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A variational expression for a generlized relative entropy

A variational expression for a generlized relative entropy. || Tsallis. Nihon University Shigeru FURUICHI. Outline. 1.Background, definition and properties 2.MaxEnt principle in Tsallis statistics 3.A generalized Fannes’ inequality 4.Trace inequality (Hiai-Petz Type)

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A variational expression for a generlized relative entropy

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  1. A variational expression for a generlized relative entropy || Tsallis Nihon University Shigeru FURUICHI

  2. Outline 1.Background, definition and properties 2.MaxEnt principle in Tsallis statistics 3.A generalized Fannes’ inequality 4.Trace inequality (Hiai-Petz Type) 5.Variational expression and its application

  3. 1.1．Background • Statistical Physics，Multifractal • Tsallis entropy 1988 (1) one-parameter extension of Shannon entropy (2) non-additive entropy

  4. 1.2.Definition Parameter is changed from to For positve matrices Tsallis relative entropy: Tsallie relative operator entropy: Consider the inequality for before the limit

  5. 1.3Properties(1) 1. (Umegaki relative entropy) 2. (Fujii-Kamei relative operator entropy) 3.

  6. 1.3 Properties(2): S.Furuichi, K.Yanagi andK.Kuriyama,J.Math.Phys., Vol.45(2004), pp. 4868-4877 1. with equality iff 2. 3. 4. 5. for trace-preserving CP linear map Completely positive map

  7. 1.3 Properties(3): Solidarity J.I.Fujii,M.Fujii,Y.Seo,Math.Japonica,Vol.35,pp.387-396(1990) 1. 2. 3. 4. 5. for a unital positive linear map 6. bounds of the Tsallis relative operator entropy :operator monotone function on S.Furuichi, K.Yanagi, K.Kuriyama,LAA,Vol.407(2005),pp.19-31.

  8. 2.Maximum entropy principle in Tsallis statistic The set of all states (density matrices) For , density and Hermitian , we denote Tsallis entropy is defined by

  9. Theorem 2.1 S.Furuichi,J.Inequal.Pure and Appl.Math.,Vol.9(2008),Art.1,7pp. Let ,where Then

  10. Proof of Theorem2.1 1. 2. 3.

  11. Remark 2.2 :concave :concave on the set The maximizer is uniquely determined :a generalized Gibbs state A generalized Helmholtz free energy: Expression by Tsallis relative entropy:

  12. 3. A generalized Fannes’ inequality Lemma 3.1 For a density operator on finite dimensional Hilbert space , we have where . Proof is done by the nonnegativity of the Tsallis relative entropy and the inequality

  13. Lemmas Lemma3.2 If is a concave function and , then we have for any and with Lemma3.3 For any real numbers and , if , then where

  14. Lemma3.4(Lemma1.7 of the book Ohya&Petz) Let and be the eigenvalues of the self-adjoint matrices and . Then we have [Ref]M.Ohya and D.Petz, Quantum entropy and its use, Springer,1993.

  15. A generalized Fannes’ inequality Theorem3.5 For two density operators and on the finite dimensional Hilbert space with and , if , then where we denote for a bounded linear operator .

  16. Proof of Theorem3.5 Let and be eigenvalues of two density operators and . Putting we have due to Lemma3.4. Applying Lemma3.3, we have

  17. By the formula , we have

  18. In the above inequality, Lemma3.1 was used for Thus we have Now is a monotone increasing function on In addition, is a monotone increasing function for Thus the proof of the present theorem is completed. □

  19. Corollary3.6(Fannes’ inequality) For two density operators and on the finite dimensional Hilbert space with , if , then where Proof Take the limit in Theorem3.5. Note that

  20. 4.Trace inequality Hiai-Petz1993 S.Furuichi, K.Yanagi and K.Kuriyama,J.Math.Phys., Vol.45(2004), pp.4868-4877. Furuichi-Yanagi-Kuriyama2004

  21. Proposition4.1 S.Furuichi,J.Inequal.Pure and Appl.Math.,Vol.9(2008),Art.1,7pp. (1)We have but (2) does nothold in general.

  22. Proof of (1) Inequality : for Hermitian (Hiai-Petz 1993) Putting in the above,we have

  23. Inequality：for (modified Araki’s inequality) implies From (a) and (b), we have (1) of Proposition4.1

  24. A counter-example of (2): Note that Then we set R.H.S. of (c) – L.H.S. of (c) approximately takes

  25. 5. Variational expression of the Tsallis relative entropy T.Furuta, LAA,Vol.403(2005),pp.24-30. Upper bound of Lower bound of Variational expression of

  26. S.Furuichi, LAA, Vol.418(2006), pp. 821-827 Theorem5.1 (1) If are positive, then (2) If is density and is Hermitian, then Proof is similar to Hiai-Petz, LAA, Vol.181(1993),pp.153-185.

  27. Proposition 5.2 If are positive, then for we have Proof: If is a monotone increase function and are Hermitian, then we have which implies the proof of Proposition 5.2

  28. Proposition 5.3 If are positive, then for , we have Proof: In Lieb-Thirring inequality： for put

  29. We want to combine the R.H.S. ofand the L.H.S. of General case is difficult so we consider : for Hermitian

  30. From (d), (e) and (f),we have Putting in (2) of Theorem5.1 Thus we have the lower bound of in thespecial case.

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