数理解析研究所講究録
1535 巻 2007 年 96-108

96

Some results on Tsallis entropies in quantum system*
Shigeru$\mathrm{F}\mathrm{u}\mathrm{r}\mathrm{u}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{i}^{\uparrow}$

Department of Electronics and Computer Science,
Tokyo University of Science, Onoda City, Yamaguchi, 756-0884, Japan

Abstract. In this survey, we review some theorems,
and applicatioo of Tsallis entropies
in classical system without proofs. See our previous papers[21,15,16,17,22,23,53] for the proofs and$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\alpha$details.

Keywords : Tsallis entropy, Tsallis relative entropy, uniqueness theorem, trace inequality, operator
inequality and entanglement

1

Tsallis entropies in quantum system

We denoteandrelations:and its inverse function
uniformly converge toand $\log x$ as A

$e_{\lambda}^{l}\equiv(1+\lambda x)^{*}$$\ln_{\lambda}x$$e_{\lambda}^{x}$, for A $\in(0,1]$ and $x>0$ . The functions, respectively. Note that wethe following$\ln_{\lambda}x\equiv a\underline{e}_{\overline{T}^{\underline{1}}}^{\lambda}$$e^{x}$$e_{\lambda}^{x+y+\lambda \mathrm{a}y}=e_{\lambda}^{x}e_{\lambda}^{y}$,$arrow 0$$-\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}$.

(1)$\ln_{\lambda}xy=\ln_{\lambda}x+\ln_{\lambda}y+\lambda\ln_{\lambda}x\ln_{\lambda}y$In this survey, we consider the linear bounded operator acting on the Hilbert space , or simply
complex matrix whose set is denoted byin the flnite quantum systern. A Herrnitian matrix
called a nonnegative matrix (and denoted by $A\geq 0$ ) if $(x, Ax)\geq 0$ for all“. A nonnegative
matrix is called a positive matrix (and denoted by $A>0$ ) if it is invertible. The set of all density
matrices (quantum states) is represented by $D_{n}\equiv\{X\in M_{n}(\mathrm{C}):X\geq 0,Tr[X]=1\}$ .
and A $\in(-1,0)\cup(0,1)$ , we denote the generalized exponential function by
the inverse function of, for $X\geq 0$ and $\lambda\in(-1,0)\cup(0,1)$ , we denote
generalized logarithmic
function by
Then the Tsallis relative entropy and Tsallis entropy for nonnegative matrices$X$ andare defined by$\mathcal{H}$$n\mathrm{x}n$$M_{n}(\mathrm{C})$$A$$\mathrm{i}\epsilon$$x\in \mathrm{C}$$A$$\mathrm{F}\mathrm{o}\mathrm{r}-I\leq X<I$$\exp(X)\equiv(I+\lambda X)^{1/\lambda^{-}}.\mathrm{A}\mathrm{s}$$\mathrm{t}\mathrm{h}\mathrm{e}\lambda$$\exp_{\lambda}(\cdot)$$\ln_{\lambda}X\equiv\underline{x}_{7^{\underline{-I}}}^{\lambda}$.

$\mathrm{Y}$$D_{\lambda}(X|\mathrm{Y})\equiv Tr[X^{1-\lambda}(\ln_{\lambda}X-\ln_{\lambda}\mathrm{Y})]$,$S_{\lambda}(X)\equiv-D_{\lambda}(X|I)$.

The entropies are generalizations of the von Neumann entropy [36] and the Umegaki relative entropy[50] in the sense that$\mathrm{e}$$\lim_{\lambdaarrow 0}S_{\lambda}(X)=S_{0}(X)\equiv-Tr[X\log X]$

and$\lim_{\lambdaarrow 0}D_{\lambda}(X|\mathrm{Y})=D_{0}(X|\mathrm{Y})$ii $Tr[X(\log X-\log \mathrm{Y})]$.

