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arXiv:0704.1844 · International Journal of Geometric Methods in Modern Physics

A NEW ALGEBRAIC STRUCTURE OF FINITE QUANTUM SYSTEMS AND THE MODIFIED BESSEL FUNCTIONS

Kazuyuki Fujii

Published Nov 1, 2007• 4 citations
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Abstract

In this paper we present a new algebraic structure (a superhyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two-level system. It fits into the so-called generalized Pauli matrices, so they play an important role in theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado.

DOI
10.1142/s021988780700251x

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In this paper we present a new algebraic structure (a superhyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two-level system. It fits into the so-called generalized Pauli matrices, so they play an important role in theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado.
In quantum physics, the choice of basis is crucial for formulation. The generalization of the Pauli matrices via Kronecker product, called Kronecker-Pauli matrices, is typically restricted to for 2n2^n2n dimensional systems. This paper explores extending this generalization to NNN-dimensional systems, where NNN is a prime integer, in order to construct N×NN\times NN×N-Kronecker-Pauli matrices. We begin by examining the specific cases of 3×33\times 33×3 and 5×55\times 55×5 Kronecker-Pauli matrices, with the goal of the purpose constructing a set of N×NN\times NN×N-Kronecker-Pauli matrices for any prime integer NNN.
In quantum physics, the choice of basis is crucial for formulation. The generalization of the Pauli matrices via Kronecker product, called Kronecker-Pauli matrices, is typically restricted to for 2n2^n2n dimensional systems. This paper explores extending this generalization to NNN-dimensional systems, where NNN is a prime integer, in order to construct N×NN\times NN×N-Kronecker-Pauli matrices. We begin by examining the specific cases of 3×33\times 33×3 and 5×55\times 55×5 Kronecker-Pauli matrices, with the goal of the purpose constructing a set of N×NN\times NN×N-Kronecker-Pauli matrices for any prime integer NNN.
This contribution summarizes work on finite, non-cyclic Hamiltonian systems —in particular the one-dimensional finite oscillator—, in conjunction with a Lie algebraic definition of the (meta-) phase space of finite systems, and a corresponding Wigner distribution function for the state vectors. The consistency of this approach is important for the strategy of fractionalization of a finite Fourier transform, and the contraction of finite unitary to continuous symplectic transformations of Hamiltonian systems.
Quantum Information is a new area of research which has been growing rapidly since the last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more "dynamical point of view" of this theory. We want to consider the concepts of entropy and pressure for "stationary systems" acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator L\mathcal{L}L acting on density matrices ρ∈MNρ\in \mathcal{M}_Nρ∈MN​ over a finite NNN-dimensional complex Hilbert space L(ρ):=∑i=1ktr(WiρWi∗)ViρVi∗,\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,L(ρ):=∑i=1k​tr(Wi​ρWi∗​)Vi​ρVi∗​, where WiW_iWi​ and ViV_iVi​, i=1,2,...ki=1,2,... ki=1,2,...k are operators in this Hilbert space. L\mathcal{L}L is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the Vi(.)Vi∗=:Fi(.)V_i (.) V_i^*=:F_i(.)Vi​(.)Vi∗​=:Fi​(.), i=1,2,...,ki=1,2,...,ki=1,2,...,k, play the role of the inverse branches (acting on the configuration space of density matrices ρρρ) and the WiW_iWi​ play the role of the weights one can consider on the IFS. We also analyze the discrete Wigner function. We suppose that for all ρρρ we have that ∑i=1ktr(WiρWi∗)=1\sum_{i=1}^k tr(W_iρW_i^*)=1∑i=1k​tr(Wi​ρWi∗​)=1. A family W:={Wi}i=1,...,kW:=\{W_i\}_{i=1,..., k}W:={Wi​}i=1,...,k​ determines a Quantum Iterated Function System (QIFS) FW\mathcal{F}_{W}FW​, FW={MN,Fi,Wi}i=1,...,k.\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.FW​={MN​,Fi​,Wi​}i=1,...,k​.
An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.