Quantum channels preserving sigma-additivity and Ulam measurable cardinals

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.

Source: arXiv:2604.25854v1 - http://arxiv.org/abs/2604.25854v1 PDF: https://arxiv.org/pdf/2604.25854v1 Original Link: http://arxiv.org/abs/2604.25854v1