In this paper, we study the local unitary (LU) classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we first introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all pairs in the $d\ensuremath{\bigotimes}d$ quantum system into ${\ensuremath{\prod}}_{j=1}^{n}({k}_{j}+1)\ensuremath{-}1$ LU-inequivalent pairs when the prime factorization of $d={\ensuremath{\prod}}_{j=1}^{n}{p}_{j}^{{k}_{j}}$. Moreover, all triples in the ${p}^{\ensuremath{\alpha}}\ensuremath{\bigotimes}{p}^{\ensuremath{\alpha}}$ quantum system for prime
p can be partitioned into $\frac{(\ensuremath{\alpha}+3)}{6}{p}^{\ensuremath{\alpha}}+O(\ensuremath{\alpha}{p}^{\ensuremath{\alpha}\ensuremath{-}1})$ LU-inequivalent triples; especially, when $\ensuremath{\alpha}=2$ and
p>2, there are exactly $\ensuremath{\lfloor}\frac{5}{6}{p}^{2}\ensuremath{\rfloor}+\ensuremath{\lfloor}\frac{p\ensuremath{-}2}{6}+{(\ensuremath{-}1)}^{\ensuremath{\lfloor}\frac{p}{3}\ensuremath{\rfloor}}\frac{p}{3}\ensuremath{\rfloor}+3$ LU-inequivalent triples.
In Phys. Rev. A 70, 032104 (2004), M. Montesinos and G. F. Torres del Castillo consider various symplectic structures on the classical phase-space of the two-dimensional isotropic harmonic oscillator. Using Dirac's quantization condition, the authors investigate how these alternative symplectic forms affect this system's quantization. They claim that these symplectic structures result in mutually inequivalent quantum theories. In fact, we show here that there exists a unitary map between the two representation spaces so that the various quantizations are equivalent.