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Mathematics and Theoretical Computer Science

Quantum channels preserving sigma-additivity and Ulam measurable cardinals

S. V. Dzhenzher

Published Jul 29, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.26907/2949-3919.2026.2.104-117

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•Jan 1, 2013•5 citations
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.
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.
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:
In this paper, the generalized quantum states, i.e. positive and normalized linear functionals on [Formula: see text]-algebras, are studied. First, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states cannot be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the Gelfand–Neumark–Segal (GNS) construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of [Formula: see text]-algebras, and theory of projections. Second, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners cannot have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand–Pettis integral. We also give physical interpretations and examples in different sections of this work.
The Bloch representation of an n-qubit channel provides a way to represent quantum channels as certain affine transformations on [special characters omitted]. In higher dimensions (n > 1), the correspondence between quantum channels and their Bloch representations is not well-understood. Partly motivated by the ability to simplify the calculation of information theoretic quantities of a qubit channel using the Bloch representation, in this thesis we investigate the correspondence between a channel and its Bloch representation with an emphasis on unital n-qubit channels, in which case the Bloch representation is linear. The thesis is divided into three main sections. First we focus our attention on qubit channels. For certain sets of quantum channels, we establish the surprising existence of a special isomorphism into the set of classical channels. We classify the sets of qubit channels with this property and show that information theoretic quantities are preserved by such classical representations. In a natural progression, we prove some well-known facts about SO(3), the proofs of which are either nonexistent or difficult to find in the literature. Some of this work is based on [12, 13]. In the next section, we consider the multi-qubit channels and show that every finite group can be realized as a subgroup of the quantum channels; this approach allows for the construction of a quantum representation for the free affine monoid over any finite group and gives a classical representation for it. We extend some fundamental results from [26, 28] to the multi-qubit case, including that the set of diagonal Bloch matrices is equal to the free affine monoid over the involution group [special characters omitted]. Some of this work appeared in [10]. Lastly, we study the extreme points for the set of n-qubit channels. There are two types of extreme points: invertible and non-invertible; invertible channels are non-singular maps for which the inverse is also a channel. We briefly study the non-invertible extreme points and then parameterize and analyze the invertiblen-qubit Bloch matrices, which form a compact connected Lie group. We calculate the Lie algebra and give an explicit generating set for the invertible Bloch matrices and a maximal torus.