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arXiv:quant-ph/9505025 · Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences

A local deterministic model of quantum spin measurement

T. N. Palmer

Published Dec 8, 1995• 29 citations
arXivHTMLPDF

Abstract

Abstract The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrized model, 'Q' for the state vector evolution of spin-1/2 particles during measurement is developed. Q draws on recent work on ‘riddled basins’ in dynamical systems theory, and is local, determin­istic, nonlinear and time asymmetric. Moreover, the evolution of the state vector to one of two chaotic attractors (taken to represent observed spin states) is effectively uncomputable. Motivation for considering this model arises from speculations about the (time asymmetric and uncomputable) nature of quantum gravity, and the (nonlinear) role of gravity in quantum state vector reduction. Although the evolution of Qs state vector cannot be determined by a numerical algorithm, the probability that initial states in some given region of phase space will evolve to one of these attractors, is itself computable. These probabilities can be made to correspond to observed quantum spin probabilities. In an ensemble sense, the evolution of the state vector to an attractor can be described by a diffusive random walk process, suggesting that deterministic dynamics may underlie recent attempts to model state vector evolution by stochastic equations. Bell’s theorem and a version of the Bell-Kochen-Specker quantum entanglement paradox, as illustrated by Penrose’s ‘magic dodecahedra’, are discussed using Q as a model of quantum spin measurement. It is shown that in both cases, proving an inconsistency with locality demands the existence of definite truth values to certain counterfactual propositions. In Q these deterministic propositions are uncomputable, and no non-algorithmic mathematical solution is either known or suspected. Adapt­ing the mathematical formalist approach, the non-existence of definite truth values to such counterfactual propositions is posited. No inconsistency with experiment is found. As a result, it is claimed that Q is not constrained by Bell’s inequality, locality and determinism notwithstanding.

