What non-additive integral for ensemble spaces?

TL;DR

The paper investigates a non-additive framework for ensemble spaces by introducing fraction capacity $\mathrm{fcap}_{\mathsf{e}}$ on a compact convex set $\mathcal{E}$ to represent both classical and quantum states. It seeks a non-additive integral $(E)\int_X$ that recovers the expectation values $F(\mathsf{e})$ for affine observables, reducing to the classical Riemann integral when additivity holds. The authors analyze the Sugeno and Choquet integrals, showing the Sugeno integral fails to converge to the Lebesgue integral in the additive limit, while the Choquet integral matches classical expectations but fails for a two-state quantum ensemble (the Bloch ball) with a concrete mismatch. This demonstrates that neither standard non-additive integral suffices, calling for new integral constructions and cross-field input from the fuzzy-measure community. The work clarifies fundamental limitations in unifying classical and quantum expectations via existing non-additive integrals and motivates further development of an appropriate integral for ensemble spaces.

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.

Figures (4)

• Figure 1: Examples of a classical discrete ensemble space over three elements (triangle) and of a two-dimensional quantum ensemble space (ball).
• Figure 2: For Choquet integral calculation, sets $A(s)$ of all pure states for which the statistical variable is greater than $s$. On the left, the Bloch ball with planes at constant $s$. On the right, the top case is when the value is greater than the maximum (empty set); the middle case is when the value is between the target point $\mathsf{o}$ and the maximum; the third case is when the value is below the target point.
• Figure 3: Geometric representation of the element $\mathsf{e}$ in $A(s)$ with the largest fraction of $\mathsf{o}$ when $\frac{1}{2} F(\mathsf{a}) + \frac{1}{2} F(\mathsf{b}) < s \leq F(\mathsf{b})$.
• Figure 4: The integrand $\mathrm{fcap}_{\mathsf{o}}(A(s))$ as a function of $s$.