In quantum physics, the density operator completely describes the state. Instead, in classical physics the mean value of physical quantities is evaluated by means of a probability distribution. We study the possibility to describe pure quantum states and events with classical probability distributions and conditional probabilities and prove that the distributions have to be nonlinear functions of the density operator. Some examples are considered. Finally, we deal with the exponential complexity problem of quantum physics and introduce the concept of classical dimension for a quantum system.
In quantum physics, the density operator completely describes the state. Instead, in classical physics the mean value of physical quantities is evaluated by means of a probability distribution. We study the possibility to describe pure quantum states and events with classical probability distributions and conditional probabilities and prove that the distributions have to be nonlinear functions of the density operator. Some examples are considered. Finally, we deal with the exponential complexity problem of quantum physics and introduce the concept of classical dimension for a quantum system.
We study the possibility to describe pure quantum states and evens with classical probability distributions and conditional probabilities and show that the distributions and/or conditional probabilities have to assume negative values, except for a simple model whose realistic space dimension is not smaller than the Hilbert space dimension of the quantum system. This gives a negative answer to a question proposed by Montina [Phys.Rev.Lett.{\bf 97}, 180401 (2006)] whether or not does there exist a classical theory whose phase-space dimension is much smaller than the Hilbert space dimension for any quantum system. Thus, any realistic theory of quantum mechanics with nonnegative probability distributions and conditional probabilities requires a number of variables grows exponentially with the physical size.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
In the standard interpretation of quantum mechanics, the state is described by an abstract wave function in the representation space. Conversely, in a realistic interpretation, the quantum state is replaced by a probability distribution of physical quantities. Bohm mechanics is a consistent example of realistic theory, where the wave function and the particle positions are classically defined quantities. Recently, we proved that the probability distribution in a realistic theory cannot be a quadratic function of the quantum state, in contrast to the apparently obvious suggestion given by the Born rule for transition probabilities. Here, we provide a simplified version of this proof.
Even after a hundred years of quantum theory, the true meaning of the quantum state has not yet been clearly understood. Does a pure quantum state, described by a wave function φ, represent a real physical quantity, comparable to position and momentum in classical mechanics, or does it, even as a pure state, only have a statistical significance, similar to a probability distribution? In this paper, we show that there is no contradiction between these two interpretations. Two concepts must be distinguished here: on the one hand, physical objects, which have a finite lifetime, and on the other hand, subsystems (i.e., tensor factors) of the Hilbert space of the universe. A complete description of an object at a point in time t is given by a linear subspace A of its Hilbert space, which, in the case of dim A = 1, can be specified by a wave function φ as A = [φ]. However, for every subsystem Q, for each point in time t, and for every condition B, the conditional quantum state W = W(Q,t|B) can be defined as a statistical operator on the Hilbert space of Q. Using W, certain (conditional) probabilities can be calculated. In the case of rank W = 1, the operator W can also be specified by a wave function φ, namely as W = |φ><φ|. It is the subsystems of the Hilbert space that play the role of "quantum systems" in the Copenhagen formalism, and the quantum states defined for them behave exactly as described by this formalism. If a wave function φ represents the description of a (quantum) object, then we are dealing with a real physical property, that is, φ plays an "ontic" role. However, if φ represents a quantum state, then its role is merely "statistical". If, in this case, condition B corresponds to the knowledge of a subject, φ plays an "epistemic" role. Additional Remarks The reasoning in this paper is largely based on a new systematic approach to solving the interpretation problem of quantum theory, which was originally published in 2018 (in German) under the title "Realistische Quantentheorie". A comprehensive account of this approach (in English) was published in February 2026 under the title "Realistic Quantum Theory: Solving the Interpretation Problem". It is available in the Zenodo repository (zenodo.18644694, see link below under "Related works" / "References"). The present document provides an English version of an earlier paper in German under the same title, which is also available in the Zenodo repository (zenodo.10694813, see link below).
We investigate epistemologically the meaning of probability as implied in quantum physics in connection with a proposed direct interpretation of the state function and of the related quantum theoretical quantities in terms of physical systems having physical properties, through an extension of meaning of the notion of physical quantity to complex mathematical expressions not reductible to simple numerical values. We show how the changes occurred in the implication of probabilities in quantum physics, from a penetration tool (probability as statistics and frequencies of occurencies of events) to a theoretical concept (probability given by a physically aimed at «amplitude of probability») actually make this view a somewhat «natural» one.