Quantum channels preserving sigma-additivity and Ulam measurable cardinals

TL;DR

This paper explores the relationship between quantum states on infinite-dimensional Hilbert spaces and set-theoretic properties of large cardinals, introducing new representations and channels involving singular sigma-additive states.

Contribution

It extends classical representation theory to non-normal states and constructs quantum channels that map normal states to singular sigma-additive states using ultrafilters.

Findings

Any sigma-additive state on the diagonal algebra can be represented as a Pettis integral.

Constructed quantum channels using ultrafilters map normal states to singular sigma-additive states.

Established a link between quantum state properties and Ulam measurability of cardinals.

Abstract

This paper investigates the interplay between the properties of quantum states on the Hilbert space \(\ell_2(\kappa)\) 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.