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arXiv:2605.24923

Singularising preserving countable additivity quantum channels on quantum measurable cardinals

Sviatoslav V. Dzhenzher

quant-phmath.FASubmitted May 24, 2026• Announced May 26, 2026• 0 citations• Revised Jun 22, 2026 (v2)
v1 · May 24, 2026v2 · Jun 22, 2026v3 · Aug 4, 2026
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Abstract

We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral. Utilizing the classical Yosida--Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states. We establish a suffucient condition under which a channel preserves $σ$-additivity and exhibits a singularising property, completely suppressing the normal component of any incoming state. In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly $σ$-additive states. Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Cesàro averages in the strong operator topology.

Comments
15 pages, no figures; minor revision was made

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We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral.
Utilizing the classical Yosida--Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states.
We establish a suffucient condition under which a channel preserves σσσ-additivity and exhibits a singularising property, completely suppressing the normal component of any incoming state.
In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly σσσ-additive states.
Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Cesàro averages in the strong operator topology.
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.
This paper investigates the interplay between the properties of quantum states on the Hilbert space ℓ2(κ)\ell_2(κ)ℓ2​(κ) and the set-theoretic nature of the cardinal κκκ. We focus on the existence of singular σσσ-additive states~--- functionals whose induced measures are σσσ-additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of κκκ, their structural and dynamical properties remain largely unexplored. We prove that any σσσ-additive state on the diagonal algebra is representable as a Pettis integral over a singular σσσ-additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using σσσ-complete ultrafilters that map normal states to singular σσσ-additive states, effectively <<archiving>> information into the singular part of the state space.
In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on M⊗ˉN\mathcal{M}\bar{\otimes}\mathcal{N}M⊗ˉ​N but on the tensor product of M\mathcal{M}M with the \emph{opposite} algebra of N\mathcal{N}N, or equivalently with a commutant; returning to N\mathcal{N}N itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is ∗*∗-anti-isomorphic to itself. Naturality cannot be dropped: whenever M\mathcal{M}M is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.