数理解析研究所講究録 1458 巻 2005 年 97-105

97

Tsallis entropy に関する作用素不等式とトレース不等式

Operator inequalities and trace inequalities
derived from Tsallis entropies
古市

茂 (Shigeru Furuicl i)
山口東京理科大(Tokyo University of Science in Yamaguchi)
柳 研二郎 (Kenjiro Yanagi)
山口大・工(Department of Applied Science, Yamaguchi University)
栗山 憲 (Ken Kuriyama)
山口大・工(Department of Applied Science, Yamaguchi University)

1$\backslash \backslash ^{\gamma}\mathrm{e}$Trace inequalities of Tsallis entropy
define -logarithm function as follow .
$\mathrm{v}\mathrm{s}_{\dot{l}}$$\mathrm{q}$$\ln_{\mathit{1}}‘ x=\frac{x^{1-(/}-1}{1-q}$,$(x\geq 0, q\geq 0, q\neq 1)$.

Then we have the following properties;. (in uniformly)(1)$1\mathrm{i}\mathrm{n}1_{\mathit{1}^{arrow 1}},111_{t\mathit{1}}X=\log x$(2)$\ln_{\mathrm{r}_{\mathit{1}}}xy=11\mathrm{T}_{\mathit{1}},X+\ln_{I},y+(1-q)\ln(\mathit{1}x\ln_{\mathrm{r}/}y$(3)$\ln_{t\mathit{1}}x$: concave in$x$.

for $q\geq 0$ .

Definition 1 (Tsatlis entropy) For density operator on a finite dt.m enis defined by
space , Tsallis entropy
sional Hilb$\rho$$?\epsilon’ rl$$\mathcal{H}$$S_{\mathrm{t}},(\rho)$$S_{l},( \rho)=\frac{Tr[\rho^{q}-p_{\mathrm{J}}^{\rceil}}{1-q}$,$(q\geq 0, q\neq 1)$.

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Proposition 1 We have the following properties;.

(1)$\lim_{\Gamma J}\underline{\backslash }\mathrm{J}S_{\mathit{1}}‘(p)=-Tr_{\lfloor}p\mathrm{l}\mathrm{r}\mathrm{o}\mathrm{g}p\rfloor\rceil$(2)$S_{\mathit{1}},(p_{1\mathrm{k}^{3}}\sigma p_{\underline{\eta}})=S_{\mathit{1}},(\rho_{1})+S_{q}(p_{)}..)+(1-q)S_{q}(p_{1})S_{(\mathit{1}}(p_{2})$Lemma 1

$S_{l}((\rho)\leq\ln_{\mathrm{r}_{j}}d,$

Proof. Since$\ln_{\mathit{1}}‘ x$(d$=\mathrm{d}\mathrm{i}_{\mathrm{J}}\mathrm{n}\mathcal{H})$.

.

is concave, we have$D,,(A|B)=- \sum_{j=1}^{d}a_{j}\ln_{q}\frac{b_{j}}{a_{j}}\geq-\ln_{\mathit{1}},(\sum_{j=1}^{d}a_{j}\frac{b_{j}}{a_{j}})=0$$\backslash 1^{\gamma}\mathrm{e}$.

put $A=\{a_{j}\}\dot,$ $B=\{u_{j}\}$ . $u_{j}= \frac{1}{d}(1\leq j\leq d)$ . Then
$D_{q}(A|B)=-d^{q-1}(S_{q}(A) -\ln_{\mathit{1}},d)\geq 0$

Tl us$S_{\{},(A)\leq\ln_{\mathit{1}},d$.

.

$\mathrm{q}.\epsilon^{\iota}.\mathrm{r}.1$.

Lennna 2 if f is a concave function satisfying $f(\mathrm{O})=f(1)=0$ , then$|f(t+s)-f(t)| \leq\max\{f(s), f(1-s)\}$ ,$n\{fieres\in[0, 1/\underline{7}]$, $t\in[0, 1]$ , $0\leq s+t\leq 1$ .

Proof. We put$r(t)=f(.\underline{\sigma})-f(t+s)+f(t)$

Then

.

$r(t)’=-f’(t+s)+f’(t)$ .

