"On the hierarchy of classicality and symmetry of quantum states" Article and 2 more works of our colleagues Scientists from our Laboratory, Arsen Khvedelidze and Astghik Torosyan, published a paper "On the hierarchy of classicality and symmetry of quantum states" in the "Notes of Scientific Seminars of the St.Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences" journal. The work was performed in cooperation with A. Razmadze Mathematical Institute of Iv. Javakhishvili Tbilisi State University, Institute of Quantum Physics and Engineering Technologies of Georgian Technical University, and A. I. Alikhanyan National Science Laboratory. Abstract: The interrelation between classicality/quantumness and symmetry of states is discussed within the phase-space formulation of finite-dimensional quantum systems. We derive representations for classicality measures QN[Hϱ] of states from the stratum of given symmetry type [Hϱ] for the Hilbert–Schmidt ensemble of qudits. The expressions for measures are given in terms of the permanents of matrices constructed from the vertices of the special Wigner function’s positivity polytope. The supposition about the partial order of classicality indicators QN[Hϱ] in accordance with the symmetry type of stratum is formulated. In the same issue of the "St. Petersburg Department of V. A. Steklov Mathematical Institute" journal (2023, Volume 528) 2 more works of our colleagues have been published: Ayryan E. A., Gambaryan M. M., Malykh M. D., Sevastyanov L. A. "Reversible differential schemes for elliptical oscillators". Abstract: For classical nonlinear oscillators, a comparison between the classical continuous theory of integration in elliptic functions and the discrete theory based on reversible difference schemes was made. These schemes are notable for the fact that the transition from layer to layer is described by Cremona transformations, which gives a large set of algebraic properties. Several properties are shown for the example of the Jacobi oscillator: 1) points of approximate trajectories fall on elliptic curves, 2) difference scheme can be written using quadrature, 3) the approximate solution is periodic. Explicit formulas to calculate the time step for which the approximate solution is a periodic sequence were found. Kornyak V. V. "Description of the evolution of finite-dimensional quantum systems by permutation group". Abstract: We consider constructive approaches to quantum theory: quantum mechanics based on permutation representations of finite groups and the Weyl–Schwinger finite phase space quantum mechanics. We show that both approaches lead to the conclusion that, at a deep level, quantum evolution is based on permutations of finite sets.
