Journal of applied mathematics & informatics

  • 제41권1호
  • /
  • Pages.123-132
  • /
  • 2023
  • /
  • 2734-1194(pISSN)
  • /
  • 2234-8417(eISSN)
한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

A RELATIVE RÉNYI OPERATOR ENTROPY

  • MIRAN, JEONG (Department of Mathematics, Chungbuk National University) ;
  • SEJONG, KIM (Department of Mathematics, Chungbuk National University)
  • 투고 : 2022.07.29
  • 심사 : 2022.10.21
  • 발행 : 2023.01.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We define an operator version of the relative Rényi entropy as the generalization of relative von Neumann entropy, and provide its fundamental properties and the bounds for its trace value. Moreover, we see an effect of the relative Rényi entropy under tensor product, and show the sub-additivity for density matrices.

키워드

과제정보

This research was supported by Chungbuk National University Korea National University Development Project (2021).

참고문헌

  1. T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203-241. https://doi.org/10.1016/0024-3795(79)90179-4
  2. R. Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, 2007.
  3. R. Bhatia, T. Jain, and Y. Lim, Strong convexity of sandwiched entropies and related optimization problems, Rev. Math. Phys. 30 (2018), 1850014.
  4. J.I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon. 34 (1989), 341-348.
  5. S. Furuichi, Inequalities for Tsallis relative entropy and generalized skew information, Linear Multilinear Algebra 59 (2011), 1143-1158. https://doi.org/10.1080/03081087.2011.574624
  6. R. Frank and E. Lieb, Monotonicity of a relative R'enyi entropy, J. Math. Phys. 54 (2013), 122201.
  7. F. Hansen and G.K. Pedersen, Jensens inequality for operators and Lowners theorem, Math. Ann. 258 (1982), 229241.
  8. F. Hiai, Log-majorization related to Renyi divergences, Linear Algebra Appl. 563 (2019), 255-276. https://doi.org/10.1016/j.laa.2018.11.004
  9. S. Kullback and R.A. Leibler, On information and sufficiency, Ann. Math. Stat. 22 (1951), 79-86. https://doi.org/10.1214/aoms/1177729694
  10. M. Muller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum Renyi entropies: a new definition, some properties, J. Math. Phys. 54 (2013), 122203.
  11. D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23 (1986), 57-65. https://doi.org/10.1016/0034-4877(86)90067-4
  12. C. Shannon, A Mathematical Theory of Communication, The Bell System Technical Journal 27 (1948), 379-423 https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
  13. M. Wilde, A. Winter and D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy, Comm. Math. Phys. 331 (2014), 593-622. https://doi.org/10.1007/s00220-014-2122-x