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:2602.01470

What non-additive integral for ensemble spaces?

Gabriele Carcassi, Christine A. Aidala, Tobias Thrien

quant-phmath-phSubmitted Feb 1, 2026• Announced Feb 3, 2026• 0 citations
arXivHTMLPDF

Abstract

In a previous work we were able to define a non-additive measure that can be used to represent both classical and quantum states in physics. We further extended this idea to work on a generic space of statistical ensembles (i.e. an ensemble space) in a way that connects to Choquet theory. The question of which non-additive integral is suitable to generalize the notion of expectation value remains open. In this paper we show that the Sugeno and Choquet integrals are not suitable.

Comments
6 pages, 4 figures, prepared for the 21st International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Rome, Italy, June 15-19, 2026

Related papers

All quantum expectation values as classical statistical mean values

Antonio Cassa

Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.

arXiv:0706.2603•Jun 18, 2007•0 citations

A probabilistic formulation of quantum theory. III

Thomas Wenning, Alexander Bach

A representation of the expectation value of quantum mechanics, which recently has been set up for normal states, is generalized to singular states. It is shown that states can be represented by finitely additive measures on the state space, which are σ-additive if and only if the states are normal.

Journal of Mathematical Physics•May 1, 1983•1 citation

Partial order and topology of Hermitian matrices and quantum Choquet integrals for density matrices with given expectation values

Apostol Vourdas

The set M of d × d Hermitian matrices (observables) is studied as a partially ordered set with the Löwner partial order. Upper and lower sets in it, define the concept of cumulativeness (used mainly with scalar quantities) in the context of Hermitian matrices. Partial order and topology are intimately related to each other and the set M of Hermitian matrices is also studied as a topological space, where open and closed sets are the upper and lower sets. It is shown that the set M of Hermitian matrices is a T 0 topological space, and its subset D of density matrices is Hausdorff totally disconnected topological space. These ideas are a prerequisite for studying quantum Choquet integrals with Hermitian matrices (as opposed to classical Choquet integrals with scalar quantities). Capacities (non-additive probabilities), cumulative quantities that involve Hermitian matrices, and Möbius transforms that remove the overlaps between non-commuting observables, are used in quantum Choquet integrals. An application of the formalism is to find a density matrix, with given expectation values with respect to n (non-commuting) observables. Examples of calculations of such a density matrix (with quantified errors in its expectation values), are presented.

arXiv:2506.06794•Journal of Geometry and Physics•Jun 3, 2025•1 citation

How Quantum Mechanics Requires Non-Additive Measures

Gabriele Carcassi, C. Aidala

Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and, therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e., no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.

arXiv:2311.01951•Entropy•Dec 18, 2023•1 citation

Wave Theory in Measure Space: From Choquet Capacities to Additive Measure Dynamics

Viktoras Rimsha, Laimontas Rimsha

This paper formulates a new framework for general wave theory designed to describe the intensity of interfering waves. Since traditional individual intensities (the squares of amplitude moduli) do not sum up to the total observed intensity during interference, classical measure theory in coordinate space breaks down. To resolve this non-additivity problem, we apply the concept of Choquet capacities, which has not been used in physics until now. It is postulated that the total observed intensity of the system is a global Choquet capacity, which simultaneously acts as a full, additive Lebesgue measure. The true intensities of individual waves are defined as additive Lebesgue measures that sum up to the total Choquet capacity. Utilizing the proportionality between the sought measures and the Choquet capacities of individual waves, analytical expressions for the true intensities are derived for an arbitrary number of alternatives. A temporal evolution via the Fokker-Planck equation is formulated for the obtained additive measures, which opens up avenues for a completely new wave analysis. In the context of quantum mechanics, these intensity-measures naturally acquire a probabilistic meaning, axiomatically resolving fundamental questions of interpretation.

