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Decomposition of a Finite Quantum System into Subsystems: Symbolic–Numerical Approach

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Abstract

Any Hilbert space with composite dimension can be represented as a tensor product of Hilbert spaces of lower dimensions. This factorization makes it possible to decompose a quantum system into subsystems. We propose a model based on finite quantum mechanics for the constructive study of decompositions of an isolated quantum system into subsystems. To study the behavior of the composite systems resulting from the decompositions, we develop algorithms based on methods of computer algebra and computational group theory.

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Notes

  1. In the strict sense, only the universe as a whole can be a closed system; otherwise, the concept of a closed system is approximate.

  2. Another approach whereby the factorization of the Hilbert space is given by the algebra of observables was proposed in [5, 6].

  3. This is a manifestation of Occam’s razor, expressed by the metaphor “the church of the larger Hilbert space” (J.A. Smolin); it allows one to obtain probabilities of all types that occur in quantum theory from the only fundamental probability that appears in Gleason’s theorem [8] and corresponds to Born’s rule.

  4. Accurate to empirically insignificant elements of the traditional formalism, mainly infinities of various kinds.

  5. This term is motivated by the physical term “ground state of a quantum-mechanical system.”

  6. Based on the current cosmological data, \(\mathcal{N} \sim {\text{Exp}}\left( {{\text{Exp}}\left( {20} \right)} \right)\) for 1 cm3 of matter and \(\mathcal{N} \sim {\text{Exp}}\left( {{\text{Exp}}\left( {123} \right)} \right)\) for the entire universe.

  7. In [13], the exact lower bound \(\mathcal{N} \geqslant 72\) was found.

  8. The need for complex numbers (nontrivial elements of cyclotomic extensions) can arise only in problems that involve certain proper subgroups of the symmetric group \({{{\text{S}}}_{\mathcal{N}}}\). A typical example is cyclic groups the irreducible representations of which (except for \({{Z}_{2}} \simeq {{S}_{2}}\)) cannot be obtained without complex numbers.

  9. Obviously, it would be more adequate to compute energy distribution for a given individual permutational evolution; however, this is a more complicated combinatorial problem.

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Authors and Affiliations

  1. Joint Institute for Nuclear Research, 141980, Dubna, Moscow oblast, Russia

    V. V. Kornyak

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  1. V. V. Kornyak
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Correspondence to V. V. Kornyak.

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Translated by Yu. Kornienko

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Kornyak, V.V. Decomposition of a Finite Quantum System into Subsystems: Symbolic–Numerical Approach. Program Comput Soft 48, 293–300 (2022). https://doi.org/10.1134/S0361768822020062

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