Since $f’$ is a monotone decreasing function, $r^{l}(t)\geq 0$ . Tl us we have $r(t)$$?’(0)=0$ . Therefore $f(t+s)-f(t)\leq f(s)$ . We also put$\geq 0$$1\supset \mathrm{y}$$P(t)=f(t|[perp] s)-f(t)+f(1-s)$ .

Th en

‘
$l$

$(t)=f’(t+s)-f’(t)$ .

is a monotone decreasing function, $l^{t}(t)\leq 0$ . Thus we have $\ell(t)\geq 0$
by $\ell(1-s)=0$ . Therefore –f$f(1-s)$ $\leq f(t+s)-f(t)$ . Thus we have theresult.
.e.cl.

Since

$f’$

$\mathrm{q}$aLemma 3 If |u $-v|\leq 1/2$ , then $|?7_{J}‘(u)-\eta_{\mathit{1}}|(v)|\leq?7_{q}(|u-v|)$ ,$\prime l\mathrm{v}fiere?7_{l}\mathit{1}(x)=\frac{x^{q}-x}{1-q}$

Proof. Since$\uparrow 7(J$, $q\in[0, 2]$ ,$?\iota$, $v\in[0,1]$ ,is a concave function with$\uparrow 7_{q}(0)=\eta_{J},(0)\}$

we have$|?7_{\mathit{1}},(t+s)-?7_{\mathit{1}}‘(t)| \leq\max\{\eta_{q}(s)_{77_{\mathit{1}}}?,(1-s)\}$and $t\in[0, 1]$ satisfying $0\leq t+s\leq 1$ by Lemma .
foris a monotone increasing function on $[0, q^{1/(1-q)}]$ andSince
for $q\in(0,2]$ .
$\underline{9}$$s \in[0, \frac{1}{arrow\gamma}]$$\eta_{q}(x)$$\max\{\eta_{J},(s), ?7_{\mathit{1}},(1-s)\}=\eta_{\mathrm{t}i}(s)$$q^{1/(1)}- \Gamma\int\leq\underline{\frac{1}{9}}$.

Thus we have the result by letting $u=t+s$ and $v=t$ .

q.e.d.

, be eigenvalues of Hermitian matrix A crnd
, be eigenvalues of Hermitian matrix B. Then we have theLemma 4 $Lei$$,\backslash _{1}\geq\lambda\underline{\gamma}\geq\cdots\geq\lambda,$$/\cdot\iota_{1}\geq_{-}\mu,\sim\circ\geq\cdots\geq l\iota_{\}}$

follow vingi$Tr[|A-B|] \geq\sum_{j=1}^{7?}|\lambda_{i}-\mu_{i}|$.

Theorem 1 (Generalized Fannes’s inequality) For two density operatorsthen,
on rvnd $q\in[0,2]$ , if$||\rho_{1}-\rho\underline{\circ}||_{1}\leq q^{1/(1-(\mathit{1})}f$$\mathcal{H}$$\rho_{1}$$\rho_{2}$$|S,,(p_{1})-S_{\mathrm{r};}(\rho_{-}.,)|\leq||p_{1}-\rho_{2}||_{1}^{q}\ln_{t\mathit{1}}cl$

1 ancl$where$ $d=$ clim

Proof. Let$t$$||A||_{1}=Tr_{\mathrm{L}}^{\lceil}|A|$$,\backslash _{1}(j)\geq\cdots\geq\lambda_{7l}^{(i)}$$+\eta_{l}:(||p_{1}-\rho_{1}.\lrcorner||_{1})$,].

be eigenvalues of$\rho_{j}$.

We set$\epsilon=\sum_{j=1}^{d}\epsilon,$,$\epsilon_{j}=|\lambda_{\mathrm{i}}^{(1)}-\lambda_{j}^{(^{r})}|\underline{)}$.

From Lemma .
$\underline{7}$.

$|S_{f\mathit{1}}( \rho_{[perp]})-S_{\mathit{1}}((\rho_{2})|\leq\sum_{j=1}^{\prime l}|\eta(\mathit{1}(\lambda_{J}^{(1)})-\eta_{q}(\lambda_{j}^{(arrow)}.)|)\leq\sum_{j=1}^{d}\eta_{q}(\epsilon_{j})$By$\ln_{1\mathit{1}}(X?/\mathrm{I}=$]$\mathrm{n}_{\mathit{1}},x+x^{1-r_{\mathit{1}}}\ln_{\mathit{1}},y$and Len ma 1. we have$\mathfrak{g}$

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I

$=$

$- \sum_{=71}^{d}$ej.

