Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics

IF 1 3区数学 Q1 MATHEMATICS Linear Algebra and its Applications Pub Date : 2024-07-17 DOI:10.1016/j.laa.2024.07.013

Aina Mayumi , Gen Kimura , Hiromichi Ohno , Dariusz Chruściński

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Abstract

By employing a weighted Frobenius norm with a positive definite matrix ω, we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm

and the standard Frobenius norm

, there are exactly five possible generalizations, labeled (i) through (v), for the bounds on the norms of the commutator

[A, B] : = A B - B A

. In this paper, we establish the tight bounds for cases (iii) and (v), and propose conjectures regarding the tight bounds for cases (i) and (ii). Additionally, the tight bound for case (iv) is derived as a corollary of case (i). All these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) (and thus also (iv)) are numerically supported for matrices up to size

n = 15

. Proofs are provided for

n = 2

and certain special cases. Interestingly, we find applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics.

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加权弗罗贝尼斯准则的博特尔-文泽尔不等式及其在量子物理学中的应用

通过使用带有正定矩阵的加权弗罗贝尼斯规范，我们引入了著名的伯特尔-文采尔（BW）不等式的自然广义。基于加权弗罗贝纽斯规范▪ 和标准弗罗贝纽斯规范▪ 的组合，换元规范的边界正好有五种可能的广义，分别标为(i)到(v)。在本文中，我们建立了 (iii) 和 (v) 两种情况的紧界，并对 (i) 和 (ii) 两种情况的紧界提出了猜想。此外，情况 (iv) 的紧界是情况 (i) 的推论。所有这些界值（i）-（v）都是对 BW 不等式的概括。情况 (i) 和 (ii) 的猜想界值（以及情况 (iv)）对于大小不超过 .我们还为某些特殊情况提供了证明。有趣的是，我们发现这些边界在量子物理学中的应用，特别是在不确定性关系和开放量子动力学的背景下。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.

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