See [37] and [1, 2, 21, 22] for details of the theory of quantum entropy and the Tsallis entropies,, respectively. Two Tsallis entropiae have non-additivitiae such that$\mathrm{r}\mathrm{e}\mathrm{s}$$S_{\lambda}(X_{1}\otimes X_{2})=S_{\lambda}(X_{1})+S_{\lambda}(X_{2})+\lambda S_{\lambda}(X_{1})S_{\lambda}(X_{2})$

,

(2)

and

,
(3)
due to the non-additivity Eq.(l) of the function. Thus the field of the study using these entropies isoften called the non-additive statistical physics and many research papers have been published in mainly$D_{\lambda}(X_{1}\otimes X_{2}|\mathrm{Y}_{1}\otimes \mathrm{Y}_{2})=D_{\lambda}(X_{1}|\mathrm{Y}_{1})+D_{\lambda}(X_{2}|\mathrm{Y}_{2})-\lambda D_{\lambda}(X_{1}|\mathrm{Y}_{1})D_{\lambda}(X_{2}|\mathrm{Y}_{2})$$\ln_{\lambda}$’Thi work$\overline{\mathrm{w}\mathrm{T}\mathrm{h}\mathrm{k}\mathrm{o}\mathrm{r}\mathrm{k}}$for$\mathrm{w}\mathrm{a}*\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{d}$$\mathrm{E}\mathrm{n}\infty \mathrm{u}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$of Youngby$\epsilon \mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\dot{\mathrm{u}}\mathrm{t}\epsilon(\mathrm{B})$$\uparrow \mathrm{E}\cdot \mathrm{m}\mathrm{a}\mathrm{i}\mathrm{l}:\mathrm{f}\mathrm{u}\mathrm{r}\mathrm{u}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{i}\copyright \mathrm{e}\mathrm{d}.\mathrm{y}\mathrm{a}\mathrm{m}\mathrm{a}.\mathrm{t}\mathrm{u}\iota.\epsilon\epsilon.\mathrm{j}\mathrm{p}$$\mathrm{t}\mathrm{h}\mathrm{o}$Japanoee Ministry of Education, Science, Sports and Culture, Grant-in-Aid, 17740068.

97
statistical physics [49]. See our previous papers $[21, 22]$ on the mathematical properties of these entropies.
See also chapter II written by A.K.Rajagopal in [49], for the quantum version of Tsallis entropies andtheir applications.

1.1

A uniqueness theorem Tsallis entropy in quantum system

As quite similar to the method in [14], we axiomatically characterize the Tsallis entropy in a finite
quantum system. Let
be a mapping the setof all density operators to the setof all positive
real values.
$T_{\lambda}$$6(\mathcal{H})$$\mathbb{R}^{+}$

Axiom 1.1 ([23]) We give the postulates which the Tsallis entropy should(T1) Continuity: For$\rho\in 6(\mathcal{H}),$$T_{\lambda}(\rho)$

is a continuous function on the eigenvalues(T2) Invariance: For unitary transformation $U,$

(T3) Generalized $m|x|ng$ condition: For$6(\mathcal{H})$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{h}^{r}$.
$x_{\mathrm{j}}$of$\rho$.

.

$T_{\lambda}(U^{\mathrm{s}}\phi U)=T_{\lambda}(\phi)$$\rho=\bigoplus_{k=1}’‘\lambda_{k\beta k}$on$\mathcal{H}=\bigoplus_{k=1}^{\hslash}\mathcal{H}_{k}$, where$x_{k}\geq 0,$ $\sum_{k=1}^{n}x_{k}=1,\rho_{k}\in$, we have the additivity:,$T_{\lambda}( \rho)=\sum_{k=1}^{n}x_{k}^{1-\lambda}T_{\lambda}(\rho_{k})+T_{\lambda}(x\iota, \cdots,x_{n})$where$(x_{1}, \cdots,x_{n})$Theorem 1.2 ([23]) If

represents the diagonal matrix$T_{\lambda}$

satisfies Axiom 1.1, then$T_{\lambda}$$(x_{k}\delta_{kj})_{k,\mathrm{j}=1,\cdots,n}$

is uniquely given by the following form

$T_{\lambda}(\rho)=\mu_{\lambda}S_{\lambda}(\rho)$with a positive constant number$\mu_{\lambda}$.