$\ln_{q}\epsilon_{j}=$$\epsilon\{-\sum_{J^{=1}}^{d}\frac{\epsilon_{j}^{q}}{\epsilon}1_{11_{\}\mathit{1}}}(\frac{\epsilon_{j}}{\epsilon}\epsilon)\}$$= \epsilon\{-.\sum_{=?1}^{d}\frac{\epsilon_{j}^{\mathit{1}}\prime}{\epsilon}\ln_{l}\frac{\epsilon_{j}}{\epsilon}J-\sum_{j=1}^{d}\frac{\epsilon_{j}^{t}\mathit{4}}{\epsilon}(\frac{\epsilon_{j}}{\epsilon})^{1-r;}\ln_{q}\epsilon\}$$= \epsilon_{(\mathit{1}}\sum_{j=1}^{d}\uparrow 7_{\mathit{1}}‘(\frac{\epsilon_{j}}{\epsilon})+_{7\prime;}?(\in)\leq\epsilon^{q}\ln_{q}d+?7q(\epsilon)$.

Therefore xvc 1ave$|S_{q}(\rho_{1})-S_{\mathit{1}}‘(\rho_{\underline{\eta}})|\leq\epsilon^{q}\ln_{\mathit{1}}‘ d+\eta_{\mathrm{r}_{\mathit{1}}}(\epsilon)$

From Lem ma 3. we have
. In addition.
have th eorem.

is monotone increase on
on $x\in[0, 2]$ . Thus weincrease
is monotone$||\rho_{1}-\rho_{9}\sim||_{1}\geq\epsilon$$g^{\alpha}\in[0, q^{1/(1-q)}]$$x^{q}$.

. And$\eta_{q}(x)$$\mathrm{q}$Since$1\mathrm{i}\mathrm{n}1_{(/-}1q^{1/(1-\prime/1}=1/e$.

$\mathrm{e}.\mathrm{c}1$.

. we have

Corollary 1 (Fannes’s inequality) For two density operators, thenif$\rho_{1}$,$\mathcal{H}$$\beta\underline{\tau}O?\mathit{1}$,$||\rho_{1}-\rho_{7}\sim||_{1}\leq 1/e$,$|S_{3}(\rho_{1})-S_{1}(\rho_{2})|\leq||\rho_{1}-\rho_{2}||_{1}\log d+\eta_{1}(||\rho_{1}-\rho_{\underline{7}}||_{1})$ulrere $S_{1}(\rho)=-T\uparrow’[\rho\log\rho]$ , ?71 $(x)=-x\log x$ .

2$\backslash \backslash ^{\gamma},\mathrm{c}$Operator inequalities of Tsallis relative operator entropychange the notation (A $=1-q$ ). That is, for$\lambda\in(0, 1]$.

$\ln_{\lambda}x=\frac{x^{\lambda}-1}{\backslash },\cdot$

Definition 2 (Tsallis relative operator entropy) For $A>0$ ,$(0, 1]$ , Tsallis relative operator entropy
is defined by$T_{\lambda}(A|B)$$T_{\lambda}(A|B)$$=A^{1/^{\mathrm{f}}\underline{)}}\ln_{\lambda}(A^{-1/2}BA^{-1/2})A^{1/2}$.

$B>0_{1},\backslash \in$

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Proposition 2 we have the follow ving properties;(I) $\lim_{\lambda 0}\underline{\backslash }T_{\lambda}(A|B)=S(A|B)=A^{1/2}\log(A^{-1/2}BA^{-1/2})A^{1/2}$ .
(2)$T_{\lambda}(C\mathrm{Y}A|\alpha B)=c\mathrm{v}T_{\lambda}(A|B)$(3) If $B\leq C$ .

$the?\mathit{7}$, a$\in \mathbb{R}^{+}$.

$T_{\lambda}(A|B)\leq T_{\lambda}(A|C)$

.

.

(4)

$T_{\lambda}(A_{1}+A_{2}|B_{1}+B_{7,arrow}.,)\geq T_{\lambda}(A_{1}|B_{1})-^{1}\ulcorner T_{\lambda}(A_{\underline{)}}\ddagger|B_{2})$(5)$T_{\backslash },(\alpha A_{1}+\beta A_{\mathit{2}}|\alpha B_{1}+\beta B_{2})\geq\alpha T_{\lambda}(A_{1}|B_{1})+\beta T_{\lambda}(A_{2}|B_{2})$.

