Constructing a Set of Kronecker-Pauli Matrices

In quantum physics, the choice of basis is crucial for formulation. The generalization of the Pauli matrices via the Kronecker product, known as Pauli strings, is typically restricted to \(2^{n}\) dimensional systems. This paper explores extending this generalization to \(N\)-dimensional systems, where \(N\) is a prime integer, to construct \(N \times N\)-Kronecker-Pauli matrices. We begin by examining the specific cases of \(3 \times 3\) and \(5 \times 5\) Kronecker-Pauli matrices, with the goal of the purpose constructing a set of \(N \times N\)-Kronecker-Pauli matrices for any prime integer \(N\).