We define quantum-like probabilistic behaviour as behaviour which is impossible to describe by using the classical probability model. We discuss the conjecture that cognitive behaviour is quantum-like. There is presented the scheme for an experimental test for quantum-like cognitive behaviour based on a generalization of the famous Bell's inequality. This generalization is an analogue of Bell's inequality, but for conditional probabilities. The use of conditional probabilities (instead of simultaneous probability distributions for pairs of observables in the original Bell's inequality) gives the possibility to separate two problems which are mixed in the original Bell's framework: nonlocality and nonclassical (quantum-like) probabilistic behaviour. Our inequality for conditional probabilities can be used for experiments with a single system (so we need not to prepare pairs of correlated systems) to find quantum-like behaviour. This possibility is extremely important in cognitive sciences where it is practically impossible to prepare pairs of precisely correlated cognitive systems.
It is shown that Marinatto's claim [Phys. Rev. Lett. 90, 258901 (2003)] that the proof of "Bell's theorem without inequalities and without probabilities for two observers" [A. Cabello, Phys. Rev. Lett. 86, 1911 (2001)] requires four spacelike separated observers rather than two is unjustified.
We argue that a certain type of many minds (and many worlds) interpretations of quantum mechanics due to Lockwood (and Deutsch) do not provide a coherent interpretation of the quantum mechanical probabilistic algorithm. By contrast, in Albert and Loewer's version of the many minds interpretation there is a coherent interpretation of the quantum mechanical probabilities. We consider Albert and Loewer's probability interpretation in the context of Bell-type and GHZ-type states and argue that it exhibits a certain form of nonlocality which is, however, much weaker than Bell's nonlocality.
A proof of Bell's theorem using two maximally entangled states of two qubits is presented. It exhibits a similar logical structure to Hardy's argument of ``nonlocality without inequalities''. However, it works for 100% of the runs of a certain experiment. Therefore, it can also be viewed as a Greenberger-Horne-Zeilinger-like proof involving only two spacelike separated regions.
Bell's inequalities can be understood in three different ways depending on whether the numbers featuring in the inequalities are interpreted as classical probabilities, classical conditional probabilities, or quantum probabilities. In the paper I will argue that the violation of Bell's inequalities has different meanings in the three cases. In the first case it rules out the interpretation of certain numbers as probabilities of events. In the second case it rules out a common causal explanation of conditional correlations of certain events (measurement outcomes) conditioned on other events (me…
The concept of asymptotic correctability of Bell-diagonal quantum states is generalised to elementary quantum systems of higher dimensions. Based on these results basic properties of quantum state purification protocols are investigated which are capable of purifying tensor products of Bell-diagonal states and which are based on $B$-steps of the Gottesman-Lo-type with the subsequent application of a Calderbank-Shor-Steane quantum code. Consequences for maximum tolerable error rates of quantum cryptographic protocols are discussed.
Quantum mechanics may be formulated as {\it Sensible Quantum Mechanics} (SQM) so that it contains nothing probabilistic except conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. Ratios of the measures for these sets of perceptions can be interpreted as frequency-type probabilities for many actually existing sets. These probabilities generally cannot be given by the ordinary quantum ``probabilities'' for a single set of alternatives. {\it Probabilism}, or ascribing pr…
The well known Bell experiment with two actors Alice and Bob is considered. First the simple deduction leading to the CHSH inequality under local realism is reviewed, and some arguments from the literature are recapitulated. Then I take up certain background themes before I enter a discussion of Alice's analysis of the situation. An important point is that her mind is limited by the fact that her Hilbert space in this context is two-dimensional. General statements about a mind's limitation during a decision process are derived from recent results on the reconstruction of quantum theory from co…
In this Comment we show that Cabello's argument [Phys. Rev. Lett. 86, 1911 (2001)] which proves the nonlocal feature of any classical model of quantum mechanics based on Einstein-Podolsky-Rosen (EPR) criterion of elements of reality, must involve at least four distant observers rather than the two employed by the author. Moreover we raise a remark on the necessity of performing a real experiment confirming Cabello's argument.
