Towards a Local Hidden Variable Theory

A local hidden variable theory of quantum mechanics is formulated by adapting Gell-Mann and Hartle’s many-histories formulation. The resulting theory solves the measurement problem by exploiting the independence loophole in Bell’s theorem; it violates the independence of hidden variable values and measuring device settings. Although the theory is problematic in some respects, it provides a concrete example via which the tenability of this approach can be better evaluated.     

A remark on the role of indeterminism and non-locality in the violation of Bell’s inequalities

Diederik Aerts was the first in the eighties to develop a concrete example of a macroscopic “classical” entity violating Bell’s inequalities (BI). In more recent years, he also developed a macroscopic model in which the amount of non-locality and indeterminism can be continuously varied, and used it to show that by increasing non-locality one increases the degree of violation of BI, whereas by increasing indeterminism one decreases the degree of violation of BI. In this article we introduce and analyze a different macroscopic model in which the amount of non-locality and indeterminism can also be parameterized, and therefore varied, and find that, in accordance with the model of Aerts, an increase of non-locality does produce a stronger violation of BI. However, differently from his model, we also find that, depending on the initial state in which the system is prepared, an increase of indeterminism can either strengthen or weaken the degree of violation of BI.     

There Is No Spooky Action at a Distance in Quantum Mechanics

Einstein became bothered by quantum mechanical action at a distance within two years of Schrödinger’s introduction of his eponymous wave equation. If the wave function represents the “real” physical state of a particle, then the measurement of the particle’s position would result in the instantaneous collapse of the wave function to the single, measured position. Such a process seemingly violates not only the Schrödinger equation but also special relativity. Einstein was not alone in this vexation; however, the dilemma eventually faded as physicists concentrated on using the Schrödinger equation to solve a plethora of pressing problems. For the next 30 years, wave function collapse, while occasionally discussed by physicists, was primarily a topic of interest for philosophers. That is, until 1964, when Bell introduced his famous inequality and maintained that its violation proved that quantum mechanics and, by implication, nature herself are nonlocal. Unfortunately, this brought the topic back to mainstream physics, where it has remained and continues to muddy the waters. To be sure, not all physicists are bothered by the apparent nonlocality of quantum mechanics. So where have those who embrace quantum nonlocality gone wrong? I argue that the answer is a gratuitous belief in the ontic nature of the quantum state.     

Quantum Pseudo-Telepathy

Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with what we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required. Pseudo-telepathy is a surprising application of quantum information processing to communication complexity. Thanks to entanglement, perhaps the most nonclassical manifestation of quantum mechanics, two or more quantum players can accomplish a distributed task with no need for communication whatsoever, which would be an impossible feat for classical players. After a detailed overview of the principle and purpose of pseudo-telepathy, we present a survey of recent and not-so-recent work on the subject. In particular, we describe and analyse all the pseudo-telepathy games currently known to the authors. Supported in Part by Canada’s Natural Sciences and Engineering Research Council (NSERC), the Canada Research Chair programme and the Canadian Institute for Advanced Research (CIAR). Supported in part by a scholarship from Canada’s NSERC. Supported in part by Canada’s NSERC Québec’s Fonds de recherche sur la nature et les technologies (FQRNT), the CIAR and the Mathematics of Information Technology and Complex Systems Network (MITACS).     

Quantum Mechanics from Focusing and Symmetry

A foundation of quantum mechanics based on the concepts of focusing and symmetry is proposed. Focusing is connected to c-variables—inaccessible conceptually derived variables; several examples of such variables are given. The focus is then on a maximal accessible parameter, a function of the common c-variable. Symmetry is introduced via a group acting on the c-variable. From this, the Hilbert space is constructed and state vectors and operators are given a definite interpretation. The Born formula is proved from weak assumptions, and from this the usual rules of quantum mechanics are derived. Several paradoxes and other issues of quantum theory are discussed.     

