The experimentally verified violation of Bell's inequalities apparently implies that at least one of two intuitive beliefs must be false: that effects propagating at infinite velocity do not exist, and that natural phenomena occur independently of being observed. Giving up any one of these two beliefs (usually known together as Local Realism) is controversial. Many theories have been proposed to reconcile the violation of Bell's inequalities with Local Realism, but none has been fully successful. In this paper, it is recalled that any theory aimed to violate Bell's inequalities must start by giving up Boolean logic. The problem is split in two: the "soft" problem is to explain the violation of Bell's inequalities within (non-Boolean) Local Realism. The "hard" problem is to predict the time values when single particles are detected. A simple hidden variables model is introduced, which solves the soft problem. This is possible thanks to the use of vectors as the hidden variables and the operation projection, which do not hold to Boolean logic. This model reconciles the violation of Bell's inequalities with Local Realism and should end decades of controversy. Regarding the hard problem, the introduced model is as incomplete as Quantum Mechanics is. It is argued that solving the hard problem involves devising a new kind of quantum computer, which should be able to accept (non-Boolean) hidden variables as input data and replace the statistical Born's rule with a deterministic threshold condition.
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.
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.
In this note, we discuss a closed-form necessary and sufficient condition for any two-qubit state to show hidden nonlocality w.r.t the Bell-CHSH inequality. This is then used to numerically compute the relative volume of states showing hidden Bell-CHSH non-locality , among all two-qubit states with one-sided reduction maximally mixed.
Explicit local hidden variables models are exhibited that assume a correlation between detection events produced in the same detector at different times. It is shown that some models give predictions closer to the Bell limit than models without time correlation.
A quantum single-error-correcting scheme can be derived from a one-way entanglement purification protocol in purifying one Bell state from a finite block of five Bell states. The main issue to be concerned with in the theory of such an error-correction is to create specific linear Boolean functions that can transform the sixteen error syndromes occurring in the error-correcting code onto their mappings so that one Bell state is corrected whenever the other four in the finite block are measured. The Boolean function is performed under the effect of its associated sequence of basic quantum unila…
We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth $n$ tree using $O(n^{2+\logω})$ queries, where $ω$ is independent of $n$ and depends only on the type of subformulas within the tree. We also prove a classical lower bound of $n^{Ω(\log\log n)}$ queries, thus showing a (small) super-polynomial speed-up.
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…
Recently, Matzkin claimed the construction of a hidden variable (HV) model which is both local and equivalent with the quantum-mechanical predictions. In this paper we will briefly present this HV model and argue, by identifying an extra non-local "hidden" HV, why this model is not local
Three classes of local hidden-variable models that violate both Bell and Leggett inequalities are presented. The models, however, do not reproduce the quantum mechanical predictions, hence they are experimentally testable. It is concluded that on one hand neither Bell or Leggett inequality fully captures the essential counterintuitiveness of quantum mechanics, while on the other hand the hypothesis of outcome independence and that of locality are uncorrelated.
