Quantum channels preserving sigma-additivity on diagonal von Neumann algebras and Ulam measurable cardinals

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 $\kappa$. We focus on the existence of singular $\sigma$-additive states~--- functionals whose induced measures are $\sigma$-additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of $\kappa$, their structural and dynamical properties remain largely unexplored. We prove that any $\sigma$-additive state on the diagonal algebra is representable as a Pettis integral over a singular $\sigma$-additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using $\sigma$-complete ultrafilters that map normal states to singular $\sigma$-additive states, effectively <<archiving>> information into the singular part of the state space.

8 pages, 1 figure; the introduction is expanded; the title is clarified