Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1512.00474

Indeterminate Probabilities and the Weak Quantum Law of Large Numbers

Fedor Herbut

Published Dec 1, 2015• 0 citations
arXivHTMLPDF

Abstract

The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes the necessary bridge between ensemble theory and experiment. The second analyzes the most important theoretical ensemble entity: the eigen-projector of the relative frequency operator. Its physical meaning is the experimental relative frequency. The third theorem formulates the quantum probabilistic convergence, which is the final result of this investigation.

DOI
10.48550/arxiv.1512.00474

Related papers

Derivation of quantum probability from measurement

Fedor Herbut

To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).

arXiv:1512.04028•Dec 13, 2015•0 citations

Questionable and Unquestionable in Quantum Mechanics

László E. Szabó, Márton Gömöri, Zalán Gyenis

According to the Kolmogorovian Censorship Hypothesis, everything that quantum theory says about the world in the language of the quantum mechanical Hilbert space formalism is actually about relationships between ordinary relative frequencies expressible in operational terms using classical Kolmogorovian probability theory. In other words, a quantum theoretical description of a system should in principle be translatable into a purely operational-probabilistic description. However, our goal in this paper is different; we do not want to deal with the problem how to translate the known theory of quantum mechanics into operational terms, or to reconstruct the theory from postulates which can be interpreted in operational terms. Our aim is somewhat broader and points in the opposite direction. We start with a general scheme for the operational description of an arbitrary physical system. The description is based solely on the notion of observable events (measurement operations and measurement results) and on general, empirically established simple laws concerning their relative frequency. These laws are so simple and fundamental that they apply equally to any physical system. In the first part of the paper, we outline the basic elements of such an operational-probabilistic theory. In the second part, we discuss how this operational-probabilistic description compares to the quantum mechanical description and to what extent the standard Hilbert space quantum mechanics can be regarded as a reformulation of the general operational-probabilistic theory.

arXiv:2309.01928•Sep 5, 2023•0 citations

Quantum mechanical probability and indeterminism

Jan von Plato

PROBABILITY IN THE OLD QUANTUM THEORY Probability was connected to quantum theory right from the start, in 1900, through the derivation of Planck's radiation law. But not much attention has been paid to the concept of probability in quantized radiation, or in the ‘quantum jumps’ from one energy level to another in an atom, and related problems. A whole chapter or book, instead of a section, could be written on the background of quantum theory in statistical mechanics and spectral analysis with this aspect in mind. Probabilistic properties were included in very many of the most important papers dealing with radiation between 1900 and 1925. It became also clear in time that Planck's law could not be derived from classical physics. Einstein admits this around 1908. Later he said, in a work of 1917, that it is a weakness of the theory of photon emission that ‘it leaves to “chance” the time and direction of the elementary processes,’ thereby, according to Pais (1982, p. 412), making explicit ‘that something was amiss with classical causality.’ It remains somewhat open how the origins and shifting interpretations of probability in the old quantum theory affected the acceptance of the probabilistic interpretation of the new quantum mechanics of 1925–1926, and of the indeterminism that found its confirmation in Heisenberg's uncertainty relation in 1927.

Cambridge University Press eBooks•Jan 28, 1994•0 citations

Foundations of a quantum probability theory

M. D. Srinivas

Statistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.

Journal of Mathematical Physics•Aug 1, 1975•50 citations

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

Carlo Pandiscia

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.

•Oct 7, 2026•0 citations

How macroscopic properties dictate microscopic probabilities

Yakir Aharonov, Benni Reznik

We argue that the quantum probability law follows, in the large N limit, from the compatibility of quantum mechanics with classical-like properties of macroscopic objects. For a finite sample, we find that likely and unlikely measurement outcomes are associated with distinct interference effects in a sample weakly coupled to an environment.

arXiv:quant-ph/0110093•Physical Review A•May 7, 2002•27 citations
The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes the necessary bridge between ensemble theory and experiment. The second analyzes the most important theoretical ensemble entity: the eigen-projector of the relative frequency operator. Its physical meaning is the experimental relative frequency. The third theorem formulates the quantum probabilistic convergence, which is the final result of this investigation.
To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).
According to the Kolmogorovian Censorship Hypothesis, everything that quantum theory says about the world in the language of the quantum mechanical Hilbert space formalism is actually about relationships between ordinary relative frequencies expressible in operational terms using classical Kolmogorovian probability theory. In other words, a quantum theoretical description of a system should in principle be translatable into a purely operational-probabilistic description. However, our goal in this paper is different; we do not want to deal with the problem how to translate the known theory of quantum mechanics into operational terms, or to reconstruct the theory from postulates which can be interpreted in operational terms. Our aim is somewhat broader and points in the opposite direction. We start with a general scheme for the operational description of an arbitrary physical system. The description is based solely on the notion of observable events (measurement operations and measurement results) and on general, empirically established simple laws concerning their relative frequency. These laws are so simple and fundamental that they apply equally to any physical system. In the first part of the paper, we outline the basic elements of such an operational-probabilistic theory. In the second part, we discuss how this operational-probabilistic description compares to the quantum mechanical description and to what extent the standard Hilbert space quantum mechanics can be regarded as a reformulation of the general operational-probabilistic theory.
PROBABILITY IN THE OLD QUANTUM THEORY Probability was connected to quantum theory right from the start, in 1900, through the derivation of Planck's radiation law. But not much attention has been paid to the concept of probability in quantized radiation, or in the ‘quantum jumps’ from one energy level to another in an atom, and related problems. A whole chapter or book, instead of a section, could be written on the background of quantum theory in statistical mechanics and spectral analysis with this aspect in mind. Probabilistic properties were included in very many of the most important papers dealing with radiation between 1900 and 1925. It became also clear in time that Planck's law could not be derived from classical physics. Einstein admits this around 1908. Later he said, in a work of 1917, that it is a weakness of the theory of photon emission that ‘it leaves to “chance” the time and direction of the elementary processes,’ thereby, according to Pais (1982, p. 412), making explicit ‘that something was amiss with classical causality.’ It remains somewhat open how the origins and shifting interpretations of probability in the old quantum theory affected the acceptance of the probabilistic interpretation of the new quantum mechanics of 1925–1926, and of the indeterminism that found its confirmation in Heisenberg's uncertainty relation in 1927.
Statistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.
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.