Miscellaneous Applications of Quons

Abstract

Abstract. This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We motivate why such algebras are interesting for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU 2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).

Bases for spin systems and qudits from angular momentum theory

“…We now define the operator v 0a (a particular case of the unitary operator v ra introduced in [21,22]…”

“…Therefore, j + , j − and j z span the Lie algebra su 2 of SU 2 and the operators v 0a and h realize a polar decomposition of su 2 [5][6][7][18][19][21][22].…”

“…Most of the material presented in this work takes its origin in papers published by the author and its collaborators [18][19][20][21][22][23][24][25][26].…”

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

“…However, deviations from the Boltzmann–Gibbs statistics are observed in quasi-particle excitations occurring in several condensed matter systems. Therefore, the study of physical systems obeying non-conventional statistics is one of the most investigated topics of contemporary statistical physics [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ]. Their interest goes from the theoretical foundation of generalized statistical mechanics [ 8 , 9 , 10 , 11 , 12 ], to high-temperature gas [ 13 ] and high- superconductivity [ 14 ], Laughlin particle with a fractional charge related to fractional quantum Hall effect [ 15 ], Josephson junctions [ 16 ], and others.…”

Classical Model of Quons

“…However, statistical deviations from bosons or fermions can be observed in quasi-particle excitations that occur in various condensed matter systems. Therefore, the study of physical systems that obey non-conventional statistics is one of the pillars of contemporary statistical physics [6][7][8][9][10][11][12]. Their interest spans from the theoretical foundation of generalized statistical mechanics [13,14], Fermi gas superfluid [15], high-temperature gas [16] and high-T c superconductivity [17], Laughlin particle with fractional charge related to fractional quantum Hall effect [18,19], Josephson junctions [20] and applications to quantum computation [21,22].…”