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)
5.Variational expression and its application
1.1．Background
• Statistical Physics，Multifractal
• Tsallis entropy 1988
Sq  X  


p  x  p  x
q
x
1 q
, q  0, q  1
Sq  X   S1  X    p  x  log p  x 
(1) lim
q1
x
one-parameter extension of Shannon entropy
(2) Sq  X  Y   Sq  X   Sq Y   1  q  Sq  X  Sq Y 
non-additive entropy
 q  1
1.2.Definition
Parameter is changed from
0    R   1    1,   0
  1 q
q to 
1   x 1/  if 1   x  0
x  1
exp   x   
,ln   x  
,  x  0

otherwise
 0
lim exp   x   exp  x  ,lim ln   x   log x
 0
 0
Consider the inequality for
exp  x  ,ln   x 
For positve matrices X , Y
Tsallis relative entropy:
D  X Y  
Tr  X  X 1Y  

before the limit
e x ,log x
 Tr  X 1  ln  X  ln  Y 
Tsallie relative operator entropy:
T  X , Y  
X # Y  X

 X 1/ 2  ln  X 1/ 2YX 1/ 2  X 1/ 2
1.3Properties(1)
D  X Y   Tr  X  log X  log Y   U  X Y 
1. lim
 0
(Umegaki relative entropy)
T  X , Y   X 1/ 2  log X 1/ 2YX 1/ 2  X 1/ 2  S  X , Y 
2. lim
 0
(Fujii-Kamei relative operator entropy)
3.  X ,Y   0  D  X Y   Tr T  X ,Y 
1.3 Properties(2):
D  X Y  ,    1,0    0,1
1. D  X Y   0 with equality iff X  Y
2. D  X 1  X 2 Y1  Y2 
S.Furuichi, K.Yanagi and
K.Kuriyama,J.Math.Phys.,
Vol.45(2004), pp. 48684877
 D  X 1 Y1   D  X 2 Y2    D  X 1 Y1  D  X 2 Y2 

3. D   p j X j
 j


j p jYj   j p j D X j Yj



4. D UXU * UYU *  D  X Y 

Completely positive map
 : S  H   S  K  is CP map
5. D    X   Y    D  X Y 
for X  0  on H  C  and
for trace-preserving CP linear map 
def

   I n  X  0  on K  Cn 

n

n.
1.3 Properties(3): T  X , Y 
1. T  X ,Y   T  X ,Y  ,   0
Solidarity
J.I.Fujii,M.Fujii,Y.Seo,
2. Y  Z  T  X ,Y   T  X , Z 
Math.Japonica,Vol.35,
3. T  X  X ,Y  Y   T  X ,Y   T  X ,Y  pp.387-396(1990)
4. T  X   X ,Y   Y   T  X ,Y    T  X ,Y 
XsY  X 1/ 2 f  X 1/ 2YX 1/ 2  X 1/ 2
5.  T  X ,Y   T    X  ,  Y  f :operator monotone function on  0,  
1
2
1
1
2
2
1
1
2
1
2
1
2
1
for a unital positive linear map
2
2

6. bounds of the Tsallis relative operator entropy
X # Y 
1

X # 1 Y   ln   X  T  X , Y 



1
1

 Y  X   ln   X # Y , 



1
S.Furuichi, K.Yanagi,
K.Kuriyama,LAA,Vol.40
7(2005),pp.19-31.
 0
2.Maximum entropy principle
in Tsallis statistic
The set of all states (density matrices)
Sn   X  M n  C : X  0, Tr  X   1
For    1,0    0,1 , density Y and Hermitian H,
we denote


C  X  S n : Tr  X 1 H   Tr Y 1 H 
Tsallis entropy is defined by
S  X    D  X I   Tr  X 1 ln  X 
Theorem 2.1
S.Furuichi,J.Inequal.Pure and
Appl.Math.,Vol.9(2008),Art.1,7pp.
 H 

 H 
Let Y  Z  exp  
 ,where Z   Tr exp   

H



 H  
1
Then X  C  S  X   S Y 
Proof of Theorem2.1
1.  H I  H  H I , 1    1  I   H  0
H
1