A time-reversal Bell test protocol is proposed. The quantum states are prepared by faraway separated partners and transferred to the third partner who carries out Bell basis measurement on them to post-select the Einstein-Podolsky-Rosen (EPR) pairs. If some loopholes open, similar as that in normal Bell test, the Bell violation in the present protocol is apt to be interpreted with local hidden variable (lhv) theory. With some modifications on the protocol, the lhvs at both sides are prevented to exchange their information. Thus they only function locally and cannot affect the behaviors of the …
We consider Bell tests in which the distant observers can perform local filtering before testing a Bell inequality. Notably, in this setup, certain entangled states admitting a local hidden variable model in the standard Bell scenario can nevertheless violate a Bell inequality after filtering, displaying so-called hidden nonlocality. Here we ask whether all entangled states can violate a Bell inequality after well-chosen local filtering. We answer this question in the negative by showing that there exist entangled states without hidden nonlocality. Specifically, we prove that some two-qubit We…
H. P. Stapp has proposed a number of demonstrations of a Bell-type theorem which dispensed with an assumption of hidden variables, but relied only upon locality together with an assumption that experimenters can choose freely which of several incompatible observables to measure. In recent papers his strategy has centered upon counterfactual conditionals. Stapp's paper in American Journal of Physics, 2004, replies to objections raised against earlier expositions of this strategy and proposes a simplified demonstration. The new demonstration is criticized, several subtleties in the logic of coun…
It is shown that no theory that satisfies certain premises can exclude faster-than-light influences. The premises include neither the existence of hidden variables, nor counterfactual definiteness, nor any premise that effectively entails the general existence of outcomes of unperformed local measurements. All the premises are compatible with Copenhagen philosophy, and the principles and predictions of relativistic quantum field theory. The present proof is contrasted with an earlier one with the same objective.
We discuss models that attempt to provide an explanation for the violation of Bell inequalities at a distance in terms of hidden influences. These models reproduce the quantum correlations in most situations, but are restricted to produce local correlations in some configurations. The argument presented in [Bancal et al. Nature Physics 8, 867 (2012)] applies to all of these models, which can thus be proved to allow for faster-than-light communication. In other words, the signalling character of these models cannot remain hidden.
Assuming that quantum mechanics is obeyed exactly after averaging over hidden variables, and considering models that obey both the hypotheses of free will and locality, we establish the form of all possible hidden-variable models that reproduce the spin-singlet.
We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(\log n) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for partial functions. A similar result was independently obtained by Gavinsky, Kempe, de Wolf [GKdW06]. Our lower bound is obtained by Fourier analysis, using the Fourier coefficients inequality of Kahn Kalai and Linial [KKL88].
Several authors have recently claimed that Bell's inequalities (BI) do not apply to certain types of generalized local hidden variables (HV) models. These claims are rejected, by means of a proof of BI valid for a very broad class of local HV models (deterministic or stochastic, with or without memory, time dependent or time independent, setting dependent or setting independent, and even allowing classical communication between the wings of the experiment). The precise meaning of the locality requirement is clarified during the proof. The exact roles of the two assumptions that amount to Bell'…
A new definition of "Realism" is proposed: it is that a gedanken "spectrograph" of hidden variables behaves as an actual (say, wavelength) spectrograph. The question is: does this definition allow, by itself, the derivation of Bell's inequalities? If it were, then such a spectrograph would be impossible, for Bell's inequalities are observed to be violated. In this short paper it is reported that, on the contrary, such spectrograph is compatible with the violation of Bell's inequalities. This result puts some new light on the controversy about the hypotheses necessary to derive Bell's inequalit…
We discuss the problem of hidden variables and the motivation for introducting them in quantum mechanics. These include determinism, and the problem of meassurement and incompleteness. We first discuss Von-Neumann's imposisbility proof and then analyse it's weakness in terms of Bell's explicit hidden variable model of spin one-half particles. We next discuss Gleason's theorem and Kochen-Specker theorem and bring out the troublems with non contextual hidden variable theories. An important role is played by Einstein locality in the discussion of hidden variable theories as was first brought out …
This paper implements in a simple but rigorous fashion a model of particle interaction involving all paths within a quantum system, both for configuration space and for spin. The model, which we call the space of all paths, leads to a locally explicable conceptual framework for quantum mechanics. Using it we present two counterexamples to Bell's theorem. Moreover, we show that the result places severe constraints on possible viable interpretations of quantum mechanics: Either an interpretation must in some form represent a quantum system in terms of all paths within the system or, alternativel…
We extend the Bell inequality known for two qubits to the four-level atom, including an artificial atom realized by the superconducting circuit, and qudit with j=3/2. We formulate the extended inequality as the inequality valid for an arbitrary Hermitian nonnegative 4x4-matrix with unit trace for both separable and entangled matrices.
