Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2151–2169 | Cite as

Noncommutative versions of inequalities in quantum information theory

  • Ali Dadkhah
  • Mohammad Sal MoslehianEmail author
  • Kenjiro Yanagi


In this paper, we aim to replace in the definitions of covariance and correlation the usual trace Tr by a tracial positive map between unital \(C^*\)-algebras and to replace the functions \(x^{\alpha }\) and \(x^{1- \alpha }\) by functions f and g satisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner–Yanase–Dyson skew information related to the tracial positive maps and functions f and g. We persent a generalization of Heisenberg’s uncertainty relation in the noncommutative framework. We extend some inequalities and properties for the generalized correlation and the generalized Wigner–Yanase–Dyson skew information. Furthermore, we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information.


Tracial positive linear map Wigner–Yanase skew information Covariance Correlation Uncertainty relation 

Mathematics Subject Classification

Primary 46L05 47A63 Secondary 81P15 



The second author (corresponding author) is supported by a grant from the Iran National Science Foundation (96006713). The third author is partially supported by JPSP KAKENHI Grant Number 19K03525.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


  1. 1.
    Arambasić, Lj, Bakić, D., Moslehian, M.S.: A treatment of the Cauchy–Schwarz inequality in \(C^*\)-modules. J. Math. Anal. Appl. 381, 546–556 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bourin, J.C.: Some inequalities for norms on matrices and operators. Linear Algebra Appl. 292, 139–154 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Choi, M.D., Tsui, S.K.: Tracial positive linear maps of \(C^*\)-algebaras. Proc. Am. Math. Soc. 87(1), 57–61 (1983)zbMATHGoogle Scholar
  4. 4.
    Dadkhah, A., Moslehian, M.S.: Quantum information inequalities via tracial positive linear maps. J. Math. Anal. Appl. 447(1), 666–680 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Friedland, S., Gheorghiu, V., Gour, G.: Universal uncertainty relations. Phys. Rev. Lett. 111, 230401 (2013)CrossRefGoogle Scholar
  6. 6.
    Fujii, J.I.: A trace inequality arising from quantum information theory. Linear Algebra Appl. 400, 141–146 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fujii, M., Izumino, S., Nakamoto, R., Seo, Y.: Operator inequalities related to Cauchy–Schwarz and Hölder–McCarthy inequalities. Nihonkai Math. J. 8, 117–122 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ilišević, D., Varošanec, S.: On the Cauchy–Schwarz inequality and its reverse in semi-inner product \(C^*\)-modules. Banach J. Math. Anal. 1, 78–84 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ko, C.K., Yoo, H.J.: Uncertainty relation associated with a monotone pair skew information. J. Math. Anal. Appl. 383, 208–214 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lance, E.C.: Hilbert \(C^*\)-Modules. London Mathematical Society Lecture Note Series, vol. 210. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  11. 11.
    Lieb, E.H.: Convex trace functions and the Wigner–Yanase–Dyson conjecture. Adv. Math. 11, 267–288 (1973)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Moslehian, M.S., Kian, M., Xu, Q.: Positivity of \(2\times 2\) block matrices of operators. Banach J. Math. Anal. (2019). CrossRefzbMATHGoogle Scholar
  13. 13.
    Schrödinger, E.: About Heisenberg uncertainty relation. Proc. Prussian Acad. Sci. Phys. Math. Sect. XIX, 293 (1930)Google Scholar
  14. 14.
    Størmer, E.: Positive Linear Maps of Operator Algebras. Springer, Berlin (2013)CrossRefGoogle Scholar
  15. 15.
    Yanagi, K., Furuichi, S., Kuriyama, K.: A generalized skew information and uncertainty relation. IEEE Trans. Inf. Theory 51, 4401–4404 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ali Dadkhah
    • 1
  • Mohammad Sal Moslehian
    • 1
    Email author
  • Kenjiro Yanagi
    • 2
  1. 1.Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran
  2. 2.Department of MathematicsJosai UniversitySakadoJapan

Personalised recommendations