It is not generally known, that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of plus and minus 1s that are writable on paper.This surprising fact is not immediately obvious from Bell's inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets.For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non…
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing a joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such a distribution experimentally, as a set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to a quantum probe, however, produces minimal back-a…
We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be define…
Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical ``existence'' and physical ``existence'' is considered from the point of view of philosophical idealism. Some of the ways in which infinitary mathematics arises in modern mathematical physics are discussed. Empirical science has led to the mathematics of quantum theory. This in turn can be taken to suggest a picture of reality involving possible minds and the physical laws which determine their probabilities. In th…
Both the quantum mechanical and classical Bells experiment are within the focus of this paper. The fact that one measures different probabilities in both experiments is traced back to the superposition of two orthogonal but nonentangled substates in the quantum mechanical case. This superposition results in an interference term that can be splitted into two additional states representing a sink and a source of probabilities in the classical event space related to Bells experiment. As a consequence, a statistical operator can be related to the quantum mechanical Bells experiment that contains a…
This paper describes a device, consisting of a central source and two widely separated detectors with six switch settings each, that provides a simple gedanken demonstration of Bell's theorem without relying on either statistical effects or the occurrence of rare events. The mechanism underlying the operation of the device is revealed for readers with a knowledge of quantum mechanics.
We study a method of generating Bell inequalities by using group actions of single-generator abelian groups. Two parties, Alice and Bob, each make one of M possible measurements on a system, with each measurement having K possible outcomes. The probabilities for the outcomes of these measurements are P(a_j = k, b_{j'}=k'), where j,j' are in the set {1,2,... M} and k,k' are in the set {0,1,... K-1}. The sums of some subsets of these probabilities have upper bounds when the probabilities result from a local, realistic theory that can be violated if the probabilities come from quantum mechanics. …
The violation of Bell inequalities is often interpreted as showing that, if hidden variables exist, they must be contextual and non local. But they can also be explained questioning the probability space employed, or the validity of the Kolmogorov axioms. In this article we explore the additional constrains which can be deduced from two widely used objetive probability theories: frequentism and propensities. One of the strongest objections in the deduction of one version of Bell inequalities goes about the probability space, which assumes the existence of values for the output of the experimen…
In a previous publication, we showed how group actions can be used to generate Bell inequalities. The group action yields a set of measurement probabilities whose sum is the basic element in the inequality. The sum has an upper bound if the probabilities are a result of a local, realistic theory, but this bound can be violated if the probabilities come from quantum mechanics. In our first paper, we considered the case of only two parties making the measurements and single-generator groups. Here we show that the method can be extended to three parties, and it can also be extended to non-Abelian…
Quantum non-locality tests have been of interest since the 1960's paper by Bell on the original EPR paradox.The present paper discusses whether the CGLMP (Bell) inequalities are possible tests for showing that quantum theory is not underpinned by local hidden variable theory (LHVT). It is found by applying Fine's theorem that the CGLMP approach involves a LHVT for the joint probabilities associated with the measurement of one observable from each of the two sub-systems, even though the underlying probabilities for joint measurements of all four observables may involve a non-local HVT. The latt…
We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely cons…
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.
We analyze optimal schemes and also propose some practical schemes for the nonlocal conversion from two shared Bell pairs to four-qubit entangled states in optical quantum networks. In the analysis, we consider two-qubit operations as nonlocal operations and minimize the number of access to ancillary qubits as possible. First, we consider two-qubit unitary operations without using ancillary qubits and derive a necessary and sufficient condition for convertible states. Second, we consider nonlocal optical systems composed of passive linear optics and postselection. For the passive linear optica…
The Bell inequalities in three and four correlations are re-derived in general forms showing that three and four data sets, respectively, identically satisfy them regardless of whether they are random, deterministic, measured, predicted, or some combination of these. The Bell inequality applicable to data is thus a purely mathematical result independent of experimental test. Correlations of simultaneously cross-correlated variable pairs do not in general all have the same form, and vary with the physical system considered and its experimental configuration. It is the form of correlations of as…
Errors in Eberly's derivation of several Bell inequalities are pointed out: (1) it is based on an equation that is incorrect; (2) it uses neither two-particle states nor locality to derive Bell's inequalities and; (3) it does not use entanglement to obtain violations of Bell's inequalities. Even leading and outstanding physicists -- as certainly is the case of Prof. Eberly -- sometimes make elementary mistakes, and this by no means diminishes the importance of their scientific contribution. In general, this is a consequence of an excessive attachment to an idea (nowadays it has become fashiona…
In this short survey article, I discuss Bell's theorem and some strategies that attempt to avoid the conclusion of non-locality. I focus on two that intersect with the philosophy of probability: (1) quantum probabilities and (2) superdeterminism. The issues they raised not only apply to a wide class of no-go theorems about quantum mechanics but are also of general philosophical interest.