In a recent article Giustina et al. [Phys. Rev. Lett. 115, 250401 (2015)] report on an advanced Bell experiment, simultaneously closing loopholes for local hidden-variable theories. The authors claim that 'local realism' has been refuted, unless 'truly exotic hypotheses' are made. Here I argue that a particularly wide and natural class of local hidden-variable theories survives, for instance when the hidden variables describe a background field. Such background-based theories exploit the freedom-of-choice loophole, which cannot be closed for this type of hidden variables. The dynamics of such …
John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can…
Bell's theorem proves only that hidden variables evolving in true physical time can't exist; still the theorem's meaning is usually interpreted intolerably wide. The concept of hidden time (and, in general, hidden space-time) is introduced. Such concept provides a whole new class of physical theories, fully compatible with current knowledge, but giving new tremendous possibilities. Those theories do not violate Bell's theorem.
The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a whole range of underlying groups. In a way, this distinguishes it from the hidden subgroup problem where more stringent requirements about the existence of a periodic subgroup have to be made. And yet, the hidden shift problem proves to be rich enough to capture interesting features of problems of algebraic, geometric, and combinatorial flavor. We present a quan…
In the above paper, it is claimed that with a particular use of the Bell inequality a simple single photon experiment could be performed to show the impossibility of any deterministic hidden variable theory in quantum optics. A careful analysis of the concept of probability for hidden variables and a detailed discussion of the hidden variable model of de Broglie-Bohm show that the reasoning and main conclusion of this paper are not correct.
This paper furthers the long historical examination of and debate on the foundations of quantum mechanics (QM) by presenting two local hidden variable (LHV) rules in the context of the EPRB experiment which violate Bell's inequality, but which are nevertheless local and deterministic under reasonable definitions of the terms, and coincide approximately with the conventional QM prediction. The theories are based on the general idea of probabilistic detection of particles depending on an interaction of hidden variables within the measuring device and particle, and relate mathematically to Fourie…
Though John Bell had claimed that his spin-1/2 example of a hidden-variable theory(HV) is an \emph{explicit} counterexample to von Neumann's proof of the non-existence of hidden variable theories empirically equivalent to quantum mechanics, such examples can be so construed only if they met all of von Neumann's requirements. In particular, that they reproduced all the observed predictions of quantum theory. To shed light on these aspects, we have, on the one hand, simplified and critically examined Bell's original example and on the other hand, constructed explicit such examples for spin-1 sys…
There are increasingly suggestions for computer simulations of quantum statistics which try to violate Bell type inequalities via classical, common cause correlations. The Clauser-Horne-Shimony-Holt (CHSH) inequality is very robust. However, we argue that with the Einstein-Podolsky-Rosen setup, the CHSH is inferior to the Bell inequality, although and because the latter must assume anti-correlation of entangled photon singlet states. We simulate how often quantum behavior violates both inequalities, depending on the number of photons. Violating Bell 99% of the time is argued to be an ideal ben…
Bell inequalities or Bell-like experiments are supposed to test hidden variable theories based on three intuitive assumptions: determinism, locality and measurement independence. If one of the assumptions of Bell inequality is properly relaxed, the probability distribution of the singlet state, for example, can be reproduced by a hidden variable model. Models that deal with the relaxation of some condition above, with more than one hidden variable, have been studied in the literature nowadays. In this work the relation between the number of hidden variables and the degree of relaxation necessa…
John Bell showed that a big class of local hidden-variable models stands in conflict with quantum mechanics and experiment. Recently, there were suggestions that empirical adequate hidden-variable models might exist, which presuppose a weaker notion of local causality. We will show that a Bell-type inequality can be derived also from these weaker assumptions.
