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

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

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

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

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

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

1.3 Properties(3): for a unital positive linear map 6. bounds of the Tsallis relative operator entropy Solidarity J.I.Fujii,M.Fujii,Y.Seo, Math.Japonica,Vol.35, pp (1990) :operator monotone function on S.Furuichi, K.Yanagi, K.Kuriyama,LAA,Vol.40 7(2005),pp

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

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

Proof of Theorem

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

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

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

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.

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.

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

By the formula, we have

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. □

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

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

Proposition4.1 (1)We have but (2) does not hold in general. S.Furuichi,J.Inequal.Pure and Appl.Math.,Vol.9(2008),Art.1,7pp.

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

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

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

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

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 S.Furuichi, LAA, Vol.418(2006), pp

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

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

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

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