some classes of states, for which all standard Bell's inequalities with two measurement settings per observer are satisfied.
We investigate the Bell inequalities derived from the graph states with violations detectable even with the presence of noises, which generalizes the idea of error-correcting Bell inequalities [Phys. Rev. Lett. 101, 080501 (2008)]. Firstly we construct a family of valid Bell inequalities tolerating arbitrary $t$-qubit errors involving $3(t+1)$ qubits, e.g., 6 qubits suffice to tolerate single qubit errors. Secondly we construct also a single-error-tolerating Bell inequality with a violation that increases exponentially with the number of qubits. Exhaustive computer search for optimal error-tol…
We examine the use of noiseless subsystems for quantum information processing between two parties who do not share a common reference frame. In particular we focus on Bell inequalities in curved spaces and outline a theoretical procedure to test a Bell inequality, demonstrating the wide applicability of noiseless subsystems.
We show that all proofs of Bell-type inequalities, as discussed in Bell's well known book and as claimed to be relevant to Einstein-Podolsky-Rosen type experiments, come to a halt when Einstein-local time and setting dependent instrument parameters are included.
Within the Dempster-Schafer theory of evidence a non-Kolmogorovian kind of epistemic uncertainty arises, which is encoded using multi-valued maps. We analyse the possible implications such non-Kolmogorovian epistemic uncertainty may have for Bell-type inequalities relating to the Einstein-Podolsky-Rosen-Bohm (EPRB) thought experiment. Our analysis leads to a notion of contextuality concerning complexes of physical measurement conditions. The use of multi-valued maps reveals an implicit link between this contextuality and counterfactual outcomes, and results in a formulation wherein the states …
We propose a method to generate analytical quantum Bell inequalities based on the principle of Macroscopic Locality. By imposing locality over binary processings of virtual macroscopic intensities, we establish a correspondence between Bell inequalities and quantum Bell inequalities in bipartite scenarios with dichotomic observables. We discuss how to improve the latter approximation and how to extend our ideas to scenarios with more than two outcomes per setting.
We present a method to derive explicit forms of tight correlation function Bell inequalities for three systems and dichotomic observables, which involve three settings for each observer. We also give sufficient conditions for quantum predictions to satisfy the new inequalities.
We study Bell scenarios with binary outcomes supplemented by one bit of classical communication. We develop a method to find facet inequalities for such scenarios even when direct facet enumeration is not possible, or at least difficult. Using this method, we partially solve the scenario where Alice and Bob choose between three inputs, finding a total of 668 inequivalent facet inequalities (with respect to relabelings of inputs and outputs). We also show that some of these inequalities are constructed from the facet inequalities found in scenarios without communication, the well known Bell ine…
In most Bell tests, the measurement settings are specially chosen so that the maximal quantum violations of the Bell inequalities can be detected, or at least, the violations are strong enough to be observed. Such choices can usually associate the corresponding Bell operators to a kind of effective observables, called pseudo Pauli operators, providing us a more intuitive understanding of Bell nonlocality in some sense. Based on that, a more general quantum-to-classical approach for the constructions of Bell inequalities is developed. Using this approach, one can not only derive several kinds o…
A set of Bell inequalities classifying the quantum entanglement of four-qubit states is presented. These inequalities involve only two measurement settings per observer and can characterize fully separable, bi-separable and tri-separable quantum states. In addition, a quadratic inequality of the Bell operators for four-qubit systems is derived.
Many issues combine for consideration when speaking of Bell's Inequalities: nonlocality, realism, hidden variables, incompatible measures, wave function collapse, other. Each of these issues then may be viewed from several viewpoints: historical, theoretical, physical, experimental, statistical, communicational, cryptographical, and mathematical. From the mathematical viewpoint, much of the Bell theory is ``just geometry''.
Local systems may appear to violate Bell's inequalities if they are observed through suitable filters. The nonlocality leading to violation is outside the system and comprises the observer comparing the outcomes of the typical two wing Bell experiment. An example based on a well known gedanken experiment by Mermin is presented, and implications for the interpretation of Bell tests are discussed.
Bell inequality with self-testing property has played an important role in quantum information field with both fundamental and practical applications. However, it is generally challenging to find Bell inequalities with self-testing property for multipartite states and actually there are not many known candidates. In this work, we propose a systematical framework to construct Bell inequalities from stabilizers which are maximally violated by general stabilizer states, with two observables for each local party. We show that the constructed Bell inequalities can self-test any stabilizer state whi…
The relation between the boolean functions and Bell inequalities for qubits is analyzed. The connection between the maximal quantum violation of a Bell inequality and the nonlinearity of the corresponding boolean function is discussed. A visualization scheme of boolean functions is proposed. An attempt to classify Bell inequalities for qubits is made, a weaker result (classification with respect to Jevons group) is obtained. The fractal structure of the classification is shown. All constructs are illustrated by Mathematica code.
It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values $\left\{ 0,1\right\} $. A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics a…
Some new temporal Bell inequalities are deduced under joint realism assumption, using some perfect correlation property. No locality condition is needed. When the measured system is a macroscopic system, joint realism assumption substitutes the non-invasive measurabilioty hypothesis advantegeously, provided that the system satisfies the perfect correlation property. The new inequalities are violated quantically. This violation can be more severe than the similar violation in the case of precedent temporal Bell inequalities. Some microscopic and mesoscopic situations in which these inequalities…
