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

A variational expression for a generlized relative entropy

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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) - PowerPoint PPT Presentation

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Page 1: A variational expression for a  generlized  relative entropy

A variational expression for a generlized relative entropy

Nihon University

Shigeru FURUICHI

  ||

Tsallis

Page 2: A variational expression for a  generlized  relative entropy

Outline1.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

Page 3: A variational expression for a  generlized  relative entropy

1.1 . Background

• Statistical Physics , Multifractal• Tsallis entropy 1988

(1) one-parameter extension of Shannon entropy

(2)

non-additive entropy

, 0, 11

q

xq

p x p xS X q q

q

11

lim logqq

x

S X S X p x p x

1q q q q qS X Y S X S Y q S X S Y

1q

Page 4: A variational expression for a  generlized  relative entropy

1.2.Definition

For positve matrices

Tsallis relative entropy:

Tsallie relative operator entropy:

1/ 2 1/ 2 1/ 2 1/ 2#, ln

X Y XT X Y X X YX X

1

1 ln lnTr X X Y

D X Y Tr X X Y

1/

0 0

0

1 1 0 1exp ,ln , 0

0

limexp exp ,li

1 1,

l g

0

m n lo

x if x xx x x

otherwise

x x x x

R

,X Y

Parameter is changed from to

q 1 q

Consider the inequality for

before the limit

exp ,lnx x

e ,logx x

Page 5: A variational expression for a  generlized  relative entropy

1.3Properties(1)

1.

(Umegaki relative entropy)

2.

(Fujii-Kamei relative operator entropy)

3.

0

lim log logD X Y Tr X X Y U X Y

1/ 2 1/ 2 1/ 2 1/ 2

0lim , log ,T X Y X X YX X S X Y

, 0 ,X Y D X Y Tr T X Y

Page 6: A variational expression for a  generlized  relative entropy

1.3 Properties(2):1. with equality iff

2.

3.

4.

5.

for trace-preserving CP linear map

, 1,0 0,1D X Y

0D X Y X Y

1 2 1 2

1 1 2 2 1 1 2 2

D X X Y Y

D X Y D X Y D X Y D X Y

j j j j j j jj j j

D p X p Y p D X Y

* *D UXU UYU D X Y

D X Y D X Y

S.Furuichi, K.Yanagi and K.Kuriyama,J.Math.Phys.,Vol.45(2004), pp. 4868-4877

Completely positive map

:

0

0 .

def nn

n

K

S H S K

H

is CP map

I X on

for X on and n

C

C

Page 7: A variational expression for a  generlized  relative entropy

1.3 Properties(3):1.

2.

3.

4.

5.

for a unital positive linear map

6. bounds of the Tsallis relative operator entropy

,T X Y

, , , 0T X Y T X Y

, ,Y Z T X Y T X Z

1 2 1 2 1 1 2 2, , ,T X X Y Y T X Y T X Y

1 2 1 2 1 1 2 2, , ,T X X Y Y T X Y T X Y

, ,T X Y T X Y

1

1 1# # ln ,

1 1ln # , 0

X Y X Y X T X Y

Y X X Y

Solidarity

J.I.Fujii,M.Fujii,Y.Seo,Math.Japonica,Vol.35,pp.387-396(1990)

1/ 2 1/ 2 1/ 2 1/ 2XsY X f X YX X f :operator monotone function on 0,

S.Furuichi, K.Yanagi, K.Kuriyama,LAA,Vol.407(2005),pp.19-31.

Page 8: A variational expression for a  generlized  relative entropy

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

: 0, 1n nX M X Tr X S C

1,0 0,1 H

1 1:nC X Tr X H Tr Y H

S

1 lnS X D X I Tr X X

Y

Page 9: A variational expression for a  generlized  relative entropy

Theorem 2.1

Let ,where

Then

1 expH

Y ZH

exp

HZ Tr

H

X C S X S Y

S.Furuichi,J.Inequal.Pure and Appl.Math.,Vol.9(2008),Art.1,7pp.

Page 10: A variational expression for a  generlized  relative entropy

Proof of Theorem2.1

1.

2.

3.

