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