Abstract
In this paper, we investigate the behavior of quantum entanglement in the process of unitary evolution in constructive models of multicomponent quantum systems. Symmetry groups of quantum systems that admit the occurrence of geometric structures associated with quantum entanglement are described. Algorithms for dynamic simulation of quantum entanglement are based on methods of computer algebra and computational group theory. Some examples of practical computations are presented.
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Notes
Any linear representation of a finite group over a field of characteristic zero is unitary.
Provided that we neglect empirically insignificant elements of the traditional formalism, e.g., infinities of various kinds.
The metaphor “church of the larger Hilbert space” (J.A. Smolin) characterizes the belief that any mixed state actually stems from a more fundamental pure state of a larger system.
An antihomomorphism is a map of groups that reverses the order of group multiplication: \(\mu \left( {ab} \right) = \mu \left( b \right)\mu \left( a \right)\).
Except for two-dimensional Hilbert spaces.
Any unitary matrix can be represented as an exponent of a Hermitian matrix, and the corresponding formula \(U = {{e}^{{{\mathbf{i}}H}}}\) can be interpreted as a unitary evolution in one time step generated by the Hamiltonian H.
Geometric structures that occur in the Hilbert space depend significantly on the quantum state. That is why, e.g., in [15], only quantum states constrained to ensure the fulfillment of the holographic principle [23] were considered; according to this principle, the information (entropy) about a physical system within a three-dimensional region is concentrated at the two-dimensional boundary of this region.
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Translated by Yu. Kornienko
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Kornyak, V.V. Dynamic Simulation of Quantum Entanglement in Finite Quantum Mechanics: A Computer Algebra Approach. Program Comput Soft 47, 124–132 (2021). https://doi.org/10.1134/S0361768821020067
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DOI: https://doi.org/10.1134/S0361768821020067