Quantum channels preserving sigma-additivity and Ulam measurable cardinals
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909040894803968 |
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| author | Dzhenzher, S. V. |
| author_facet | Dzhenzher, S. V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum channels preserving sigma-additivity and Ulam measurable cardinals Dzhenzher, S. V. Quantum Physics Functional Analysis 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. |
| title | Quantum channels preserving sigma-additivity and Ulam measurable cardinals |
| topic | Quantum Physics Functional Analysis |
| url | https://arxiv.org/abs/2604.25854 |