Quantum channels preserving sigma-additivity and Ulam measurable cardinals
https://doi.org/10.26907/2949-3919.2026.2.104-117
Abstract
We study the interplay between the properties of quantum states on the Hilbert space ℓ2(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.
About the Author
S. V. Dzhenzher
Moscow Institute of Physics and Technology
Russian Federation
Russian Federation
Dzhenzher Sviatoslav Vadimovich,
9, Institutskii per., Dolgoprudny 141701.
References
1. K.Yosida, E.Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1), 46–66 (1952). https://doi.org/10.1090/S0002-9947-1952-0045194-X
2. G.G.Amosov, V.Zh.Sakbaev, On analogs of spectral decomposition of a quantum state, Math. Notes 93 (3), 351–359 (2013). https://doi.org/10.1134/S0001434613030012
3. G.Amosov, V.Sakbaev, On dynamics of quantum states generated by averaging of random shifts, Adv. Oper. Theory 10 (3), paper No. 51 (2025). https://doi.org/10.1007/s43036-025-00440-2
4. I.V.Volovich, V.Zh.Sakbaev, Universal boundary value problem for equations of mathematical physics, Proc. Steklov Inst. Math. 285 (1), 56–80 (2014). https://doi.org/10.1134/S0081543814040063
5. I.V.Volovich, V.Zh.Sakbaev, On quantum dynamics on C∗-algebras, Proc. Steklov Inst. Math. 301 (1), 25–38 (2018). https://doi.org/10.1134/S008154381804003X
6. I.V.Volovich, V.Zh.Sakbaev, Diffusion of quantum states generated by a classical random walk, CMFD 71 (2), 275–286 (2025) [in Russian]. https://doi.org/10.22363/2413-3639-2025-71-2-275-286
7. E.A.Dzhenzher, S.V.Dzhenzher, V.Zh.Sakbaev, Banach and counting measures, and dynamics of singular quantum states generated by averaging of operator random walks, Lobachevskii J. Math. (2026), to appear.
8. D.P.Blecher and N. Weaver, Quantum measurable cardinals, J. Funct. Anal. 273 (5), 1870–1890 (2017). https://doi.org/10.1016/j.jfa.2017.05.006
9. T.J.Jech, Measurable cardinals, in: Set theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg, 125–138 (2003). https://doi.org/10.1007/3-540-44761-X_10
10. N.Dunford, J.Schwartz, Linear operators. Part 1. General theory, Interscience Publ., New York, 1958. URL: https://books.google.ru/books?id=x94XzQEACAAJ
Review
For citations:
Dzhenzher S.V. Quantum channels preserving sigma-additivity and Ulam measurable cardinals. Mathematics and Theoretical Computer Science. 2026;4(2):104-117. (In Russ.) https://doi.org/10.26907/2949-3919.2026.2.104-117
Views:
14
JATS XML
Email this article 