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Carpathian Mathematical Publications

Algebras of symmetric functions on Banach spaces and their applications in quantum physics

S. V. Sharyn, Taras Vasylyshyn

Published Jun 26, 2026• 0 citations
Publisher

Abstract

We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of $N$ noninteracting identical bosons. In general, the system is described by the $N$-fold symmetric tensor product $\mathcal{E}^{\odot N},$ where $\mathcal{E}$ is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of $\mathcal{E}^{\odot N}.$ We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries $S$ such that the partition function of the system is $S$-symmetric.

DOI
10.15330/cmp.18.1.250-257

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We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of NNN noninteracting identical bosons. In general, the system is described by the NNN-fold symmetric tensor product E⊙N,\mathcal{E}^{\odot N},E⊙N, where E\mathcal{E}E is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of E⊙N.\mathcal{E}^{\odot N}.E⊙N. We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries SSS such that the partition function of the system is SSS-symmetric.
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A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral
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We explore the connection between quantum entanglement and the exchange symmetry of the states of N identical particles. Each particle has n-levels. The N particles span the nN dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry that are found to have a measure of entanglement which is related to the representation of the permutation group. The maximally entangled states are generated from the linear combinations of fully correlated but unentangled states. The states of lower entanglement are generated from the manifold of partially correlated states. The degree of exchange symmetry is found to be related to a group theoretical measure of entanglement. Email: Jagdish_Rai@hotmail.com, jrai@iitk.ac.in Email: srai56@hotmail.com
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in n i , i = 1,..., κ . Their Hilbert spaces are given by κ -fold tensor products of Hilbert spaces of dimensions n 1 ,..., n κ . Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.