Hermite张量的秩R正Hermite逼近算法与正Hermite分解  

Rank-R positive Hermitian approximation algorithms and positive Hermitian decompositions of Hermitian tensors

作　　者：杨博 李颖[1] 倪谷炎[1] 张梦石 Bo Yang;Ying Li;Guyan Ni;Mengshi Zhang

机构地区：[1]国防科技大学理学院数学系,长沙410073

出　　处：《中国科学：数学》2023年第8期1125-1144,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11871472和12201633);湖南省自然科学基金(批准号:2021JJ40662)资助项目。

摘　　要：Hermite张量是Hermite矩阵的高阶推广,可以用于表示量子混合态.在量子信息中,量子混合态的可分性判别和分解问题仍然是一个重要而棘手的问题.本文推导逼近函数的梯度,进而提出3种算法:Hermite张量的秩R正Hermite逼近的负梯度算法和BFGS(Broyden-Fletcher-Goldfarb-Shanno)算法,以及Hermite张量可分性判别和分解的BFGS算法.基于Taylor公式和凸分析,本文证明BFGS算法的有效性.数值算例进一步验证理论分析的正确性和算法的有效性.结果表明,BFGS算法可用于Hermite张量的可分性判别和正Hermite分解,并可得到其正Hermite秩分解.与半定松弛算法相比,BFGS算法能够分解高阶或高维Hermite张量且运行时间短.The Hermitian tensor is regarded as an extension of the Hermitian matrix and can be used to represent a quantum mixed state.In quantum information,the problem of separability discrimination and decomposition of a quantum mixed state is still important and hard.In this paper,we deduce the gradient of the approximation function,and propose three algorithms:A negative gradient algorithm and a BFGS algorithm for rank-R positive Hermitian approximation of Hermitian tensors,and a separability discrimination and decomposition algorithm for Hermitian tensors.According to the Taylor formula and the convexity analysis,we prove the effectiveness of the algorithm.Numerical examples also verify the correctness of the theoretical analysis and the effectiveness of algorithms.They show that the BFGS algorithm can be used for the separability discrimination and the positive Hermitian decomposition,as well as to obtain a rank-positive Hermitian decomposition.Compared with the semidefinite relaxation algorithm,the BFGS algorithm has the advantages of less running time and solving the decomposition of higher-order or higher-dimensional Hermitian tensors.

关 键 词：Hermite张量 正Hermite分解 秩R逼近 BFGS算法 量子混合态 

分 类 号：O183.2[理学—数学]