DOI
10.1098/rspa.1995.0145

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The quantum theory (QT) and new stochastic approaches have no deterministic prediction for a single measurement or for a single time‐series of events observed for a trapped ion, electron or any other individual physical system. The predictions of QT being of probabilistic character apply to the statistical distribution of the results obtained in various experiments. The Copenhagen interpretation (CI) of QT acknowledged the abstract and statistical character of the predictions of QT but at the same time claimed that a state vector Ψ provided complete description of each individual physical system. The assigning the state vector to an individual physical system together with a postulate of its instantaneous reduction in the measurements was shown by Einstein Podolski and Rosen to lead to so called EPR paradox for the experiments with the entangled pairs of the particles. EPR concluded that a state vector could not provide a complete description of the individual systems and the question arose whether the probabilistic predictions of QT could be derived from some more fundamental spatio‐temporal deterministic description of the invisible sub‐phenomena by introduction of supplementary parameters. The experimental violation of the Bell inequalities in the spin polarization correlation experiments (SPCE) which were the implementations of Bohm and EPR gedanken experiments eliminated so called local and realistic models of the sub‐phenomena. Quite often the violation of these inequalities has been incorrectly interpreted as a proof of the completeness of QT or as the violation of the locality and causality in the micro‐world. The local and realistic models overlooked the fact that an experimental outcome is only the information about a particular system‐system or system‐instrument interaction. Namely the quantum phenomena are described in terms of the probabilities. It is well known that the probability distribution is not an attribute of a dice but it is a characteristic of a whole random experiment: “rolling a dice”. In this paper we will underline: that the quantum probabilities do not describe the lack of knowledge about the physical systems but the lack of knowledge about the experiments and therefore they are “contextual” and that the statistical long range correlations between two random variables X and Y are not a proof of any causal relation between these variables. Moreover any probabilistic model used to described a random experiment is consistent only with a specific protocol telling how the random experiment has to be performed. The probabilistic model used to prove Bell inequalities implied a protocol completely inappropriate and impossible to implement for SPCE . We conclude that the meaningful question about the predictable completeness of QT is still open and it should be answered by a careful and unconventional analysis of the experimental data. The correct understanding of statistical and contextual character of QT is essential for the research in the domain of quantum information and quantum computing.
Abstract Quantum states in superposition cannot be observed, suggesting that they merely embody information on possible measurement results. Yet, we think of them as describing physical systems that evolve in time according to given mathematical equations. Furthermore, evolution takes place in physical space-time, and local causality is imposed on space-time by Special Relativity (a sequence of cause and effect that constitutes a fundamental principle by which we think about and do scientific work). It is therefore strange that there is non–locality and violation of causality embedded in Quantum Mechanics. We try to give here a first step to see whether a deterministic and local interpretation is possible. We will show that Bell’s inequality is not conclusive about the non–local nature of Quantum Mechanics, in that there exists an interpretation of reality (which we put forward) which is local and in which Bell’s inequality cannot be derived. Assuming determinism and locality a new Bell-like inequality is derived, which is always satisfied by Quantum Mechanics and which allows for local hidden variables [1].
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Eighty years ago Einstein, Podolsky and Rosen demonstrated that instantaneous reduction of wave function, believed to describe completely a pair of entangled physical systems, led to EPR paradox. The paradox disappears in statistical interpretation of quantum mechanics (QM) according to which a wave function describes only an ensemble of identically prepared physical systems. QM predicts strong correlations between outcomes of measurements performed on different members of EPR pairs in far-away locations. Searching for an intuitive explanation of these correlations John Bell analysed so called local realistic hidden variable models and proved that correlations consistent with these models satisfy Bell inequalities which are violated by some predictions of QM and by experimental data. Several different local models were constructed and inequalities proven. Some eminent physicists concluded that Nature is definitely nonlocal and that it is acting according to a law of nonlocal randomness. According to these law perfectly random, but strongly correlated events, can be produced at the same time in far away locations and a local and causal explanation of their occurrence cannot be given. We strongly disagree with this conclusion and we prove the contrary by analysing in detail some influential finite sample proofs of Bell and CHSH inequalities and so called Quantum Randi Challenges. We also show how one can win so called Bell's game without violating locality of Nature. Nonlocal randomness is inconsistent with local quantum field theory, with standard model in elementary particle physics and with causal laws and adaptive dynamics prevailing in the surrounding us world. The experimental violation of Bell-type inequalities does not prove the nonlocality of Nature but it only confirms a contextual character of quantum observables and gives a strong argument against counterfactual definiteness and against a point of view according to which experimental outcomes are produced in irreducible random way.
The entanglement and the violation of Bell and CHSH inequalities in spin polarization correlation experiments (SPCE) is considered to be one of the biggest mysteries of Nature and is called quantum nonlocality . In this paper we show once again that this conclusion is based on imprecise terminology and on the lack of understanding of probabilistic models used in various proofs of Bell and CHSH theorems. These models are inconsistent with experimental protocols used in SPCE. This is the only reason why Bell and CHSH inequalities are violated. A probabilistic non-signalling description of SPCE, consistent with quantum predictions, is possible and it depends explicitly on the context of each experiment. It is also deterministic in the sense that the outcome is determined by supplementary local parameters describing both physical signals and measuring instruments. The existence of such description gives additional arguments that quantum theory is emergent from some more detailed theory respecting causality and local determinism. If quantum theory is emergent then there exist perhaps some fine structures in time-series of experimental data which were not predicted by quantum theory. In this paper we explain how a systematic search for such fine structures can be done. If such reproducible fine structures were found it would show that quantum theory is not predictably complete, which would be a major discovery.
A realistic measurement-free theory for the quantum physics of multiple qubits is proposed. This theory is based on a symbolic representation of a fractal state-space geometry which is invariant under the action of deterministic and locally causal dynamics. This symbolic representation is constructed from self-similar families of quaternionic operators. Using number-theoretic properties of the cosine function, the statistical properties of the symbolic representation of the invariant set are shown to be consistent with the contextual requirements of the Kochen-Specker theorem, are not constrained by Bell inequalities, and mirror the statistics of entangled qubits. These number-theoretic properties in turn reflect the sparseness of the invariant set in state space, and relate to the metaphysical notion of counterfactual incompleteness. Using the concept of probability, the complex Hilbert Space can be considered the completion of this symbolic representation into the state space continuum. As a result, it is proposed that the complex Hilbert Space should merely be considered a computational convenience in the light of the algorithmic intractability of the invariant set geometry, and consequently the superposed state should not be considered a fundamental aspect of physical theory. The physical basis for the proposed theory is relativistic gravity; for example the symbols used to describe the invariant set themselves label gravitationally distinct cosmological space-times. This implies that the very notion of a `quantum theory of gravity' may be profoundly misguided - erroneously putting the quantum cart before the gravitational horse. Here some elements of an alternative `gravitational theory of the quantum' are proposed, based on a deterministic and locally causal theory of gravity which extends general relativity by being geometric in both space-time and state space.