Quantum channels preserving sigma-additivity and Ulam measurable cardinals

• URL: http://arxiv.org/abs/2604.25854v1
• Date: Tue, 28 Apr 2026 16:57:17 GMT
• Title: Quantum channels preserving sigma-additivity and Ulam measurable cardinals
• Abstract summary: We focus on the existence of singular $$-additive states.<n>We prove that any $$-additive state on the diagonal algebra is representable as a Pettis integral over a singular $$-additive measure.<n>We construct a class of quantum channels using $$-complete ultrafilters that map normal states to singular $$-additive states.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: This paper investigates the interplay between the properties of quantum states on the Hilbert space \(\ell_2(κ)\) 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.

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