(6) $T_{l}\backslash (UAU’|UBU^{*})=UT_{\lambda}(A|B)U’$ .
(7) $\Phi(T_{\lambda}(A|B))\leq T_{\lambda}(\Phi(A)|\Phi(B))$ ,

have

$U$

is an unital positive linear map.

Remark 1 Same properties are shown for a more general case by Fujii et. .
$c\iota l$

Solodarity AsB $=A^{1/2}f(A^{-1/2}BA^{-1/\underline{7}})A^{1/2}$ for operator monotone f.

Since$\frac{x^{-\lambda}-1}{-\backslash },\leq\log x\leq\frac{x^{\lambda}-1}{\lambda}$

for $x>0$ , A $>0$ . we have the following.

Proposition 3 For A $>0$ , B $>0$ , A $\in(0,$1], toe have th e following;$T_{-\lambda}(A|B)\leq S(A|B)\leq T_{\lambda}(A|B)$

Sinc

.

$\mathrm{e}$$1- \frac{1}{x}\leq\ln_{\lambda}x\leq x-1$

for $x>0$ ,$0<\lambda$$\leq 1$

. we have the follo wing.

Proposition 4 For A $>0$ , B $>0$ ,$’\backslash \in(0,$1], we have the followving;

A $-AB^{-1}A\leq T_{\backslash (}’,A|B)\leq B-A$ .

Since$x_{\backslash }^{\lambda_{(}}1- \frac{1}{\alpha x})+\ln_{\lambda}\frac{1}{\alpha}\leq\ln_{\backslash },x\leq\frac{x}{\alpha}-1-x^{\lambda}111_{\lambda}\frac{1}{\alpha}$for$\alpha$$>0$ ,$\iota?$. $>0,0<$ A$\leq 1$,

we have the following.

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Theorem 2 For $A>0$ , $B>0$ ,$A\#\lambda B$

$wh$

are

$- \frac{1}{\alpha}A\#\lambda-lB$$A\#\lambda B$$+( \ln_{\lambda}\frac{1}{\alpha})A\leq T_{\lambda}(A|B)\leq\frac{1}{\alpha}B-A-(\ln_{\lambda}\frac{1}{\alpha})A\mathrm{Q}_{\lambda}B$taking$1\supset \mathrm{y}$A $>0,$ B $>0$ ,$\eta\sim For$

we have the following;$>0,$ $/\backslash \in(0,1]_{f}$$=A^{1/2}(A^{-1/2}BA^{-1/2})^{\lambda}A^{1/2}$

We have the following
Corollary$\alpha$.

$,\backslash arrow 0_{J}.\alpha=1$$\alpha$$>0$ ,,, respectively;

we have the following;$(1- \log\alpha)A-\frac{1}{\alpha}AB^{-1}A\leq S(A|B)\leq(\log\alpha -1)A+\frac{1}{\alpha}B$.

For $A>0$ , $B>0$ . we have the following;$A-AB^{-1}A\leq S(A|B)\leq B-A$ .

Lemma 5 For X $>0$ , Y $>0$ , a$\in \mathbb{R}_{7}$we have$(X\otimes \mathrm{I}’)^{a}=X^{o}\otimes$

Theorem 3 For$A_{1}$,$A_{2}$,$B_{1}$$Y^{rx}$.

, $B_{2}>0$ , A $\in(0, 1]f$ we have th e following;.

$T_{\backslash },(A_{1’}\cap.A_{2}|B_{1}\mathrm{c}\mathrm{r}_{\backslash }D_{2})=T_{\lambda}(A_{1}|B_{1})^{\gamma}\langle_{\Leftrightarrow\sim}A_{)}+A_{1}\otimes T_{\lambda}(A_{2}|B_{-},)\neq,\backslash T_{\lambda}(A_{1}|B_{1})\otimes T_{\backslash },(A_{\underline{)}}.|B)arrow)$

Proof. From$\mathrm{L}\mathrm{e}1211\mathfrak{n}$ina . we have for X $>0$ , Y $>0,$$\mathrm{D}\sim$$\backslash (X\otimes Y)$$=$($1\mathrm{n}\mathrm{A}$

$+$

A(

A)
$\ln_{\lambda}$$\otimes$$/\backslash \in(0,$1].