We analyze the assumptions that are made in the proofs of Bell-type inequalities for the results of Einstein-Podolsky-Rosen type of experiments. We find that the introduction of time-like random variables permits the construction of a broader mathematical model which accounts for all correlations of variables that are contained in the time dependent parameter set of the backward light cone. It also permits to obtain the quantum result for the spin pair correlation, a result that contradicts Bell's inequality. Two key features of our mathematical model are (i) the introduction of time operators…
Franson's Bell experiment with energy-time entanglement [Phys. Rev. Lett. 62, 2205 (1989)] does not rule out all local hidden variable models. This defect can be exploited to compromise the security of Bell inequality-based quantum cryptography. We introduce a novel Bell experiment using genuine energy-time entanglement, based on a novel interferometer, which rules out all local hidden variable models. The scheme is feasible with actual technology.
For a bipartite entangled state shared by two observers, Alice and Bob, Alice can affect the post-measured states left to Bob by choosing different measurements on her half. Alice can convince Bob that she has such an ability if and only if the unnormalized postmeasured states cannot be described by a local-hidden-state (LHS) model. In this case, the state is termed steerable from Alice to Bob. By converting the problem to construct LHS models for two-qubit Bell diagonal states to the one for Werner states, we obtain the optimal models given by Jevtic \textit{et al.} [J. Opt. Soc. Am. B 32, A4…
What classical resources are required to simulate quantum correlations? For the simplest and most important case of local projective measurements on an entangled Bell pair state, we show that exact simulation is possible using local hidden variables augmented by just one bit of classical communication. Certain quantum teleportation experiments, which teleport a single qubit, therefore admit a local hidden variables model.
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…
Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when…
Experiments motivated by Bell's theorem have led some physicists to conclude that quantum theory is nonlocal. However, the theoretical basis for such claims is usually taken to be Bell's Theorem, which shows only that if certain predictions of quantum theory are correct, and a strong hidden-variable assumption is valid, then a certain locality condition must fail. This locality condition expresses the idea that what an experimenter freely chooses to measure in one spacetime region can have no effect of any kind in a second region situated spacelike relative to the first. The experimental resul…
We analyze and compare the mathematical formulations of the criterion for separability for bipartite density matrices and the Bell inequalities. We show that a violation of a Bell inequality can formally be expressed as a witness for entanglement. We also show how the criterion for separability and a description of the state by a local hidden variable theory, become equivalent when we restrict the set of local hidden variable theories to the domain of quantum mechanics. This analysis sheds light on the essential difference between the two criteria and may help us in understanding whether there…
Contrary to what Lambare [arXiv:2008.00369] assumes, in non-Newtonian calculus (a calculus based on non-Diophantine arithmetic) an integral is typically given by a nonlinear map. This is the technical reason why all the standard proofs of Bell-type inequalities fail if non-Newtonian hidden variables are taken into account. From the non-Newtonian perspective, Bell's inequality is a property of a limited and unphysical class of hidden-variable models. An explicit counterexample to Bell's theorem can be easily constructed.
We derive a set of Bell-type inequalities for arbitrarily high-dimensional systems, based on the assumption of partial separability in the hybrid local-nonlocal hidden variable model. Partially entangled states would not violate the inequalities, and thus upon violation, these Bell-type inequalities are sufficient conditions to detect the full $N$-particle entanglement and validity of the hybrid local-nonlocal hidden variable description.