,

depending on parameter$\lambda$.

1.2

Properties of Tsallis relative entropy
Here we review some information-theoretical properties of the Tsallis relative entropy deflned in quantum
system. For the quantum Tsallis relative entropy
and the quantum relative entropy, thefollowing relations are known.
$D_{\lambda}(\rho|\sigma)$$U(\rho|\sigma)$

Proposition 1.3 (Ruskai-Stillinger [43] (see also [37])) For density operators(1)$D_{1-\lambda}(\rho|\sigma)\leq U(\rho|\sigma)\leq D_{1+\lambda}(\rho|\sigma)$

for

(2)

$D_{1+\lambda}(\rho|\sigma)\leq U(\rho|\sigma)\leq D_{1-\lambda}(\rho|\sigma)$

for-l$0<\lambda\leq 1$$\rho$

and a, we have,.

$\leq\lambda<0$.

Note that the both sides in the both inequalities converge to.
asWe next consider another relation on the quantum Tsallis relative entropy. In [12], the relative
operator entropy was defined by$\lambdaarrow 0$$U(\rho|\sigma)$$S(A|B)\equiv A^{1/2}\log(A^{-1/2}BA^{-1/2})A^{1/2}$,

for two strictly positive operators and . If and are commutative, then we have $U(A|B)=$$-Tr[S(A|B)]$ . For this relative operator entropy and the quantum relative entropy $U(A|B)$ , Hiai andPetz proved the following relation:
$A$

$B$

$A$

$B$

$U(A|B)\leq-Tr[S(A|B)]$ ,(4)

in [28] (see also [29]).
In our previous papers [53], we introduced the Tsallis relative operator entropy
metric extension of the relative operator entropy $S(A|B)$ such as,$T_{\lambda}(A|B) \equiv\frac{A^{1/2}(A^{-1/2}BA^{-1/2})^{\lambda}A^{1/2}-A}{\lambda}$$T_{\lambda}(A|B)$

as a para-

98
for $0<\lambda\leq 1$ and strictly positive operators

$A$

and

$B$

, in the sense that$\lim_{\lambdaarrow 0}T_{\lambda}(A|B)=S(A|B)$.

Also we now have that$\lim_{\lambdaarrow 0}D_{\lambda}(A|B)=U(A|B)$

(5)

.

(6)

These relations Eq.(4), Eq.(5) and Eq.(6) naturally lead us to show the following theorem as a parametricextension of Eq.(4).
Theorem 1.4 ([21]) For $0<\lambda\leq 1$ and any strictly positive operators

$A$

and

$B$

, we have(7)$D_{\lambda}(A|B)\leq-Tr[T_{\lambda}(A|B)]$

Thus the result proved by Hiai and Petz in $[28, 29]$ is recovered as a special case of Theorem 1.4 inthe limit. On the quantum Tsallis relative entropy
for density operators and a, we havethe following properties.
$\lambdaarrow 0$$D_{\lambda}(\rho|\sigma)$$\rho$

Proposition 1.5 ([21, 1, 40]) For $0<\lambda\leq 1$ and any density operators
entropy
has the following properties.

$\rho$

and , the quantum relative$\sigma$$D_{\lambda}(\rho|\sigma)$(1) (Non-negativity)(2) (Non-additivity)$D_{\lambda}(\rho|\sigma)\geq 0$.

$D_{\lambda}(\rho_{1}\emptyset\rho_{2}|\sigma_{1}\otimes\sigma_{2})=D_{\lambda}(\rho_{1}|\sigma_{1})+D_{\lambda}(\rho_{2}|\sigma_{2})-\lambda D_{\lambda}(\rho_{1}|\sigma_{1})D_{\lambda}(\rho_{2}|\sigma_{2})$(3) (Joint convexity)$D_{\lambda}( \sum_{\mathrm{j}}\lambda_{j\rho_{j}}|\sum_{j}\lambda_{j}\sigma_{j})\leq\sum_{j}\lambda_{j}D_{\lambda}(\rho_{j}|\sigma_{f})$.

.

