A note on a conjecture from distillability of quantum entanglement

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-11-19 DOI:10.1016/j.laa.2024.11.012

Weng Kun Sio, Che-Man Cheng

引用次数: 0

Abstract

A conjecture from the distillability of quantum entanglement is that when A and B are

4 \times 4

trace zero complex matrices and

{‖ A ‖}^{2} + {‖ B ‖}^{2} = 1 / 4

(where

‖ \cdot ‖

is the Frobenius norm), the sum of squares of the largest two singular values of

A \otimes I_{4} + I_{4} \otimes B

does not exceed 1/2. In this paper, the conjecture is proved when

(i)
A or B is unitarily similar to a direct sum of $2 \times 2$ trace zero matrices;
(ii)
A and B are unitarily similar to matrices, when partitioned into $2 \times 2$ blocks, having zero diagonal blocks.

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关于量子纠缠可蒸馏性猜想的说明

量子纠缠的可提炼性的一个猜想是：当 A 和 B 是 4×4 痕量为零的复矩阵且‖A‖2+‖B‖2=1/4（其中‖⋅‖是弗罗贝尼斯规范）时，A⊗I4+I4⊗B 的最大两个奇异值的平方和不超过 1/2 。本文证明了以下猜想：(i) A 或 B 与 2×2 痕零矩阵的直接和具有单位相似性；(ii) A 和 B 被分割成 2×2 块时与矩阵具有单位相似性，且对角块为零。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.