  Classification is a common method to study quantum entanglement, and local
unitary equivalence (LU-equivalence) is an effective classification tool. The
purpose of this work is to show how to determine the LU-equivalence of sets of
generalized Bell states (GBSs) in an arbitrary dimensional bipartite quantum
system $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ ($d$ is an integer no less than
3). The idea is that, for a given GBS set $\mathcal{M}$, try to find all the
GBS sets that are LU-equivalent to $\mathcal{M}$, then we can determine whether
another GBS set is LU-equivalent to $\mathcal{M}$ by comparison. In order to
accomplish this intention, we first reduce the LU-equivalence of two GBS sets
to the unitary conjugate equivalence (UC-equivalence) of two generalized Pauli
matrix (GPM) sets. Then we give the necessary and sufficient conditions for a
2-GPM set UC-equivalent to a special 2-GPM set $\{ X^{a}, Z^{b} \}$ ($a, b$ are
nonnegative integers and factors of $d$). The general case, that is, the
UC-equivalence of two general GPM sets, follows by the particular case.
Moreover, these results are programmable, that is, we provide programs that can
give all standard GPM sets that are UC-equivalent to a given GPM set, as well
as programs that can determine all standard GPM sets that are unitary
equivalent (U-equivalent) to a given GPM set, and then the U-equivalence of two
arbitrary GPM sets (or LU-equivalence of two GBS sets) can be determined by
comparison. To illustrate the role of the programs, we provide two examples,
one showing that all 4-GBS sets (58905 items) in $\mathbb{C}^{6}\otimes
\mathbb{C}^{6}$ can be divided into 31 LU-equivalence classes, and the other
providing a complete LU-equivalent classification of all 4-GBS sets in
$\mathbb{C}^{4}\otimes \mathbb{C}^{4}$.
