Any Hilbert space of composite dimension can be decomposed into a tensor product of Hilbert spaces of lower dimensions. Such a factorization makes it possible to decompose a quantum system into subsystems. Using a modification of quantum mechanics in which the continuous unitary group in a Hilbert space is replaced with a permutation representation of a finite group, we suggest a model for the constructive study of decompositions of a closed quantum system into subsystems. To investigate the behavior of composite systems resulting from decompositions, we develop algorithms based on methods of computer algebra and of computational group theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. N. Page and W. K. Wootters, “Evolution without evolution: Dynamics described by stationary observables,” Phys. Rev. D, 27, 2885–2892 (1983).
S. M. Carroll and A. Singh, “Quantum mereology: Factorizing Hilbert space into subsystems with quasiclassical dynamics,” Phys. Rev. A, 103, No. 2, 022213 (2021).
Ch. J. Cao, S. M. Carroll, and S. Michalakis, “Space from Hilbert space: Recovering geometry from bulk entanglement,” Phys. Rev. D, 95, No. 2, 024031 (2017).
M. Woods, “The Page–Wootters mechanism 36 years on: a consistent formulation which accounts for interacting systems,” Quantum Views, 3, 16 (2019).
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge Univ. Press (2016).
V. V. Kornyak, “Quantum models based on finite groups,” J. Phys.: Conf. Series, 965, 012023 (2018).
V. V. Kornyak, “Modeling quantum behavior in the framework of permutation groups,” EPJ Web of Conferences, 173, 01007 (2018).
V. V. Kornyak, “Mathematical modeling of finite quantum systems,” Lect. Notes Comput. Sci., 7125, 79–93 (2012).
T. Banks, “Finite deformations of quantum mechanics,” arXiv:2001.07662 [hep-th] (2020).
G. ’t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, Springer (2016).
M. J. Collins, “On Jordan’s theorem for complex linear groups,” J. Group Theory, 10, No. 4, 411–423 (2007).
A. Rényi, “On measures of entropy and information,” in: Proc. 4th Berkeley Symp. Math. Stat. Probab., Vol. 1 (1961), pp. 547–561.
Č. Brukner and A. Zeilinger, “Conceptual inadequacy of the Shannon information in quantum measurements,” Phys. Rev. A, 63, No. 2, 022113 (2001).
M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” Gen. Rel. Grav., 42, 2323–2329 (2010).
J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortschr. Phys., 61, No. 9, 781–811 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 507, 2021, pp. 183–202.
Rights and permissions
About this article
Cite this article
Kornyak, V.V. Subsystems of a Closed Quantum System in Finite Quantum Mechanics. J Math Sci 261, 717–729 (2022). https://doi.org/10.1007/s10958-022-05783-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05783-2