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Qeios

Kronecker-Pauli Operators

Christian Rakotonirina

Published Mar 9, 2025• 0 citations
Publisher

Abstract

In quantum mechanics, there are some classical bases which are generalized to higher dimensional matrices. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a Kronecker-Pauli matrices.

DOI
10.32388/bsbyu8

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In quantum mechanics, there are some classical bases which are generalized to higher dimensional matrices. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a Kronecker-Pauli matrices.
In quantum mechanics, classical matrix bases such as the Pauli matrices are often generalized to higher dimensions. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a set of Kronecker-Pauli matrices.
In quantum mechanics, classical matrix bases such as the Pauli matrices are often generalized to higher dimensions. So, it is useful to express their corresponding operators using the Dirac Bra and Ket. In this paper, to express the corresponding operators we review the Kronecker-Pauli matrices and how to construct them for an N-dimensional system, with N a prime integer, N>2. Then, we give the expression of the Kronecker-Pauli operators and show that their matrices with respect to the standard basis fulfill the conditions to form a set of Kronecker-Pauli matrices. Relationship between the Kronecker-Pauli operators and the Weyl operators has been studied.
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.
Within the framework of quantum theory, we review the Chrestenson operator, the Weyl operator basis, and the Kronecker-Pauli operator basis in ddd-dimensional Hilbert spaces using Dirac notation, where ddd is a prime integer strictly greater than 2. We establish a new algebraic relation connecting these operators and present the cases d=3d=3d=3 and d=5d=5d=5 as illustrative examples.
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 2n2^{n}2n dimensional systems. This paper explores extending this generalization to NNN-dimensional systems, where NNN is a prime integer, 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. Another possible method for constructing a set of Kronecker-Pauli matrices is discussed.