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Fluctuation and Noise Letters

QUANTUM CORRELATIONS FROM CLASSICAL GAUSSIAN RANDOM VARIABLES: FUNDAMENTAL ROLE OF VACUUM NOISE

Andrei Yu. Khrennikov

Published Jun 29, 2010• 5 citations
Publisher

Abstract

We show that, in spite of rather common opinion, correlations of observables on subsystems of a composite quantum system can be represented as correlations of classical Gaussian variables. We restrict our model to the finite dimensional case (which is important, e.g., in quantum information theory). Here quantum correlations are represented by integrals with Gaussian densities which can be directly calculated. In particular, the EPR–Bell correlations for polarizations of entangled photons can be represented as correlations of Gaussian random variables. The tricky point of our construction is the necessity to introduce the Gaussian noise ("vacuum fluctuations") to obtain positively defined covariance matrices for "prequantum random variables." Quantum mechanics can be considered as a method to proceed by subtracting the influence of this Gaussian background noise.

DOI
10.1142/s0219477510000265

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We show that, in spite of rather common opinion, correlations of observables on subsystems of a composite quantum system can be represented as correlations of classical Gaussian variables. We restrict our model to the finite dimensional case (which is important, e.g., in quantum information theory). Here quantum correlations are represented by integrals with Gaussian densities which can be directly calculated. In particular, the EPR–Bell correlations for polarizations of entangled photons can be represented as correlations of Gaussian random variables. The tricky point of our construction is the necessity to introduce the Gaussian noise ("vacuum fluctuations") to obtain positively defined covariance matrices for "prequantum random variables." Quantum mechanics can be considered as a method to proceed by subtracting the influence of this Gaussian background noise.
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
We developed a purely field model of microphenomena—prequantum classical statistical field theory (PCSFT). This model not only reproduces important probabilistic predictions of quantum mechanics (QM) including correlations for entangled systems, but also gives a possibility to go beyond QM, i.e. to make predictions of phenomena that could be observed at the subquantum level. In this paper, we discuss one such prediction—the existence of nonlocal correlations between prequantum random fields corresponding to all quantum systems. (And by PCSFT, quantum systems are represented by classical Gaussian random fields and quantum observables by quadratic forms of these fields.) The source of these correlations is the common background field. Thus all prequantum random fields are 'entangled', but in the sense of classical signal theory. On the one hand, PCSFT demystifies quantum nonlocality by reducing it to nonlocal classical correlations based on the common random background. On the other hand, it demonstrates total generality of such correlations. They exist even for distinguishable quantum systems in factorizable states (by PCSFT terminology—for Gaussian random fields with covariance operators corresponding to factorizable quantum states).
In a series of previous papers we developed a purely field model of microphenomena, so called prequantum classical statistical field theory (PCSFT). This model not only reproduces important probabilistic predictions of QM including correlations for entangled systems, but it also gives a possibility to go beyond quantum mechanics (QM), i.e., to make predictions of phenomena which could be observed at the subquantum level. In this paper we discuss one of such predictions - existence of nonlocal correlations between prequantum random fields corresponding to {\it all} quantum systems. (And by PCSFT quantum systems are represented by classical Gaussian random fields and quantum observables by quadratic forms of these fields.) The source of these correlations is the common background field. Thus all prequantum random fields are "entangled", but in the sense of classical signal theory. On one hand, PCSFT demystifies quantum nonlocality by reducing it to nonlocal classical correlations based on the common random background. On the other hand, it demonstrates total generality of such correlations. They exist even for distinguishable quantum systems in factorizable states (by PCSFT terminology - for Gaussian random fields with covariance operators corresponding to factorizable quantum states).
One of the crucial differences between mathematical models of classical and quantum mechanics (QM) is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems, one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and the quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein dreamed of. QM is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem. We present the basic physical interpretation of prequantum classical statistical field theory in Sec. II. However, this is only the first step toward real physical theory.
The idea that quantum mechanics (QM) is simply a version of classical field theory is very old. Recently this idea has been realized in a new mathematical framework - on the basis of theory of random fields (L2-valued random variables). Surprisingly (for at least orthodox Copenhagenist) fundamental predictions of QM can be reproduced on the basis of a purely wave model (prequantum classical statistical field theory, PCSFT). In particular, all quantum correlations (including correlations of composite systems in entangled states) can be represented as correlations of classical random signals. These signals fluctuate at the space-time scale which is essentially finer than the scale of quantum measurements. At the moment we are not able to monitor such signals. However, one can expect that increasing of the precision of measurements will provide such a possibility. In this paper we show that bosonic and fermionic correlations can be obtained in the classical field framework. Finally, we stress that QM can be reduced to theory of classical random fields only in the presence of a relatively strong background field.
We show that QM can be represented as a natural projection of a classical statistical model on the phase space Ω = H × H , where H is the real Hilbert space. Statistical states are given by Gaussian measures on Ω having zero mean value and dispersion of very small magnitude α (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps f : Ω → R (functions of classical fields). The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Taylor expansion (up to the terms of the second order at the vacuum field point ψ vacuum ≡ 0) of variables f : Ω → R with respect to the small parameter √α The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space Ω. A Gaussian measure (statistical state) is represented in QM by its covariation operator. Equations of Schrödinger, Heisenberg and von Neumann are images of Hamiltonian dynamics on Ω. The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.