Classical signal model for quantum channels

Recently it was shown that the main distinguishing features of quantum mechanics (QM) can be reproduced by a model based on classical random fields, the 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. We consider some consequences of the PCSFT for quantum information theory. They are based on our previous observation 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 transforms of classical Gaussian signals.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

Prequantum Classical Statistical Field Theory: Schrödinger Dynamics of Entangled Systems as a Classical Stochastic Process

The idea that quantum randomness can be reduced to randomness of classical fields (fluctuating at time and space scales which are essentially finer than scales approachable in modern quantum experiments) is rather old. Various models have been proposed, e.g., stochastic electrodynamics or the semiclassical model. Recently a new model, so called prequantum classical statistical field theory (PCSFT), was developed. By this model a “quantum system” is just a label for (so to say “prequantum”) classical random field. Quantum averages can be represented as classical field averages. Correlations between observables on subsystems of a composite system can be as well represented as classical correlations. In particular, it can be done for entangled systems. Creation of such classical field representation demystifies quantum entanglement. In this paper we show that quantum dynamics (given by Schrödinger’s equation) of entangled systems can be represented as the stochastic dynamics of classical random fields. The “effect of entanglement” is produced by classical correlations which were present at the initial moment of time, cf. views of Albert Einstein.     

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.     

Emergence of Quantum Mechanics from Theory of Random Fields

Our aim in this paper is to enlighten the possibility to treat quantum mechanics as emergent from a kind of classical physical model, in spite of recent remarkable experiments demonstrating a violation of the Bell inequality. To proceed in a rigorous way, we use the methodology of ontic–epistemic modeling of physical phenomena. This methodology is rooted in the old Bild conception about theoretical and observational models in physics. This conception was elaborated in the fundamental works of Hertz, Boltzmann, and Schrödinger. Our ontic model (generating the quantum model) is of the random field type, prequantum classical statistical field theory (PCSFT). We present a brief review of its basic features without overloading the presentation by mathematical details. Then we show that the Bell inequality can be violated not only at the epistemic level, i.e., for observed correlations, but even at the ontic level, for classical random fields. We devote the important part of the paper to an analysis of the internal energy structure of prequantum random fields and their coupling with the background field of subquantum fluctuations. Finally, we present a unified picture of the microworld based on the composition of prequantum random fields from elementary fluctuations. Since quantum systems are treated as the symbolic representation of prequantum fields, this picture leads to a unifying treatment of all quantum systems as special blocks of elementary fluctuations carrying negligibly small energies.     

Nonlinear Schrödinger equations from prequantum classical statistical field theory

We derive some important features of the standard quantum mechanics from a certain classical-like model—prequantum classical statistical field theory, PCSFT. In this approach correspondence between classical and quantum quantities is established through asymptotic expansions. PCSFT induces not only linear Schrödinger's equation, but also its nonlinear generalizations. This coupling with “nonlinear wave mechanics” is used to evaluate the small parameter of PCSFT.     

Quantum-State Dynamics as Linear Representation of Classical (Nonlinear) Stochastic Dynamics

We show that the basic equation of the theory of open systems, the Gorini–Kossakowski–Sudarshan–Lindblad equation, as well as its linear and nonlinear generalizations have a natural classical probabilistic interpretation – within the framework of prequantum classical statistical field theory. The latter gives an example of the classical probabilistic model (with random fields as subquantum variables) reproducing the basic probabilistic predictions of quantum mechanics.     

Interplay between classical and quantum signals: partial trace and measurement channels

We continue the study of similarities between quantum information theory and theory of classical Gaussian signals. The possibility of using quantum entropy for classical Gaussian signals was explored a long time ago. Recently we demonstrated that some basic quantum channels can be represented as linear transforms of classical Gaussian signals. Here we consider bipartite quantum systems and show that an important quantum channel given by the partial trace operation has a simple classical representation, namely, a coordinate projection of a classical “prequantum signal.” We also consider the classical signal realization of quantum channels corresponding to state transforms in the process of measurement. The latter induces a difficult interpretational problem — the output signal corresponding to one system depends on a measurement that has been done on the second system. This situation might be interpreted as a sign of quantum nonlocality, action at a distance. Although we do not exclude such a possibility, i.e., that, in complete accordance with Bell, the creation of a realistic prequantum model is impossible without action at a distance, we found another interpretation of this situation that is not related to quantum nonlocality.     

Quantum Randomness as a Result of Random Fluctuations at the Planck Time Scale?