The categorization of quantum states for composite systems as either separable or entangled, or alternatively as Bell local or Bell non-local states based on local hidden variable theory is reviewed in Sections 1 and 2, focusing on simple bipartite systems. The significance of states demonstrating Bell non-locality for settling the long standing controversy between the Copenhagen interpretation of the quantum measurement process involving the collapse of the wave-function and the alternative interpretation based on pre-existing hidden variables is emphasized. Although experiments demonstrating…
While entanglement and violation of Bell inequalities were initially thought to be equivalent quantum phenomena, we now have different examples of entangled states whose correlations can be described by local hidden--variable models and, therefore, do not violate any Bell inequality. We provide an up to date overview of the existing local hidden--variable models for entangled quantum states, both in the bipartite and multipartite case, and discuss some of the most relevant open questions in this context. Our review covers twenty five years of this line of research since the seminal work by Wer…
The problem of computing the local hidden variable (LHV) value of a Bell inequality plays a central role in the study of quantum nonlocality. In particular, this problem is the first step towards characterizing the LHV polytope of a given scenario. In this work, we establish a relation between the LHV value of bipartite Bell inequalities and the mathematical notion of excess of a matrix. Inspired by the well developed theory of excess, we derive several results that directly impact the field of quantum nonlocality. We show infinite families of bipartite Bell inequalities for which the LHV valu…
In a old paper by G. Lochak, it is claimed that the Bell definition of a hidden variable is in conflict with the formalism of quantum mechanics. This result implies that it is not necessary to invoke non locality to explain the violation of the Bell inequality. A careful analysis of the concept of probability for hidden variables, as defined differently by Bell and Lochak, show that the reasoning and main conclusions of are not correct.
One way to deal with the fact that many versions of reality simultanenously exist in the wave function is to suppose there are hidden variables that single out one version for perception. Bell showed theoretically and the Aspect experiment confirmed that there could be no hidden variable theory which satisfied a locality condition. We show here that the locality condition can be derived from the well-established principle of the relativity of simultaneity. Thus virtually all hidden variable theories, not just obviously local ones, are in conflict with the results of the Aspect experiment and a…
As is well known, quantum mechanical behavior cannot, in general, be simulated by a local hidden variables model. Most -if not all- the proofs of this incompatibility refer to the correlations which arise when each of two (or more) systems separated in space is subjected to a single ideal measurement. This setting is good enough to show contradictions between local hidden variables models and quantum mechanics in the case of pure states. However, as shown here, it is not powerful enough in the case of mixtures. This is illustrated by an example. In this example, the correlations which arise wh…
An alternative method of detection-loophole-free Bell test is proposed using local hidden variable (LHV) models with optimal detection efficiencies. A framework for constructing such optimal LHV models is presented. Optimal LHV models for maximally and non-maximally entangled twopartite states are constructed to reproduce the quantum correlations within the critical detection efficiencies. The LHV models are shown to be completely equivalent with the existing twopartite Bell inequalities in their optimized setups, and to have even lower critical efficiencies in the LHV modes' own optimized set…
University of Waterloo（滑铁卢大学）；IBM TJ Watson Research Center（IBM TJ沃森研究中心）；NEC Laboratories America（NEC美国实验室）
We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends strongly on f. We demonstrate that the easiest instances of this problem correspond to bent functions, in the sense that an exact one-query algorithm exists if and only if the function is bent. We partially characterize the hardest instances, which include delta functions. Moreover, we show that the problem is easy for random functions, since two queries suffice.…
We analyse the proof of Bell's inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics, 1933. We found a (numerical) statistical correction to Bell's inequality. Such an additional term in the right hand side of Bell's inequality can be considered as a probability invariant of a quantum state. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell's inequality can be considered as just experiments to estimate this constant. It seem…
We discuss a class of proofs of Bell-type inequalities that are based on tables of potential outcomes. These proofs state in essence: if one can only imagine (or write down in a table) the potential outcome of a hidden parameter model for EPR experiments then a contradiction to experiment and quantum mechanics follows. We show that these proofs do not contain hidden variables that relate to time or, if they do, lead to logical contradictions that render them invalid.
Although skeptical of the prohibitive power of no-hidden-variables theorems, John Bell was himself responsible for the two most important ones. I describe some recent versions of the lesser known of the two (familar to experts as the "Kochen-Specker theorem") which have transparently simple proofs. One of the new versions can be converted without additional analysis into a powerful form of the very much better known "Bell's Theorem", thereby clarifying the conceptual link between these two results of Bell.