Nonlocality Without Nonlocality

Bell’s theorem is purported to demonstrate the impossibility of a local “hidden variable” theory underpinning quantum mechanics. It relies on the well-known assumption of ‘locality’, and also on a little-examined assumption called ‘statistical independence’ (SI). Violations of this assumption have variously been thought to suggest “backward causation”, a “conspiracy” on the part of nature, or the denial of “free will”. It will be shown here that these are spurious worries, and that denial of SI simply implies nonlocal correlation between spacelike degrees of freedom. Lorentz-invariant theories in which SI does not hold are easily constructed: two are exhibited here. It is conjectured, on this basis, that quantum-mechanical phenomena may be modeled by a local theory after all. This paper is dedicated to the memory of John A. Wheeler.     

Quantum fluctuations as deviation from classical dynamics of ensemble of trajectories parameterized by unbiased hidden random variable

A quantization method based on replacement of c-number by c-number parameterized by an unbiased hidden random variable is developed. In contrast to canonical quantization, the replacement has a straightforward physical interpretation as a statistical modification of classical dynamics of an ensemble of trajectories, and implies a unique operator ordering. We then apply the method to develop quantum measurement without wave function collapse and external observer a la pilot-wave theory.     

Probabilistic Teleportation via Entanglement

With an arbitrary bi-particle entangled mixed state which is shared by Alice (the sender) and Bob (the receiver) acted as a quantum channel, at first, a teleportation protocol that Alice successfully transmits an unknown mixed state to Bob based on a positive operator-valued measurement (POVM) is presented. The upper bound of probability to teleport successfully an unknown mixed state is then investigated, and conclude that it completely depends on the entanglement degree of the bi-particle entangled mixed state as a resource.     

Kochen-Specker theorem for a three-qubit system: A state-dependent proof with seventeen rays

We consider Kochen-Specker theorem for three-qubit system with eight-dimensional state space. Reexamining the proof given by Kernaghan and Peres, we make some clarifications on the orthogonality of rays and rank-two projectors found by them. Basing on their five groups of orthogonal octad, we then show a proof that requires only seventeen rays.     

A simplified basis for Bell–Kochen–Specker theorems

We show that a reduced form of the structural requirements for deterministic hidden variables used in Bell–Kochen–Specker theorems is already sufficient for the no-go results. Those requirements are captured by the following principle: an observable takes a spectral value x if and only if the spectral projector associated with x takes the value 1. We show that the “only if” part of this condition suffices. The proof identifies an important structural feature behind the no-go results; namely, if at least one projector is assigned the value 1 in any resolution of the identity, then at most one is.     

Relational Interpretation of the Wave Function and a Possible Way Around Bell’s Theorem

The famous “spooky action at a distance” in the EPR-scenario is shown to be a local interaction, once entanglement is interpreted as a kind of “nearest neighbor” relation among quantum systems. Furthermore, the wave function itself is interpreted as encoding the “nearest neighbor” relations between a quantum system and spatial points. This interpretation becomes natural, if we view space and distance in terms of relations among spatial points. Therefore, “position” becomes a purely relational concept. This relational picture leads to a new perspective onto the quantum mechanical formalism, where many of the “weird” aspects, like the particle-wave duality, the non-locality of entanglement, or the “mystery” of the double-slit experiment, disappear. Furthermore, this picture circumvents the restrictions set by Bell’s inequalities, i.e., a possible (realistic) hidden variable theory based on these concepts can be local and at the same time reproduce the results of quantum mechanics. PACS: 03.65.Ud, 04.60.Nc     

Local Causality and Completeness: Bell <Emphasis Type="Italic">vs.</Emphasis> Jarrett

J.S. Bell believed that his famous theorem entailed a deep and troubling conflict between the empirically verified predictions of quantum theory and the notion of local causality that is motivated by relativity theory. Yet many physicists continue to accept, usually on the reports of textbook writers and other commentators, that Bell’s own view was wrong, and that, in fact, the theorem only brings out a conflict with determinism or the hidden-variables program or realism or some other such principle that (unlike local causality), allegedly, nobody should have believed anyway. Moreover, typically such beliefs arise without the person in question even being aware that the view they are accepting differs so radically from Bell’s own. Here we try to shed some light on the situation by focusing on the concept of local causality that is the heart of Bell’s theorem, and, in particular, by contrasting Bell’s own understanding with the analysis of Jon Jarrett which has been the most influential source, in recent decades, for the kinds of claims mentioned previously. We point out a crucial difference between Jarrett’s and Bell’s own understanding of Bell’s formulation of local causality, which turns out to be the basis for the erroneous claim, made by Jarrett and many others, that Bell misunderstood the implications of his own theorem.     

