关于矩阵数字特征的一些注记
Some Notes on the Numerical Characteristics of Matrix
摘要: 根据矩阵数字特征的定义，结合矩阵Lie积的定义，利用矩阵迹的性质，矩阵数字特征的性质，以及密度矩阵的性质，对由Hermite矩阵组成的实线性空间Hn上的一些数字特征进行了数值计算并且给出了一个例子，其次对WY相关系数进行改造使其成为矩阵空间上的半内积并且利用矩阵范数、矩阵迹及矩阵数字特征的性质得出了改造后的WY相关系数与原WY相关系数、斜信息以及协方差的关系，获得了原WY相关系数和斜信息的一些性质，并给出改造后的WY相关系数的一个应用。这些结果将在量子信息论中具有一定的应用。
Abstract: According to the definition of matrix numerical characteristics, combined with the definitions of Lie product of matrix, using the properties of matrix traces, matrix numerical characteristics, and density matrix, numerical calculations were performed on some numerical characteristics on real linear space Hn composed of Hermite matrices, and an example was given, secondly, the WY corre-lation coefficient was modified to become a semi inner product in the matrix space, and the rela-tionship between the modified WY correlation coefficient and the original WY correlation coefficient and skew information, as well as some properties of covariance, the original WY correlation coefficient and skew information, were obtained using the properties of matrix norm, matrix trace, and matrix numerical characteristics, and an application of the modified WY correlation coefficient was also presented. These results will have certain applications in quantum information theory.
文章引用：禹鸥洋, 任芳国. 关于矩阵数字特征的一些注记[J]. 理论数学, 2024, 14(2): 482-491. https://doi.org/10.12677/PM.2024.142047

参考文献

[1] Watrous, J. (2018) Theory of Quantum Information. Cambridge University Press, Cambridge. [Google Scholar] [CrossRef] 
[2] Horn, R.A. and Johnson, C.R. (1985) Matrix Analysis. Cambridge University Press, London. [Google Scholar] [CrossRef] 
[3] Nielsen M A,Chuang I L, Quantum Computation and Quantum Information[M]. Cambridge University Press, Cambridge, 2000.
[4] Wiger, E.P. and Yanase, M.M. (1963) Infor-mation Contents of Distributions. Proceedings of the National Academy of Sciences, 49, 910-918. [Google Scholar] [CrossRef] [PubMed]
[5] Luo, S.L. (2003) Wigner-Yanase Skew Information and Uncertainty Relations. Physical Review Letters, 91, Article 180403. [Google Scholar] [CrossRef] 
[6] Luo, S. and Zhang, Q. (2004) On Skew Information. IEEE Transactions on Information Theory, 50, 1778-1782. [Google Scholar] [CrossRef] 
[7] Luo, S.L. (2005) Heisenberg Uncertainty Relation for Mixed States. Physics Letters A, 72, 440-450. [Google Scholar] [CrossRef] 
[8] Yanagi, K., Furuichi, S. and Kuriyama, K. (2005) A General-ized Skew Information and Uncertainty Relation. IEEE Transactions on Information Theory, 51, 4401-4404. [Google Scholar] [CrossRef] 
[9] Luo, S.L. (2006) Quantum Uncertainty of Mixed States Based on Skew Information. Physical Review A, 73, 457-460. [Google Scholar] [CrossRef] 
[10] Li, D., Li, X., Wang, F., Huang, H., Li, X. and Kwek, L.C. (2009) Uncertainty Relation of Mixed States by Means of Wigner-Yanase-Dyson Information. Physical Review A, 79, Article 052106. [Google Scholar] [CrossRef] 
[11] Furuichi, S., Yanagi, K. and Kuriyama, K. (2012) Trace Ine-qualities on a Generalized Wigner-Yanase Skew Information. Journal of Mathematical Analysis & Applications, 356, 179-185. [Google Scholar] [CrossRef] 
[12] Ko, C.K. (2010) Comments on Conjectures of Trace Inequalities on a Generalized Wigner-Yanase Skew Information. Journal of Mathematical Analysis & Applications, 369, 164-167. [Google Scholar] [CrossRef] 
[13] Yanagi K. (2010) Uncertainty Relation on Wig-ner-Yanase-Dyson Skew Information. Journal of Mathematical Analysis and Applications, 365, 12-18. [Google Scholar] [CrossRef] 
[14] Chen, B. and Fei, S.M. (2015) Uncertainty Relations Based on Mutually Unbiased Measurements. Quantum Information Processing, 14, 2227-2238. [Google Scholar] [CrossRef] 
[15] Chen, B., Fei, S.M. and Long, G.L. (2016) Sum Uncertainty Re-lations Based on Wigner-Yanase Skew Information. Quantum Information Processing, 15, 2639-2648. [Google Scholar] [CrossRef] 
[16] Zhang, Y. and Luo, S.L. (2021) Wigner Function, Wigner-Yanase Skew Information, and Parity Asymmetry. Physics Letters A, 395, Article ID: 127222. [Google Scholar] [CrossRef] 

为你推荐