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A Relation Between the Chrestenson Operator, Weyl Operator Basis, and Kronecker-Pauli Operator Basis

Mickaya A. Razanaparany, Christian Rakotonirina

Abstract

Within the framework of quantum theory, we review the Chrestenson operator, the Weyl operator basis, and the Kronecker-Pauli operator basis in $d$-dimensional Hilbert spaces using Dirac notation, where $d$ is a prime integer strictly greater than 2. We establish a new algebraic relation connecting these operators and present the cases $d=3$ and $d=5$ as illustrative examples.

A Relation Between the Chrestenson Operator, Weyl Operator Basis, and Kronecker-Pauli Operator Basis

Abstract

Within the framework of quantum theory, we review the Chrestenson operator, the Weyl operator basis, and the Kronecker-Pauli operator basis in -dimensional Hilbert spaces using Dirac notation, where is a prime integer strictly greater than 2. We establish a new algebraic relation connecting these operators and present the cases and as illustrative examples.
Paper Structure (4 sections, 1 theorem, 23 equations)

This paper contains 4 sections, 1 theorem, 23 equations.

Table of Contents

  1. Introduction
  2. Operator Definitions
  3. Chrestenson, Weyl, and Kronecker-Pauli Relationship
  4. Conclusion and Outlook

Key Result

Proposition 7

Let $d>2$ be a prime integer, let $C_d$ denote the Chrestenson transform, $U_{nm}$ the Weyl operators, and $\Pi_\ell$ the Kronecker-Pauli operators. Then, for every $n,m\in \{0,1,\,\dots, d-1\}$ there exist integers $k\in \{0,1,\,\dots, d-1\}$ and $\ell\in \{1,2,\,\dots, d^2\}$ such that where $w$ is the $d$-th root of unity.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Example 3
  • Definition 4
  • Definition 5
  • Example 6
  • Proposition 7
  • proof
  • Example 8
  • Example 9