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Journal of Physics Conference Series

Kronecker product and a related useful conjecture

Xiling Wang

Published Feb 1, 2025• 0 citations
Publisher

Abstract

Abstract This paper not only explores in detail the basic properties of the Kronecker product, such as the application of the union law, the distributive law, the transpose and the conjugate transpose, as well as the trace and the rank of the Kronecker product, but also rigorously proves these properties by means of mathematical induction or direct proofs. In addition, this paper extends these properties to Kronecker n-matrices and explores the applicability of inverse matrices and determinants to Kronecker products. More importantly, we introduce a conjecture related to the long-standing problem of distillability, which is partially solved when the number of nonzero elements is restricted. By showing the framework of the proof of the conjecture and a special case, we provide new perspectives for further research in this area.

DOI
10.1088/1742-6596/2964/1/012032

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Abstract This paper not only explores in detail the basic properties of the Kronecker product, such as the application of the union law, the distributive law, the transpose and the conjugate transpose, as well as the trace and the rank of the Kronecker product, but also rigorously proves these properties by means of mathematical induction or direct proofs. In addition, this paper extends these properties to Kronecker n-matrices and explores the applicability of inverse matrices and determinants to Kronecker products. More importantly, we introduce a conjecture related to the long-standing problem of distillability, which is partially solved when the number of nonzero elements is restricted. By showing the framework of the proof of the conjecture and a special case, we provide new perspectives for further research in this area.
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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.
Abstract Tensor products or, more specifically, Kronecker products play a fundamental role in quantum computing and can also help with modeling certain combinatorial problems. In this chapter, we therefore discuss Kronecker products and their properties. To segue into this discussion, we first introduce the important notion of pseudo Boolean functions and look at their relations to multilinear algebra. Given this use case, we can easily understand the role of different kinds of tensor products and investigate algebraic properties especially of Kronecker products.
Quantum information theory has been greatly developed in the past decades, and many theoretical problems are related to matrix theory. We study the equality condition for a matrix inequality, [Formula: see text] rank([Formula: see text] ⊗[Formula: see text]) [Formula: see text] rank([Formula: see text] ⊗[Formula: see text] where [Formula: see text]’s are linearly independent matrices of the same size, and [Formula: see text]’s are linearly independent matrices of the same size. The inequality used to be a conjecture since 2013 and has recently been proven in the paper [Z. Song, L. Chen, Y. Sun and M. Hu, IEEE Trans. Inform. Theory 69 (2023) 2385]. We study several cases such as that [Formula: see text]’s are column vectors and [Formula: see text]’s are of various sizes. It turns out that some cases never satisfy the equality condition.