•Jul 11, 2026•0 citations

Quantum statistics as Choquet exponential distributions

Negin Bayrami, Hosein Mohammadzadeh, Hamzeh Agahi +2 more

Abstract The classical Maxwell–Boltzmann distribution is an exponential law. Using the Choquet integral framework, one can introduce nonlinear (non-additive) capacities and thereby define new families of distributions belonging to the class of Choquet–exponential distributions. In this paper, we assume that a suitable nonlinear measure can be constructed such that the quantum Bose–Einstein and Fermi–Dirac distributions emerge naturally as Choquet–exponential forms. After calculating this measure, we analyze its mathematical and physical properties. According to our adopted convention, the thermodynamic curvature is positive for intrinsic attractive correlations and negative for repulsive ones. Considering the concavity and convexity of the nonlinear Choquet measure, we propose a direct interpretation of statistical interactions in terms of the geometric behavior of the capacity itself. Finally, the same approach is applied to Tsallis quantum statistics, and the proposed interpretation is verified up to second order in the deviation from the standard cases.

Journal of Physics A Mathematical and Theoretical•Feb 24, 2026•0 citations
In a previous work we were able to define a non-additive measure that can be used to represent both classical and quantum states in physics. We further extended this idea to work on a generic space of statistical ensembles (i.e. an ensemble space) in a way that connects to Choquet theory. The question of which non-additive integral is suitable to generalize the notion of expectation value remains open. In this paper we show that the Sugeno and Choquet integrals are not suitable.
Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.
The set M of d × d Hermitian matrices (observables) is studied as a partially ordered set with the Löwner partial order. Upper and lower sets in it, define the concept of cumulativeness (used mainly with scalar quantities) in the context of Hermitian matrices. Partial order and topology are intimately related to each other and the set M of Hermitian matrices is also studied as a topological space, where open and closed sets are the upper and lower sets. It is shown that the set M of Hermitian matrices is a T 0 topological space, and its subset D of density matrices is Hausdorff totally disconnected topological space. These ideas are a prerequisite for studying quantum Choquet integrals with Hermitian matrices (as opposed to classical Choquet integrals with scalar quantities). Capacities (non-additive probabilities), cumulative quantities that involve Hermitian matrices, and Möbius transforms that remove the overlaps between non-commuting observables, are used in quantum Choquet integrals. An application of the formalism is to find a density matrix, with given expectation values with respect to n (non-commuting) observables. Examples of calculations of such a density matrix (with quantified errors in its expectation values), are presented.
Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and, therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e., no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.
This paper formulates a new framework for general wave theory designed to describe the intensity of interfering waves. Since traditional individual intensities (the squares of amplitude moduli) do not sum up to the total observed intensity during interference, classical measure theory in coordinate space breaks down. To resolve this non-additivity problem, we apply the concept of Choquet capacities, which has not been used in physics until now. It is postulated that the total observed intensity of the system is a global Choquet capacity, which simultaneously acts as a full, additive Lebesgue measure. The true intensities of individual waves are defined as additive Lebesgue measures that sum up to the total Choquet capacity. Utilizing the proportionality between the sought measures and the Choquet capacities of individual waves, analytical expressions for the true intensities are derived for an arbitrary number of alternatives. A temporal evolution via the Fokker-Planck equation is formulated for the obtained additive measures, which opens up avenues for a completely new wave analysis. In the context of quantum mechanics, these intensity-measures naturally acquire a probabilistic meaning, axiomatically resolving fundamental questions of interpretation.
Abstract The classical Maxwell–Boltzmann distribution is an exponential law. Using the Choquet integral framework, one can introduce nonlinear (non-additive) capacities and thereby define new families of distributions belonging to the class of Choquet–exponential distributions. In this paper, we assume that a suitable nonlinear measure can be constructed such that the quantum Bose–Einstein and Fermi–Dirac distributions emerge naturally as Choquet–exponential forms. After calculating this measure, we analyze its mathematical and physical properties. According to our adopted convention, the thermodynamic curvature is positive for intrinsic attractive correlations and negative for repulsive ones. Considering the concavity and convexity of the nonlinear Choquet measure, we propose a direct interpretation of statistical interactions in terms of the geometric behavior of the capacity itself. Finally, the same approach is applied to Tsallis quantum statistics, and the proposed interpretation is verified up to second order in the deviation from the standard cases.