We show that the mathematical formalism of the quantum statistical model can be interpreted as a method for approximation of classical (measure-theoretic) averages on the infinite-dimensional phase space. The technique of approximation is based on the Taylor expansion of functionals of classical fields. To find the order of the deviation of quantum statistical predictions from the classical predictions, we use the time-scaling arguments. We show that quantum randomness might be considered as the result of random fluctuations at the Planck time-scale.     

Where is an object before you look at it?

According to the standard interpretation of quantum mechanics (QM), no meaning can be assigned to the statement that a particle has a precise value of any one of the variables describing its physical propertes before having interacted with a suitable measuring instrument. On the other hand, it is well known that QM tends to classical statistical mechanics (CSM) when a suitable classical limit is performed. One may ask therefore how is it that in this limit, the statement, meaningless in QM, that a given variable has always a precise value independently of having been measured, gradually becomes meaningful. In other words, one may ask how can it be that QM, which is a theory describing the intrinsically probabilistic properties of a quantum object, becomes a statistical theory describing a probabilistic knowledge of intrinsically well determined properties of classical objects.In the present paper we try to answer to this question and show that an inconsistency arises between the conventional interpretation of CSM which presupposes objectively existing Newtonian trajectories, and the standard interpretation of QM. We conclude that the latter needs revisiting unnless we wish to adopt a strictly subjective conception of the world around us, implying that macroscopic objects as well are not localized anywhere before we look at them.     

Minimal Coupling in Koopman-von Neumann Theory

Classical mechanics (CM), like quantum mechanics (QM), can have an operatorial formulation. This was pioneered by Koopman and von Neumann (KvN) in the 1930s. They basically formalized, via the introduction of a classical Hilbert space, earlier work of Liouville who had shown that the classical time evolution can take place via an operator, nowadays known as the Liouville operator. In this paper we study how to perform the coupling of a point particle to a gauge field in the KvN version of CM. So we basically implement at the classical operatorial level the analog of the minimal coupling of QM. We show that, differently than in QM, not only the momenta but also other variables have to be coupled to the gauge field. We also analyze in detail how the gauge invariance manifests itself in the Hilbert space of KvN and indicate the differences with QM. As an application of the KvN method we study the Landau problem proving that there are many more degeneracies at the classical operatorial level than at the quantum one. As a second example we go through the Aharonov-Bohm phenomenon showing that, at the quantum level, this phenomenon manifests its effects on the spectrum of the quantum Hamiltonian while at the classical level there is no effect whatsoever on the spectrum of the Liouville operator.     

Born's rule from classical random fields

This Letter is an attempt to go beyond QM. In our approach 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.     

A Classical Analog of Random Quantum States

We examine the statistical properties of a pure quantum state randomly chosen with respect to the uniform measure in a Hilbert space. Namely, we consider the distribution of outcomes of a fixed measurement performed on the random quantum state. We show that such distribution is completely analogous to the distribution of measurement outcomes of an a priori unknown classical random system. In particular, Shannon entropies of both distributions coincide. We study this correspondence between quantum and classical random systems and clarify its origin.     

Entanglement and Classical Correlations in the Quantum Frame

The frame of classical probability theory can be generalized by enlarging the usual family of random variables in order to encompass nondeterministic ones. This leads to a frame in which two kinds of correlations emerge: the classical correlation that is coded in the mixed state of the physical system and a new correlation, to be called probabilistic entanglement, which may occur also at pure states. We examine to what extent this characterization of correlations can be applied to quantum mechanics. Explicit calculations on simple examples outline that a same quantum state can show only classical correlations or only entanglement depending on its statistical content; situations may also arise in which the two kinds of correlations compensate each other.     

On the probabilistic treatment of fields

Some basic problems of the probabilistic treatment of fields are considered, proceeding from the fundamentals of the complete probability theory. Two essentially equivalent definitions of random fields related to continuous objects are suggested. It is explained why the conventional classical probabilistic treatment generally is inapplicable to fields in principle. Two types of finite-dimensional random variables created by random fields are compared. Some general regularities related to Lagrangian and Hamiltonian partial equations, obtainable proceeding from the corresponding sets of ordinary differential equations, are revealed by using the functional derivative defined anew. It is shown that Hamiltonian random fields give rise to two types of Hamiltonian random variables, variables of the second type being those considered in the author's previous paper and immediately suited to the quantum approach. The results obtained are illustrated by some general examples. Critical remarks concerning second quantization are made, demonstrating the artificiality of this method. It is emphasized that the given probabilistic consideration of fields cannot be directly applied to, for instance, the electromagnetic field, which needs a special treatment.