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Fluctuation and Noise Letters

QUANTUM PROBABILITIES FROM DETECTION THEORY FOR CLASSICAL RANDOM FIELD

Andrei Yu. Khrennikov

Published Dec 1, 2008• 4 citations
Publisher

Abstract

We present a model of detectors operating with classical random fields and producing probabilities described by the formalism of quantum mechanics. The basic assumption of this measurement theory is that the probability of detection of field (= signal) is proportional to the power of this field. (Therefore we call this approach "power signal-field detection theory" – PFSDT.) Our approach has some similarities with stochastic electrodynamics. However, formalisms describing random fields – "quantum fluctuations" – are different. Moreover, our approach all particles are represented by random fields, e.g., electronic or neutronic field. Thus we have various types of "quantum fluctuations".

DOI
10.1142/s0219477508005148

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•Aug 14, 2016•28 citations
We present a model of detectors operating with classical random fields and producing probabilities described by the formalism of quantum mechanics. The basic assumption of this measurement theory is that the probability of detection of field (= signal) is proportional to the power of this field. (Therefore we call this approach "power signal-field detection theory" – PFSDT.) Our approach has some similarities with stochastic electrodynamics. However, formalisms describing random fields – "quantum fluctuations" – are different. Moreover, our approach all particles are represented by random fields, e.g., electronic or neutronic field. Thus we have various types of "quantum fluctuations".
This paper is an attempt to reproduce predictions of quantum mechanics (QM) on probability of detection by starting with classical random field (e.g. random electromagnetic field). Thus it is an attempt to go beyond QM. The crucial point of our approach is that density operators of QM can be represented as covariance operators of classical random fields. Born’s rule can be obtained from measurement theory for classical random field under the assumption that the probability of detection of field is proportional to the power of this field.
The very old problem of the statistical content of quantum mechanics is studied in a novel framework. We show that quantum mechanics (QM) can be considered as a measurement model of prequantum classical statistical field theory (PCSFT): a pure field theory reproducing quantum correlations. We present a measurement scheme for classical signals which reproduces the basic rule of QM, the Born's rule. This is the scheme of threshold type detection. Calibration of detectors plays a crucial role. The size of the detection threshold is determined by the mean energy of a prequantum signal. QM is interpreted as theory of measurements of classical random signals with threshold type detectors which are calibrated in a proper way. The essence of this paper is the creation of a rigorous mathematical model of threshold detection of classical (ergodic) stochastic processes.
We present quantum mechanics (QM) as theory of special classical random signals. On one hand, this approach provides a possibility to go beyond conventional QM: to create a finer description of micro processes than given by the QM-formalism. In fact, we present a model with hidden variables of the wave-type. On the other hand, our approach establishes coupling between quantum and classical information theories. We recall that quantum information theory has already been used for description of the entropy of Gaussian input signals for noisy channels. The entropy of a classical random input was invented as the entropy of the quantum density operator corresponding to the covariance operator of the input process.1 In this paper, we proceed the other way around: we apply classical signal theory to create a measurement model which reproduces quantum probabilities.
The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The Schrödinger's dynamics is a special form of the linear dynamics of classical fields. Measurements are described as interactions of classical fields with detectors. Discontinuity, the "collapse of the wave function", has the trivial origin: usage of threshold type detectors. The von Neumann projection postulate can be interpreted as the formal mathematical encoding of the absence of coincidence detection in measurement on a single quantum system, e.g., photon's polarization measurement. Our model, prequantum classical statistical field theory (PCSFT), in combination with measurements by threshold detectors satisfies the quantum restriction on coincidence detections: the second order coherence is less than one (opposite to all known semiclassical and classical feld models). The basic rule of quantum probability, the Born's rule, is derived from properties of prequantum random felds interacting with threshold type detectors. Comparison with De Broglie's views to quantum mechanics as theory of physical waves with singularities is presented.
We present a purely wave model (based on classical random field) which reproduces quantum probabilities (given by the fundamental law of quantum mechanics, Born's rule) including probabilities for joint detection of a pair of quantum observables (e.g., spin or polarization projections). The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. The presence of the background field (vacuum fluctuations) is also the key-element of our prequantum model. It is of the classical signal type and the methods of classical signal theory (including statistical radiophysics) are used for its development. We stress that our prequantum model is not objective, i.e., the values of observables (clicks of detectors) cannot be assigned in advance, i.e., before measurement. Hence, the dilemma, nonobjectivity or nonlocality, is resolved in favor of nonobjectivity (our model is local of the classical field type). In particular, we reproduce the probabilities for the EPR-experiment for photon polarization and, hence, violate CHSH inequality for classical random signals (measured by the threshold type and properly calibrated detectors acting in the presence of the background field).
Recently it was shown that the main distinguishing features of quantum mechanics (QM) can be reproduced by a model based on classical random fields, so called prequantum classical statistical field theory (PCSFT). This model provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system (e.g., entangled systems), as averages with respect to fluctuations of classical (Gaussian) random fields. In this note we consider some consequences of PCSFT for quantum information theory. They are based on the observation \cite{W} of two authors of this paper that classical Gaussian channels (important in classical signal theory) can be represented as quantum channels. Now we show that quantum channels can be represented as classical linear transformations of classical Gaussian signal