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Empirical frequencies and asymptotic density: a Koopman-von Neumann approach to probability

Carlo Pandiscia

Published Oct 7, 2026• 0 citations
Publisher

Abstract

We analyze the sequence of empirical frequencies resulting from repeated measurements of an observable in a fixed state. We propose that the appropriate mathematical formulation of the empirical law of probability is not classical pointwise convergence, but rather convergence in asymptotic density: deviations from the limiting probability can occur infinitely often, but their relative frequency becomes negligible. Using the Koopman-von Neumann theorem, we show that, under the ergodic hypothesis, the theoretically assigned probability must result compatible with the Ces\'aro limit of the empirical frequencies. This compatibility expresses a requirement of agreement between the assigned probability and the empirical data, not an operational definition of probability: the Ces\'aro limit is an ideal limiting value that is not empirically accessible from finite data.

DOI
10.5281/zenodo.23220476

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We analyze the sequence of empirical frequencies resulting from repeated measurements of an observable in a fixed state. We propose that the appropriate mathematical formulation of the empirical law of probability is not classical pointwise convergence, but rather convergence in asymptotic density: deviations from the limiting probability can occur infinitely often, but their relative frequency becomes negligible. Using the Koopman-von Neumann theorem, we show that, under the ergodic hypothesis, the theoretically assigned probability must result compatible with the Ces'aro limit of the empirical frequencies. This compatibility expresses a requirement of agreement between the assigned probability and the empirical data, not an operational definition of probability: the Ces'aro limit is an ideal limiting value that is not empirically accessible from finite data.
In this work, operator valued measures are used to study finite and infinite sequences of measurements. It is shown that to each such process qΔ there is uniquely associated a probability operator measure OqΔ which contains all the statistical properties of the process. In order to make this association for infinite processes, the operator valued equivalent of the Kolmogorov extension theorem is needed. This theorem is given and proved. It is then shown that for each qΔ and each set E of possible outcome sequences, there are two ways to find the probability that carrying out qΔ on a system in state ρ gives an outcome sequence φ in E. The usual method of repeating qΔ on ρ over and over again generates a sequence α of outcome sequences φ. The probability is obtained as the limit relative frequency that α(j) is in E, for j = 0, 1, …. The other, new, method is the repeated measurement of OEqΔ on ρ. The remarkable aspect of this equivalence is that the mathematical procedures of the usual method for determining if α(j) is in E or not `disappear' into the operators OEqΔ of the new method. This is discussed in some detail and examples are given.
Before the thermodynamic limit, macroscopic averages need not commute for a quantum system. As a consequence, aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis, when dealing with several observables. We propose an implementation of ideas that go back to John von Neumann’s writing about the macroscopic measurement. We apply our scheme to the relation between macroscopic autonomy and an H-theorem, and to the problem of equivalence of ensembles. In particular, we show how the latter is related to the asymptotic equipartition theorem. The main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate, as the size of the system gets larger, on the macroscopic values for the different macroscopic observables. Deviations from that law are governed by the entropy.
In this work, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo-Fraenkel set theory. The main aim of this work is to show that this definition, unlike that of earlier work, is sufficiently powerful to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory. We also extend, to the more powerful τ, some earlier results on randomness and the empirical determinability of the probability measure which a physical theory assigns to the outcome set of an infinite sequence of experiments.
The study of mathematical connections between operator-theoretic formulations of classical dynamics and quantum mechanics began at least as early as the 1930s in work of Koopman and von Neumann and was developed in later decades by many authors, often independently, into a framework now broadly known as Koopman-von Neumann representation of classical dynamics. This article surveys aspects of this framework for measure-preserving ergodic dynamical systems and connects it with recent approximation techniques for Koopman and transfer operators that are amenable to data-driven numerical implementation. In broad terms, these methods are based on representations of (i) classical observables as elements of an algebra of operators acting on a Hilbert space; and (ii) classical probability measures as elements of the state space of that algebra, with lifted versions of the Koopman and transfer operators inducing dynamical evolution of observables and states, respectively. A common theme underlying the techniques surveyed here is the use of reproducing kernel Hilbert spaces with coalgebra structure (so-called "reproducing kernel Hilbert algebras'') that aids the quantum representation of classical objects, as well as the use of Fock spaces to build approximation schemes with high expressivity and structure preservation properties (notably, preservation of positivity and multiplicativity of composition operators). Applications to quantum algorithms for approximating the Koopman evolution of observables in systems with pure point spectra are also discussed.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
We formulate an operator-algebraic mechanism by which exact Boolean records can arise from local completely positive quantum operations without being imposed as microscopic structure. The kinematic input is an algebraic process functional assigning probabilities to local normal completely positive operations in a finite operational context. From the predual response of a target algebra to source interventions, relative to a background strategy class, we define an influence algebra; exact events are then, by definition, the projections in its center. The dynamical question is whether nontrivial centers can be generated by coarse-graining rather than inserted through split-record laboratories. We address this question using state-preserving normal unital completely positive coarse-graining channels. If the Cesaro means of such a channel converge to a Choi-Effros infrared range and the range is asymptotically abelian in the GNS seminorm, then the represented infrared algebra is a commutative von Neumann algebra. Its projection lattice is therefore a complete Boolean algebra. We also give a finite-sector block-primitive criterion, motivated by locality and scrambling, which implies this asymptotic abelianness with exponential suppression of off-sector coherences and intra-sector fluctuations. The result is a conservative mathematical statement: classical facts are not identified with arbitrary projections of a Type-III local algebra, but with central projections selected by an asymptotically abelian completely positive infrared limit.