 H 
H 
I 
  exp  
  0  Z  0
H 

 H 




2. ln  x1Y  ln  Y  ln  x 1 Y  , for Y  0 and x R
1
1
3. X  C  Tr  X H   Tr Y H 



Tr  X 1 ln  Y   Tr  X 1 ln  Z 1 exp   H / H  


 Tr  X 1  H / H  ln  Z 1  I   H / H  




 ln Z  I  Z H / H 
 ln Z  I  Z H / H 
 H / H  ln Z  I   H / H 
ln Z exp   H / H 

 Tr  X 1

 Tr Y 1

 Tr Y 1

 Tr Y 1


1




1





1

1


 Tr Y 1 ln  Y 
D  X Y   0  Tr  X 1 ln  Y   Tr  X 1 ln  X 

S  X   Tr  X 1 ln  X   Tr  X 1 ln  Y   Tr Y 1 ln  Y   S Y 
Remark 2.2
f   x    x1 ln  x,  1    1 :concave
 S :concave on the set C
 The
maximizer Y is uniquely determined
 Y :a generalized Gibbs state
 A generalized Helmholtz free energy:
F  X , H   Tr  X 1 H   H S  X 
 Expression by Tsallis relative entropy:
F  X , H   H D  X Y   ln  Z 1Tr  X 1  H   H  
3. A generalized Fannes’ inequality
Lemma 3.1
For a density operator  on finite dimensional
Hilbert space H, we have
S     ln  d ,  0    1
where d  dim H , ln  
x  1

.
Proof is done by the nonnegativity of the Tsallis
relative entropy and the inequality
ln  z  z  1  0    1, z  0 
Lemmas
Lemma3.2
If f is a concave function and f  0   f 1  0 ,
then we have
f  t  s   f  t   max  f  s  , f 1  s 
for any s  0,1/ 2 and t  0,1 with 0  s  t  1.
Lemma3.3
For any real numbers u, v  0,1 and    1,1,
if u  v  1/ 2 , then   u    v     u  v 
where
x1  x
  x  

Lemma3.4(Lemma1.7 of the book Ohya&Petz)
Let 1  2   n and 1   2    n be the
eigenvalues of the self-adjoint matrices A and B .
Then we have
n
Tr  A  B     j   j .
j 1
[Ref]M.Ohya and D.Petz, Quantum entropy and
its use, Springer,1993.
A generalized Fannes’ inequality
Theorem3.5
For two density operators 1 and  2 on the finite
dimensional Hilbert space H with dim H  d  
1/ 




1


  ,
2 1
and    1,1 , if 1
then
1
S  1   S  2   1  2 1 ln  d    1  2 1 
where we denote
1/ 2
*

A 1  Tr  A A 


for a bounded linear operator A .
Proof of Theorem3.5
1
1
1
2
2
2
Let 1   2    d   and 1   2    d  
be eigenvalues of two density operators 1 and  2 .
Putting
d
    j ,  j   j 1   j  2
j 1
we have
1/ 
 j    1   2 1  1     1/ 2
due to Lemma3.4. Applying Lemma3.3, we have
d
         .
S  1   S   2      j
j 1
1

 2
j
d
j 1

j
By the formula ln   xy   ln  x  x ln  y , we have
1







  j  j ln   j
d
d
j 1
j 1
 d  j1
 j 
   
ln   
 
 j 1 

1
d
 d  j1

j
j j 
  
ln   
  ln   
 j 1    
 j 1 

d
j 
1
         
j 1
 
  1 ln  d     
In the above inequality, Lemma3.1 was used for
  diag 1 /  , ,  d /  . Thus we have
S  1   S  2    1 ln  d     .
Now   x  is a monotone increasing function on
1/ 

x  0, 1     . In addition, x1 is a monotone


increasing function for    1,1. Thus the proof
of the present theorem is completed.
□
Corollary3.6(Fannes’ inequality)
For two density operators 1 and  2 on the finite
dimensional Hilbert space H with dim H  d  ,
if 1  2 1  1/ e , then
S0  1   S0   2   1   2 1 ln d  0  1   2 1 
where
S0     Tr 0    ,0  x    x ln x.
Proof Take the limit   0 in Theorem3.5.
1/ 
Note that lim 1     1/ e.
 0
4.Trace inequality
Hiai-Petz1993
U  X Y   Tr  S  X , Y    Tr  X  log X 1/ 2YX 1/ 2  

Furuichi-Yanagi-Kuriyama2004
S.Furuichi, K.Yanagi and
K.Kuriyama,J.Math.Phys.,
Vol.45(2004), pp.4868-4877.
D  X Y   Tr T  X ,Y 
 p/2 p
 p / 2 1/ p 

U  X Y   Tr X log  X
Y X
  , p  0

Proposition4.1
S.Furuichi,J.Inequal.Pure and
Appl.Math.,Vol.9(2008),Art.1,7pp.
(1)We have
 p/2 p
 p / 2 1/ p 

D  X Y   Tr X ln   X
Y X
  , p  1

but
(2) D  X Y   Tr  X ln   X  p / 2Y p X  p / 2 

does not hold in general.
1/ p
 ,0  p  1

Proof of (1)
Inequality :
Tr  e pA # e pB 

1/ p
  Tr e1  A B 



for Hermitian A, B (Hiai-Petz 1993)
Putting A  log X , B  log Y in the above,we have
Tr  X p # Y p 

1/ p
  Tr e log X 1 log Y  



1

 Tr elog X elog Y   by G.T . ineq.