                                

1

, 1 1 0

exp 0 0

HH I H H I I

H

H HI Z

H H

1 1ln ln ln , 0x Y Y x Y for Y and x

R

1 1X C Tr X H Tr Y H

Page 11: A variational expression for a  generlized  relative entropy

1 1 1

1 1

1 1

1 1

1 1

1 1

1

ln ln exp /

/ ln /

ln /

ln /

/ ln /

ln exp /

ln

Tr X Y Tr X Z H H

Tr X H H Z I H H

Tr X Z I Z H H

Tr Y Z I Z H H

Tr Y H H Z I H H

Tr Y Z H H

Tr Y Y

1 10 ln lnD X Y Tr X Y Tr X X

1 1 1ln ln lnS X Tr X X Tr X Y Tr Y Y S Y

Page 12: A variational expression for a  generlized  relative entropy

Remark 2.2                    :conc

ave

     :concave on the set

   The maximizer    is uniquely determined

     :a generalized Gibbs state

    A generalized Helmholtz free energy:

    Expression by Tsallis relative entropy:

 

1 ln , 1 1f x x x S C

Y Y

1,F X H Tr X H H S X

1 1, lnF X H H D X Y Z Tr X H H

Page 13: A variational expression for a  generlized  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

ln , 0 1S d

1dim , ln

xd H

H

ln 1 0 1, 0z z z

Page 14: A variational expression for a  generlized  relative entropy

LemmasLemma3.2

If is a concave function and ,

then we have

for any and with

Lemma3.3

For any real numbers and ,

if , then

where

f 0 1 0f f

max , 1f t s f t f s f s

0,1/ 2s 0,1t 0 1.s t

, 0,1u v 1,1 1/ 2u v u v u v

1x x

x

Page 15: A variational expression for a  generlized  relative entropy

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.

1 2 n 1 2 n A B

1

.n

j jj

Tr A B

Page 16: A variational expression for a  generlized  relative entropy

A generalized Fannes’ inequalityTheorem3.5

For two density operators and on the finite

dimensional Hilbert space with

and , if ,

then

where we denote

for a bounded linear operator .

1 2

H dimH d

1,1 1/1 2 11

1

1 2 1 2 1 21 1lnS S d

1/ 2*

1A Tr A A

A

Page 17: A variational expression for a  generlized  relative entropy

Proof of Theorem3.5Let and

be eigenvalues of two density operators and .

Putting

we have

due to Lemma3.4. Applying Lemma3.3, we have

1 1 11 2 d 2 2 2

1 2 d

1 2

1 2

1

,d

j j j jj

1/1 2 11 1/ 2j

1 21 2

1 1

.d d

j j jj j

S S

Page 18: A variational expression for a  generlized  relative entropy

By the formula , we have

ln ln lnxy x x y

1

1 1

1

1

1 1

1 1

1

1

1

ln

ln

ln ln

ln

d d

j j jj j

dj j

j

d dj j j j

j j

dj

j

d

Page 19: A variational expression for a  generlized  relative entropy

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 proofof the present theorem is completed.      

 □

1 / , , / .d diag

11 2 ln .S S d

x

1/0, 1 .x

1x

1,1 .

Page 20: A variational expression for a  generlized  relative entropy

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

1 2

H dimH d

1 2 11/ e

0 1 0 2 1 2 0 1 21 1lnS S d

0 0 0, ln .S Tr x x x

0

1/0

lim 1 1/ .e

Page 21: A variational expression for a  generlized  relative entropy

4.Trace inequalityHiai-Petz1993

1/ 2 1/ 2, logU X Y Tr S X Y Tr X X YX

Furuichi-Yanagi-Kuriyama2004

,D X Y Tr T X Y

1// 2 / 2log , 0pp p pU X Y Tr X X Y X p

S.Furuichi, K.Yanagi and K.Kuriyama,J.Math.Phys.,Vol.45(2004), pp.4868-4877.

Page 22: A variational expression for a  generlized  relative entropy

Proposition4.1

(1)We have

but

(2)

does not   hold in general.

1// 2 / 2 1ln ,pp p pD X Y Tr X pX X Y

1// 2 / 2ln ,0 1pp p pD X Y Tr X X Y X p

S.Furuichi,J.Inequal.Pure and Appl.Math.,Vol.9(2008),Art.1,7pp.