$I+I\otimes(\ln_{\lambda}Y)$A)$\otimes$$(\ln_{\lambda}Y)$.

and by multiplying, $Y=A^{-1/2},.B_{2}A_{\sim}^{-1/2}i$
By putting’. from both sides, we have the theorem.
$.1^{r}=A_{1}^{-1/7}.\lrcorner B_{1}A_{1}^{-1/2}$$A_{1}^{1/}\underline{\urcorner}|\log$$A_{\gamma}^{1/2}.$

Corollary 3 For$A_{13}$

A2,$B_{1}$.

$B_{2}>0$, we have$S(A_{1\mathcal{L}\backslash } (\mathrm{q}A_{2}|B_{1}\otimes B_{\underline{9}})=S(A_{1}|B_{1})\otimes A‘).+A_{1}\otimes$$S(A_{-},|B_{2})$

.

. .1.
$\langle$$\mathrm{c}1\cdot \mathrm{e}$103Since we put$B_{1}=B_{\mathrm{J}}..=I$,$A_{j}=\rho_{i}$.

have the following;$1\mathrm{k}’\mathrm{e}$Corollary 4 (pseudo additivity) For$p_{1}$, p.t, eve have

$S_{\lambda}(p_{1}\otimes\rho_{2})=S_{\lambda}(\rho_{1})+S_{\lambda}(\rho_{\underline{7}}.)+\lambda S_{\lambda}(p_{1})S_{\lambda}(\rho_{2})$.

Corollary 5 From Theorem 3 we have the following inequalities;(1) For A $\in(0, 1]$ and $0<A_{i}\leq B_{j}(\mathrm{i}=1,2)$ , we have(a).

$T_{\backslash },(A_{1/}\mathrm{F}_{-|B_{1}\otimes B_{\sim})}0A\underline{\mathrm{f})}|)\geq\lambda T_{\lambda}(A_{1}|B_{1})\otimes T_{\lambda}(A_{2}|B_{2})$(1) Tx (Ai(2) For A 6 (3$\otimes$$A_{2}|B_{1}\otimes$ $B_{2}$)$\geq T_{\lambda}(A_{1}|B_{1})$$\otimes$.

$A_{9}$$+A_{1}\otimes$$T_{\lambda}(A\circ|\sim B_{2})$and $0<B_{i}\leq A_{j}(\mathrm{i}=1,2)$ , we have

$0_{\mathrm{t}}1]$(c)$T_{r}\backslash (A_{1}rs A_{\underline{?}}|B_{1}\otimes B_{\underline{9}})\leq\lambda T_{\lambda}(A_{1}|B_{1})\otimes$(d)$T_{\lambda}(A_{1}\otimes A_{7,arrow},|B_{1}\otimes B_{2})\geq T\mathrm{T}\mathrm{x}$(Ai$|B_{1}$)$\otimes$$T_{\lambda}(A_{2}|B_{\underline{7}})$$A_{2}+A_{1}\otimes$.

$T_{\lambda}(A_{1}|B_{\sim}.))$.

Trace inequalities of Tsallis relative entropy

Definition 3 (Tsallis relative entropy) For density operators ,entropy is defined by$\rho$$7^{\cdot}efnl\uparrow.l\mathit{1}G$$D_{\lambda}( \rho|\sigma)=’\frac{T?[\rho-p^{1-\lambda}\sigma^{\lambda}|}{\lambda}$

Theorem 4

Proof.

$\mathrm{t}1^{\gamma}\mathrm{e}$$D_{\lambda}(\rho|\sigma)\leq-T\tau^{\backslash }[T_{\lambda}(p|\sigma)]$,$\lambda\in(0, 1]$.

.

remark that.
$A\#,\nu B$$=A^{1/2}(A^{-1/2}BA^{-1/2})^{\alpha}A^{1/_{\vee}^{\gamma}}$

is a power mean. By Theorem 3.4 in Hiai-Petz$\lfloor\lceil 3$] we have$\backslash$$Tr[e^{\Lambda}\#\alpha e^{\prime\}}]\leq T_{7^{\sim}}[e^{(1-\prime \mathrm{v})\Lambda+\mathrm{r}\mathrm{v}B}]$

for any$\mathit{0}^{\mathrm{J}}\in[0, 1]$.