Non-locality without inequality is an elegant argument introduced by L. Hardy for two qubit systems, and later generalised to $n$ qubits, to establish contradiction of quantum theory with local realism. Interestingly, for $n=2$ this argument is actually a corollary of Bell-type inequalities, viz. the CH-Hardy inequality involving Bell correlations, but for $n$ greater than 2 it involves $n$-particle probabilities more general than Bell-correlations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for $n$ qubits, and then, asso…
Violations of Bell inequalities in classical optics have been demonstrated in terms of field mean intensities and correlations, however, the quantum meaning of violations point to statistics and probabilities. We present a violation of Bell inequalities for classical fields in terms of probabilities, where we convert classical-field intensities into probabilities via the standard photon-counting equation. We find violation for both, entangled and separable field states. We conclude that any obtained quantum effect might be fully ascribed to the quantum nature of the detector rather than the fi…
Bounds on quantum probabilities and expectation values are derived for experimental setups associated with Bell-type inequalities. In analogy to the classical bounds, the quantum limits are experimentally testable and therefore serve as criteria for the validity of quantum mechanics.
Within Guney-Hillery approach a number of examples of classical and quantum bounds on sum of probabilities resulting from two orbits of $S_4$ is considered. It is shown that the violation of Bell's inequalities is rather rare and gentle.
For the classical mind, quantum mechanics is boggling enough; nevertheless more bizarre behavior could be imagined, thereby concentrating on propositional structures (empirical logics) that transcend the quantum domain. One can also consistently suppose predictions and probabilities which are neither classical nor quantum, but which are subject to subclassicality; that is, the additivity of probabilities for mutually exclusive, co-measurable observables, as formalized by admissibility rules and frame functions.
In this essay a quantum-dualistic, perspectival and synchronistic interpretation of quantum mechanics is further developed in which the classical world-from-decoherence which is perceived (decoherence) and the perceived world-in-consciousness which is classical (collapse) are not necessarily identified. Thus, Quantum Reality or "{\it unus mundus}" is seen as both i) a physical non-perspectival causal Reality where the quantum-to-classical transition is operated by decoherence, and as ii) a quantum linear superposition of all classical psycho-physical perspectival Realities which are governed b…
A Bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers, measurement settings or measurement outcomes. In this article, such "liftings" of Bell inequalities are studied. It is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope.
Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s. A function p from S to [0,1] is called a numerical event or alternatively an S-probability. If a set P:={p(s)|s in S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances i…
Wigner's marginal probability theory is revisited, and systematically applied to n-particle correlation measurements. A set of Bell inequalities whose corollaries are Hardy contradiction and its generalisation are derived with intuitive graphical analysis.
Probabilities of measurement outcomes of two-particle entangled states give a physically transparent interpretation of the concurrence and of the I-concurrence as entanglement measures. The (I)-concurrence can thus be measured experimentally. The tight connection between these measures and Bell inequalities is highlighted.
Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann's 'no hidden variables' proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense 'like Bohm's theory.' His seminal result provides a positive answer to the question. I argue that Bell's analysis misconstrues von Neumann's argument. What von Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (de…
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…
We present a Theorem that all generalized Greenberger-Horne-Zeilinger states of a three-qubit system violate a Bell inequality in terms of probabilities. All pure entangled states of a three-qubit system are shown to violate a Bell inequality for probabilities; thus, one has Gisin's theorem for three qubits.
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.
We present a homodyne detection scheme to verify Bell's inequality on correlated optical beams at the output of a nondegenerate parametric amplifier. Our approach is based on tomographic measurement of the joint detection probabilities, which allows high quantum efficiency at detectors. A self-homodyne scheme is suggested to simplify the experimental set-up.