 Tr  X 1Y  
a
Inequality： Tr  X aY a   Tr Y 1/ 2 XY 1/ 2 

a
 ,  0  a  1

for X , Y  0 (modified Araki’s inequality)
implies
Tr  X # Y

p

p 1/ p


1/ p



p
/
2

p
/
2
p

p
/
2
p
/
2
  Tr X  X
Y
X
X






 p/2 p
 p/2 / p 

 Tr X  X
Y X
   b 

From (a) and (b), we have (1) of Proposition4.1
A counter-example of (2):
Note that
 p/2 p
 p / 2 1/ p 

D  X Y   Tr X  ln  X
Y X
 

 p/2 p
 p/2 / p 

 Tr X  X
Y X
 Tr  X 1Y  



Then we set
c
10 3 
5 4
p  0.3,   0.9, X  
,Y  


3
9
4
5




R.H.S. of (c) – L.H.S. of (c) approximately takes
0.00309808
5. Variational expression of the
Tsallis relative entropy
Upper bound of
D  X Y 
T.Furuta,
LAA,Vol.403(2005),pp.24-30.
 1 K   
1

Tr
X
Tr
Y
 D  X Y 








D  X Y   Tr T  X ,Y   
Lower bound of
?
D  X Y 
 D  X Y 

Variational expression of D  X Y 
Theorem5.1
S.Furuichi, LAA, Vol.418(2006),
pp. 821-827
(1) If A, Y are positive, then
ln  Tr exp   A  ln  Y  


 max Tr  X 1 A  D  X Y  : X  0, Tr  X   1
(2) If X is density and B is Hermitian, then

D X exp  B 



 max Tr  X 1 A  ln  Tr exp  A  B   : A  0
Proof is similar to Hiai-Petz, LAA, Vol.181(1993),pp.153-185.
Proposition 5.2
If X , Y are positive, then for 0    1 we have
Tr exp  X  Y    Tr exp  X  Y  Y 1/ 2 XY 1/ 2  
Proof: If f : R  R is a monotone increase function and
A, B are Hermitian, then we have
A  B  Tr  f  A  Tr  f  B 
which implies the proof of Proposition 5.2
Proposition 5.3
If X , Y are positive, then for 0    1, we have
Tr exp  X  Y   XY   Tr exp  X  exp Y 
Proof: In Lieb-Thirring inequality：

Tr  AB    Tr  A B  for A  0, B  0,   1


put A  I   X , B  I  Y ,  1

We want to combine the R.H.S. of
Tr exp  X  Y    Tr exp  X  Y  Y 1/ 2 XY 1/ 2  
d 
and the L.H.S. of
Tr exp  X  Y   XY   Tr exp  X  exp Y 
General case is difficult so we consider
 Tr  HZHZ   Tr  H 2 Z 2  for Hermitian H , Z

1
2
e
:
2
2
 Tr  I  A  B  B1/ 2 AB1/ 2    Tr  I  A  B  AB   ,  A  0, B  0 




2
2


1
1
1 1/ 2 1/ 2  
1
1
1
1
1 
  
 Tr  I  X  Y  Y XY    Tr  I  X  Y  XY   ,  A  X , B  Y 
2
2
4
2
2
4
2
2 
 
  


2
2
 1 
 1 
1 1/ 2 1/ 2   
1
 
 Tr   I   X  Y  Y XY     Tr   I   X  Y  XY   
2
2
  
  
  2 
  2 


1
1




 Tr exp1/ 2  X  Y  Y 1/ 2 XY 1/ 2    Tr exp1/ 2  X  Y  XY  
2
2






f
From (d), (e) and (f),we have
Tr exp1/ 2  X  Y   Tr exp1/ 2  X  exp1/ 2 Y 
g
Putting B  ln1/ 2 Y , A  ln1/ 2 Y 1/ 2 XY 1/ 2 in (2) of Theorem5.1

D1/ 2  X Y   D1/ 2 X exp1/ 2  ln1/ 2 Y 

 Tr  X 1/ 2 A  ln1/ 2 Tr exp1/ 2  A  B  
 Tr  X 1/ 2 A  ln1/ 2 Tr exp1/ 2  A  exp1/ 2  B  
by g  
 Tr  X 1/ 2 ln1/ 2 Y 1/ 2 XY 1/ 2   ln1/ 2 Tr Y 1/ 2 XY 1/ 2Y 
 Tr  X 1/ 2 ln1/ 2 Y 1/ 2 XY 1/ 2 
Thus we have the lower bound of D  X Y 
Tr  X 1/ 2 ln1/ 2 Y 1/ 2 XY 1/ 2   D1/ 2  X Y 
in the special case.