Page 23: A variational expression for a  generlized  relative entropy

Proof of (1)Inequality :

for Hermitian (Hiai-Petz 1993)

Putting in the above,we have

1/ 1#p A BpA pBTr e e Tr e

,A B

log , logA X B Y

1

1

1/ log log

log log

1

#

. . .

pp p X Y

X Y

Tr X Y Tr e

Tr e e by GT ineq

T ar X Y

Page 24: A variational expression for a  generlized  relative entropy

Inequality :

for (modified Araki’s inequality)

implies

From (a) and (b), we have (1) of Proposition4.1

1/ 2 1/ 2 , 0 1aa aTr X Y Tr Y XY a

, 0X Y

1/1/ / 2 / 2 / 2 / 2

// 2 / 2

#ppp p p p p p p

pp p p

Tr X Y Tr X X Y X X

Tr X X Y X b

Page 25: A variational expression for a  generlized  relative entropy

A counter-example of (2):Note that

Then we set

R.H.S. of (c) – L.H.S. of (c) approximately takes

1// 2 / 2

// 2 / 2 1

lnpp p p

pp p p

D X Y Tr X X Y X

Tr X cX Y X Tr X Y

10 3 5 40.3, 0.9, ,

3 9 4 5p X Y

0.00309808

Page 26: A variational expression for a  generlized  relative entropy

5. Variational expression of the Tsallis relative entropy

Upper bound of

Lower bound of

Variational expression of

,D X Y Tr T X Y

D X Y

D X Y

? D X Y

D X Y

T.Furuta,

LAA,Vol.403(2005),pp.24-30. 11 K

Tr X Tr Y D X Y

Page 27: A variational expression for a  generlized  relative entropy

Theorem5.1 (1) If are positive, then

(2) If is density and is Hermitian, then

,A Y

1

ln exp ln

max : 0, 1

Tr A Y

Tr X A D X Y X Tr X

X B

1

exp

max ln exp : 0

D X B

Tr X A Tr A B A

Proof is similar to Hiai-Petz, LAA, Vol.181(1993),pp.153-185.

S.Furuichi, LAA, Vol.418(2006), pp. 821-827

Page 28: A variational expression for a  generlized  relative entropy

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

,X Y 0 1

1/ 2 1/ 2exp expTr X Y Tr X Y Y XY

:f R R

,A B

A B Tr f A Tr f B

Page 29: A variational expression for a  generlized  relative entropy

Proposition 5.3 If are positive, then for , we have

Proof: In Lieb-Thirring inequality : for

put

,X Y 0 1

exp exp expTr X Y XY Tr X Y

Tr AB Tr A B 0, 0, 1A B

1, ,A I X B I Y

Page 30: A variational expression for a  generlized  relative entropy

We want to combine the R.H.S. of

and the L.H.S. of

General case is difficult so we consider :

for Hermitian

1/ 2 1/ 2e exp xpTr XTr X Y Y Y XY d

exp exp expTTr Y eYX Y XX r

1

2

2 2Tr HZHZ Tr H Z ,H Z

2 21/ 2 1/ 2

2 21/ 2 1/ 2

2

1/ 2 1/ 2

, 0, 0

1 1 1 1 1 1 1 1, ,

2 2 4 2 2 4 2 2

1 1 1 1

2 2 2 2

Tr I A B B AB Tr I A B AB A B

Tr I X Y Y XY Tr I X Y XY A X B Y

Tr I X Y Y XY Tr I X Y XY

1/ 2 1/ 21/ 2

2

1/ 2

1 1exp exp

2 2Tr X Y Y XY Tr X XY fY

Page 31: A variational expression for a  generlized  relative entropy

From (d), (e) and (f),we have

Putting in (2) of Theorem5.1

Thus we have the lower bound of

in the special case.

1/ 2 1/ 2 1/ 2exp exp expTr X Y Tr X gY

1/ 2 1/ 21/ 2 1/ 2ln , lnB Y A Y XY

1/ 2 1/ 2 1/ 2 1/ 2

1/ 21/ 2 1/ 2

1/ 21/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2 1/ 2 1/ 21/ 2 1/ 2

1/ 2 1/ 2 1/ 21/ 2

exp ln

ln exp

ln exp exp

ln ln

ln

D X Y D X Y

Tr X A Tr A B

Tr X A Tr A B by

Tr X Y XY Tr Y XY Y

Tr X Y XY

g

1/ 2 1/ 2 1/ 21/ 2 1/ 2lnTr X Y XY D X Y

D X Y