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Variational representations related to Tsallis relative entropy

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Abstract

We develop variational representations for the deformed logarithmic and exponential functions and use them to obtain variational representations related to the quantum Tsallis relative entropy. We extend Golden–Thompson’s trace inequality to deformed exponentials with deformation parameter \( q\in [0,1], \) thus complementing the second author’s previous study of the cases with deformation parameter \( q\in [1,2]\) and \( q\in [2,3]\).

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References

  1. Ando, T.: Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26, 203–241 (1979)

    Article  MathSciNet  Google Scholar 

  2. Bekjan, T.: On joint convexity of trace functions. Linear Algebra Appl. 390, 321–327 (2004)

    Article  MathSciNet  Google Scholar 

  3. Carlen, E.A., Frank, R.L., Lieb, E.H.: Inequalities for quantum divergences and the Audenaert–Datta conjecture. J. Phys. A 51(48), 483001 (2018)

    Article  MathSciNet  Google Scholar 

  4. Carlen, E.A., Lieb, E.H.: A Minkowsky type trace inequality and strong subadditivity of quantum entropy II: convexity and concavity. Lett. Math. Phys. 83(2), 107–126 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  5. Furuichi, S.: Trace inequalities in nonextensive statistical mechanics. Linear Algebra Appl. 418, 821–827 (2006)

    Article  MathSciNet  Google Scholar 

  6. Furuichi, S., Yanagi, K., Kuriyama, K.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45(12), 4868–4877 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  7. Hansen, F.: Multivariate extensions of the Golden–Thompson inequality. Ann. Funct. Anal. 6(4), 301–310 (2015)

    Article  MathSciNet  Google Scholar 

  8. Hansen, F.: Golden–Thompson’s inequality for deformed exponentials. J. Stat. Phys. 159(5), 1300–1305 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  9. Hansen, F., Liang, J., Shi, G.: Peierls–Bogolyubov’s inequality for deformed exponentials. Entropy 19(6), 271 (2017)

    Article  ADS  Google Scholar 

  10. Hiai, F.: Concavity of certain matrix trace functions. Taiwan. J. Math. 5, 535–554 (2001)

    Article  MathSciNet  Google Scholar 

  11. Hiai, F., Petz, D.: The Golden–Thompson trace inequality is complemented. Linear Algebra Appl. 181, 153–185 (1993)

    Article  MathSciNet  Google Scholar 

  12. Lieb, E.H.: Convex trace functions and the Wigner–Yanase–Dyson conjecture. Adv. Math. 11, 267–288 (1973)

    Article  MathSciNet  Google Scholar 

  13. Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. With an appendix by B. Simon. J. Math. Phys. 14(12), 1938–1941 (1973)

    Article  ADS  Google Scholar 

  14. Lieb, E.H., Seiringer, R.: Stronger subadditivity of entropy. Phys. Rev. A 71(6), 062329 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  15. Petz, D.: A variational expression for the relative entropy. Commun. Math. Phys. 114, 345–349 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  16. Tropp, J.: From joint convexity of quantum relative entropy to a concavity theorem of Lieb. Proc. Am. Math. Soc. 140(5), 1757–1760 (2012)

    Article  MathSciNet  Google Scholar 

  17. Tropp, J.: User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12(4), 389–434 (2012)

    Article  MathSciNet  Google Scholar 

  18. Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  19. Umegaki, H.: Conditional expectation in an operator algebra, IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)

    Article  MathSciNet  Google Scholar 

  20. Yu, Y.L.: Note on a concavity theorem of Lieb. cs.uwaterloo.ca (2011)

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Acknowledgements

The authors are grateful for receiving a very detailed and carefully written referee report with many useful comments. The first author acknowledges supports from the Natural Science Foundation of Jiangsu Province for Youth, Grant No: BK20190874, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China, Grant No: 18KJB110033. The second author acknowledges support from the Japanese Government Grant-in-Aid for scientific research 17K05267.

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Authors and Affiliations

  1. School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu, China

    Guanghua Shi

  2. Department of Mathematical Sciences, Copenhagen University, Copenhagen, Denmark

    Frank Hansen

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  1. Guanghua Shi
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  2. Frank Hansen
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Correspondence to Frank Hansen.

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Shi, G., Hansen, F. Variational representations related to Tsallis relative entropy. Lett Math Phys 110, 2203–2220 (2020). https://doi.org/10.1007/s11005-020-01289-7

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  • DOI: https://doi.org/10.1007/s11005-020-01289-7

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