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arXiv:2508.14326 · Fundamenta Mathematicae

Non-representable quantum measures

Alexandru Chirvăsitu

Published Sep 15, 2026• 0 citations
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Abstract

Grade-d measures on a σ-algebra A⊆2X over a set X are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum mechanics. Every signed polymeasure λ on (X,A)d produces a grade-d measure as its diagonal λ˜(A):=λ(A,…,A), and we prove that as soon as d≥2, measures (as opposed to polymeasures) do not suffice: the separate σ-additivity of a λ producing μ=λ˜ cannot, generally, be amplified to global σ-additivity. This amends a result in the literature, asserting the contrary in case d=2.

DOI
10.4064/fm250910-3-4

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Grade-d measures on a σ-algebra A⊆2X over a set X are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions initially introduced as interference operators in quantum mechanics. Every signed polymeasure λ on (X,A)d produces a grade-d measure as its diagonal λ˜(A):=λ(A,…,A), and we prove that as soon as d≥2, measures (as opposed to polymeasures) do not suffice: the separate σ-additivity of a λ producing μ=λ˜ cannot, generally, be amplified to global σ-additivity. This amends a result in the literature, asserting the contrary in case d=2.
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Denoted by M(A), QM(A) and SQM(A) the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F(A) of all real-valued functions on A and prove that M(A) is a face of SQM(A). It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix M µ to a quantum measure µ on the power set A n of an n-element set X, it is proved that µ ≪ ν (resp. µ ⊥ ν) if and only if M µ ≪ M ν (resp. M µ M ν =0). Also, it is shown that two nontrivial measures µ and ν are mutually absolutely continuous if and only if µ·ν∈QM(A n ). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on A n is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved.
Quantum measure spaces possess a certain “quantum weirdness” and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as “quantum coins” and particle-an tip article pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don't need quantum mechanics or measure theory to understand this article.
We study the interplay between the properties of quantum states on the Hilbert space ℓ 2 ( κ ) and the set-theoretic nature of the cardinal κ . We focus on the existence of singular σ -additive states – functionals whose induced measures are σ -additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of κ , their structural and dynamical properties remain largely unexplored. We prove that any σ -additive state on the diagonal algebra is representable as a Pettis integral over a singular σ -additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using σ -complete ultrafilters that map normal states to singular σ -additive states, effectively «archiving» information into the singular part of the state space.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B \mathcal{B} B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B \mathcal{B} B X constitute a convex set whose extreme points are degenerate probability measures.
ly R is the Klein operator ±exp[iπ(Ĥ − E0)] while in Schrodinger coordinate representation, first investigated by Yang, R is realised by ±P where P is the parity operator: P|x >= ±|x >, P−1 = P, P = 1, PxP−1 = −x. (7) The basic (anti-)commutation relation (1) and (3) together with their derived relation (4) will be referred to here as constituting the WH algebra which is in fact a parabose algebra for one degree of freedom. We shall assume here in after, without loss of generality, that c is positive, i.e. c =| c |> 0. Thus, in coordinate representation the generalized quantization a la Wigner requires that x = x, ∞− < x < ∞, p = −i d dx + ic 2x P, R = P. (8) Indeed, following Yang representation [6] we obtain the coordinate representation for the ladder operators as given by â± −→ a±c 2 = 1 √ 2 ( ± d dx ∓ c 2x P − x ) . (9) Yang’s wave mechanical description was further investigated in [7, 8]. The Wigner-Heisenberg Algebra in Quantum Mechanics http://dx.doi.org/10.5772/55994 479 4 Advances in Quantum Mechanics The present author have applied a super-realization so that R = Σ3 to illustrate the first application of our operator method to the cases of the Hamiltonian of an isotonic oscillator (harmonic plus a centripetal barrier) system. To obtain a super-realisation of the WH algebra, we introduce, in addition to the usual bosonic coordinates (x,−i d dx ), the fermionic ones b∓(= (b±)†) that commute with the bosonic set and are represented in terms of the usual Pauli matrices Σi, (i = 1, 2, 3). Indeed, expressing a ±( c 2 ) in the following respective factorised forms: a( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( 1 2 x)( d dx )exp( −1 2 x2)x−(1/2)cΣ3 (10) a−( c 2 ) = 1 √ 2 Σ1x (1/2)cΣ3 exp( −1 2 x)( −d dx )exp( 1 2 x2)x−(1/2)cΣ3 , (11) where these lader operators satisfy the algebra of Wigner-Heisenberg. From (1), (11) and (10) the Wigner Hamiltonian becomes H( c 2 ) = 1 2 [ a( c 2 ), a−( c 2 ) ] + = ( H−( c 2 − 1) 0 0 H+( c 2 − 1) = H−( c 2 ) ) , (12) where the even and odd sector Hamiltonians are respectively given by H−( c 2 − 1) = 1 2 { − d 2 dx2 + x + 1 x2 ( c 2 )( c 2 − 1) }