$\backslash ,\backslash t_{8}$.

put $A=\log p_{7}B=\log\sigma$ .
$T?^{\mathrm{Y}}[\rho\#,,\sigma]\leq T\uparrow^{\backslash }[e^{\mathrm{l}\mathrm{o}^{1-0}+\log\sigma^{\alpha}}]\mathrm{g}/’$$\backslash h^{\gamma}\mathrm{e}$.

apply Golden-Thompson inequality$Tr’[e^{\Lambda+t\mathrm{I}}]\leq Tr^{\backslash }[e^{\Lambda}e^{\Pi}]$.

$\sigma$,

Tsallis104

for any Hermitian operators$A$, $B$ . Then we 1ave

$T?\cdot[e^{\mathrm{I}\mathrm{o}\mathrm{g}\rho^{1-\mathrm{t}\mathrm{Y}}+\log\sigma^{\alpha}}]\leq T?^{\mathrm{s}}[e^{\log\rho^{1-\alpha}}e^{\log\sigma^{\alpha}}]=T_{7^{\mathrm{s}}}[\rho^{1-\alpha}\sigma^{\mathrm{r}\mathrm{v}}]$Thus we have
$Tr[\rho^{1/2}(p^{-1/2}\sigma\rho^{-1/2})^{\cap}.\rho^{1/2}]\leq Tr_{\lfloor}^{\lceil}\rho^{1-(\gamma}\sigma^{\alpha}]$.

.
$\mathrm{q}.\mathrm{e}.\mathrm{d}$Corollary 6 (Hiai-Petz)$Tr[p(\mathrm{f}\mathrm{o}\mathrm{g} p 〕og \sigma)]$$\leq-Tr[\rho\log(\rho^{-1/\underline{\gamma}}\sigma p^{-1/2})]$

Definition 4 (Tsallis relative entropy) For positive operators, we define
A

$A$

,

$B$

.

.

rrnd $0<$$\leq 1$

$D_{\lambda}(A||B)=’ \frac{T?[A-A^{1-\lambda}B^{\lambda}]}{/\backslash }$.

Theorem 5 (Generalized Bogoliub ov inequality) For positive operators, and $0<$ A. we have the follow ing;

$A$

$B$

$\leq 1$.

$D_{\lambda}(A||B) \geq’\frac{T\uparrow[A]-(Tr[A])^{1-\lambda}(Tr^{\mathrm{r}}[B])^{\lambda}}{\lambda}$

Proof. It follows$1\supset \mathrm{y}$the application of the Holder’s inequality:$|Tr[XY]|\leq Tr[|X|5]^{1/s}Tr[|Y|^{t}]^{1/\ell}$for $1<s$ , $t<\infty$ , $1/s+1_{/}’t=1$ .

$\mathrm{q}.‘ 3.\mathrm{c}1$

Corollary 7 (Peierls-Bogoliubov inequality) For positive operators ,;have the foflon$A$$ei\prime P$$l^{1}mg$[ ($Tr$ $A$$\log$A-$\log$ $B)$

] $\geq Tr[A](\log Tr[A] -\log T?^{\backslash }[B])$ .

$B$

.
.

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Referen ces[1] S.Furuichi. K.Yanagi and .Kuriyama. Fundamental properties for Tsallis
relative entropy. J.Math.Phys, vo1.45, pp.4868-4877, 2004.
$\mathrm{I}[searrow]’$$\lfloor^{\underline{9}}\lceil]$

S.Furuichi. K.Yanagi and K.Kuriyama. A note on operator inequalities of
Tsallis relative operator entropy. Linear Algebra AppL. vo1.407, pp.19-31.

2005.
inequality is
F.Hiai and D.Petz, The Golden-Thompson
vo1.181,1993.
153-185.
pp.
mented. Linear Algebra AppL.
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$[3\lceil]$$[\lceil 4]$$\mathrm{c}\mathrm{o}\mathfrak{m}1^{3}1\mathrm{e}-$K.Yanagi. S.Furuichi and K.Kuriyama. Generalized Shannon inequalities
based on Tsallis relative operator entropy, Linear Algebra AppL. vo1.394.

pp.109-118. 2005