On the basis of local realism theory, nonlocal information is necessary for violation of Bell's inequality. From a theoretical point of view, nonlocal information is essentially the mutual information on distant outcome and measurement setting. In this work we prove that if the measurement is free and unbiased, the mutual information about the distant outcome and setting is both necessary for the violation of Bell's inequality in the case with unbiased marginal probabilities. In the case with biased marginal probabilities, we point out that the mutual information about distant outcome cease to…
The majority of recent works investigating the link between non-locality and randomness, e.g. in the context of device-independent cryptography, do so with respect to some specific Bell inequality, usually the CHSH inequality. However, the joint probabilities characterizing the measurement outcomes of a Bell test are richer than just the degree of violation of a single Bell inequality. In this work we show how to take this extra information into account in a systematic manner in order to optimally evaluate the randomness that can be certified from non-local correlations. We further show that t…
Three extensions and reinterpretations of nonclassical probabilities are reviewed. (i) We propose to generalize the probability axiom of quantum mechanics to self-adjoint positive operators of trace one. Furthermore, we discuss the Cartesian and polar decomposition of arbitrary normal operators and the possibility to operationalize the corresponding observables. Thereby we review and emphasize the use of observables which maximally represent the context. (ii) In the second part, we discuss Pitowsky polytopes for automaton logic as well as for generalized urn models and evaluate methods to find…
We analyze the correlation structure of bipartite arbitrary-dimensional Bell inequalities via novel conditions of correlations in terms of differences of joint probabilities called correlators. The conditions of correlations are shown to be necessary for the multi-level Bell state. In particular, we find that the bipartite arbitrary-dimensional Bell-type inequalities introduced by Collins-Gisin-Linden-Massar-Popescu [Phys. Rev. Lett. 88, 040404 (2002)] and Son-Lee-Kim [Phys. Rev. Lett. 96, 060406 (2006)] are composed of correlators, and we reveal that the maximal violations by the Bell state j…
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…
In this paper we analyze the existence of joint probabilities for the Bell-type and GHZ entangled states. We then propose the usage of nonmonotonic upper probabilities as a tool to derive consistent joint upper probabilities for the contextual hidden variables. Finally, we show that for the extreme example of no error, the GHZ state allows for the definition of a joint upper probability that is consistent with the strong correlations.
The celebrating theorem of A. Fine implies that the CHSH inequality is violated if and only if the joint probability distribution for the quadruples of observables involved the EPR-Bohm-Bell experiment does not exist, i.e., it is impossible to use the classical probabilistic model (Kolmogorov, 1933). In this note we demonstrate that, in spite of Fine's theorem, the results of observations in the EPR-Bohm-Bell experiment can be described in the classical probabilistic framework. However, the "quantum probabilities" have to be interpreted as conditional probabilities, where conditioning is with …
We consider dynamics of hidden variables for measurements in a generalized bell-type model for a single spin using natural assumptions. The evolution of the system, which can be expressed as dynamic chaos is studied. The equilibrium state that the system evolves to asymptotically is consistent with the predictions of quantum theory. The thesis of incompleteness of quantum mechanics in dynamic interpretation and completeness in statistical interpretation are developed. Conceptual problems of quantum mechanics such as violations of Bell inequalities, negative probabilities, complementarity princ…
Neutrino oscillations provide evidence for the mode entanglement of neutrino mass eigenstates in a given flavour eigenstate. Given this mode entanglement, it is pertinent to consider the relation between the oscillation probabilities and other quantum correlations. In this work, we show that all the well-known quantum correlations, such as the Bell's inequality, are directly related to the neutrino oscillation probabilities. The results of the neutrino oscillation experiments, which measure the neutrino survival probability to be less than unity, imply Bell's inequality violation.
We address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e. those corresponding to complete commuting sets of observables. For four-dimensional phase space with position variables qi and momentum variables pj, we establish the two following points: i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2) and (p1,p2), there does not always exist a positive phase s…
By analysing probabilistic foundations of quantum theory we understood that the so called quantum calculus of probabilities (including Born's rule) is not the main distinguishing feature of "quantum". This calculus is just a special variant of a contextual probabilistic calculus. In particular, we analysed the EPR-Bohm-Bell approach by using contextual probabilistic models (e.g., the frequency von Mises model). It is demonstrated that the EPR-Bohm-Bell consideration are not so much about "quantum", but they are merely about contextual. Our conjecture is that the "fundamental quantum element" i…
Bell's theorem is supposed to exclude all local hidden-variable models of quantum correlations. However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states. Observable probabilities are consistent with the usual arithmetic employed by macroscopic observers, but counterfactual aspects of Bell's theorem are sensitive to the choice of hidden-variable arithmetic and calculus. The model is classical in the sense of Einste…
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of non-commuting projectors. We present local and non-local examples of violation of generalized Bell inequalities in phase spac…
We calculate the joint probabilities and the correlation function in Einstein--Podolsky--Rosen type experiments with a massive vector boson in the framework of quantum field theory. We report on the strange behavior of the correlation function (and the probabilities) -- the correlation function, which in the relativistic case still depends on the particle momenta, for some fixed configurations has local extrema. We also show that relativistic spin-1 particles violate some Bell inequalities more than nonrelativistic ones and that the degree of violation of the Bell inequality is momentum depend…