Quantum nonlocality for two-mode correlated states based on generalized pseudospin operators

In this Letter, we investigate the quantum nonlocality of two-mode correlated states. We find that the pseudospin formalism [Z.B. Chen, J.W. Pan, G. Hou, Y.D. Zhang, Phys. Rev. Lett. 88 (2002) 040406] generally fails to depict the nonlocality of the states when the photon number difference between the two modes is odd. The formalism is then generalized such that the nonlocality of a two-mode correlated state can be well revealed without regard to the difference. Later we consider the nonlocality of the two-mode intelligent SU(1,1) states in the generalized formalism and compare our results with the entanglement of the corresponding states.     

The Bell Theorem as a Special Case of a Theorem of Bass

The theorem of Bell states that certain results of quantum mechanics violate inequalities that are valid for objective local random variables. We show that the inequalities of Bell are special cases of theorems found 10 years earlier by Bass and stated in full generality by Vorob’ev. This fact implies precise necessary and sufficient mathematical conditions for the validity of the Bell inequalities. We show that these precise conditions differ significantly from the definition of objective local variable spaces and as an application that the Bell inequalities may be violated even for objective local random variables.     

Comments on Shimony’s “An Analysis of Stapp’s ‘A Bell-Type Theorem without Hidden Variables’ ”

The hidden-variable theorems of Bell and followers depend upon an assumption, namely the hidden-variable assumption, that conflicts with the precepts of quantum philosophy. Hence from an orthodox quantum perspective those theorems entail no faster-than-light transfer of information. They merely reinforce the ban on hidden variables. The need for some sort of faster-than-light information transfer can be shown by using counterfactuals instead of hidden variables. Shimony’s criticism of that argument fails to take into account the distinction between no-faster-than-light connection in one direction and that same condition in both directions. The argument can be cleanly formulated within the framework of a fixed past, open future interpretation of quantum theory, which neatly accommodates the critical assumptions that the experimenters are free to choose which experiments they will perform. The assumptions are compatible with the Tomonaga–Schwinger formulation of quantum field theory, and hence with orthodox quantum precepts, and with the relativistic requirement that no prediction pertaining to an outcome in one region can depend upon a free choice made in a region spacelike-separated from the first.     

Quantum mechanics needs no consciousness

It has been suggested that consciousness plays an important role in quantum mechanics as it is necessary for the collapse of wave function during the measurement. Here we formulated several predictions that follow from this hypothetical relationship and that can be empirically tested. Experimental results that are already available suggest falsification of these predictions. Thus, the suggested link between human consciousness and collapse of wave function does not seem viable. We discuss the implications of these conclusions on the role of the human observer for quantum mechanics and on the role of quantum mechanics for the observer's consciousness.     

Spatial Degrees of Freedom in Everett Quantum Mechanics

Stapp claims that, when spatial degrees of freedom are taken into account, Everett quantum mechanics is ambiguous due to a “core basis problem.” To examine an aspect of this claim I generalize the ideal measurement model to include translational degrees of freedom for both the measured system and the measuring apparatus. Analysis of this generalized model using the Everett interpretation in the Heisenberg picture shows that it makes unambiguous predictions for the possible results of measurements and their respective probabilities. The presence of translational degrees of freedom for the measuring apparatus affects the probabilities of measurement outcomes in the same way that a mixed state for the measured system would. Examination of a measurement scenario involving several observers illustrates the consistency of the model with perceived spatial localization of the measuring apparatus.This work was sponsored by the Air Force under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the U.S. Government.     

Bell’s Theorem without Inequalities and without Unspeakable Information

A proof of Bell’s theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.