(4) (Unitary invariance) The quantum Tsallis relative entropy is invariant under the unitary transformation :$U$$D_{\lambda}(U\rho U^{*}|U\sigma U^{\cdot})=D_{\lambda}(\rho|\sigma)$.

(5) (Monotonicity) For any trace-preserving completely positive linear map
and and $0<\lambda\leq 1$ , we have$\Phi$, any density operators$\rho$$\sigma$$D_{\lambda}(\Phi(\rho)|\Phi(\sigma))\leq D_{\lambda}(\rho|\sigma)$

Putting$\sigma=\frac{1}{n}I$.

(8)in Eq.(8), we have the following corollary.

Corollary 1.6 For any
and $0<\lambda\leq 1$ , we have$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\triangleright \mathrm{p}\mathrm{r}\mathrm{e}\epsilon \mathrm{e}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}$completely positive linear unital map$\Phi$, any density operator$\rho$$S_{\lambda}(\Phi(\rho))\geq S_{\lambda}(\rho)$.

We note that Eq.(8) for the fixed , namely the monotonicity of the quantum Tsallis relative entropy, was proved in [4] to establish Clausius’ inequality.
in the case of$\sigma$$\Phi(\sigma)=\sigma$

Remark 1.7 It is known [33] (see also $[u]$ ) that there is an equivalent relation between the monotonicity
for the quantum relative entropy and the strong subadditivity for the quantum entropy. However in ourcase, we have not yet found such a relation. Because the non-additivity of -logarithm function$\lambda$$\ln_{\lambda}xy=\ln_{\lambda}x+\ln_{\lambda}y+\lambda\ln_{\lambda}x\ln_{\lambda}y$disturbs us to derive the beautiful relation such as$D_{\lambda}(p(x,y)|p(x)p(y))=S_{\lambda}(p(x))+S_{\lambda}(p(y))-S_{\lambda}[p(x,y))$for the Tsallis relative entropy, the Tsallis entropy
joint entropy, even if our stage is in classical system.
$D_{\lambda}(p(x,\mathrm{y})|p(x)p(y))$$S_{\lambda}(p(x,y))$$S_{\lambda}(p(x)),$ $S_{\lambda}(p(y))$

and the Tsallis

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1.3

Maximum entropy principle in Tsallis statistics

Here, we review the derivation of the maximum entropy principle for a density matrix in Tsallis statistics.

Theorem 1.8 ([16]) Let $Y=Z_{\lambda}^{-1}\exp_{\lambda}(-H/||H||)$ , where $Z_{\lambda}\equiv Tr[\exp_{\lambda}(-H/||H||)]$ , for $\lambda\in(-1,0)\cup$$(0,1)$ and a Hermitian matrix $H$ . We denote$C_{\lambda}\equiv\{X\in D_{n’}.

If$X\in C_{\lambda}$, then$S_{\lambda}(X)\leq S_{\lambda}(\mathrm{Y})$Tr[X^{1-\lambda}H]\leq Tr[\mathrm{Y}^{1-\lambda}H]\}$.

.

Remark 1.9
is a strictly concave function,
is a strictly concave function on the set. This means that the maximizer is uniquely determined so that we may regard as a generalized
Gibbs state. Thus we may deflne a generalized Helmholtz free energy such by$\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{e}-x^{1-\lambda}\ln_{\lambda}x$$S_{\lambda}$$\mathrm{Y}$$C_{\lambda}$$\mathrm{Y}$.

$F_{\lambda}(X, H)\equiv Tr[X^{1-\lambda}H]-||H||S_{\lambda}(X)$This can be also represented by the Tsallis relative entropy such as.

$F_{\lambda}(X, H)=||H||D_{\lambda}(X|\mathrm{Y})+\ln_{\lambda}Z_{\lambda}^{-1}Tr[X^{1-\lambda}(||H||-\lambda H)]$

We straightforwardly have the following corollary by taking the limit as$\lambdaarrow 0$.

Corollary 1.10 ([46]) Let $\mathrm{Y}=Z_{0}^{-1}\exp(-H/||H||)$ , where $Z_{0}\equiv Tr[\exp(-H/||H||)]$ , for a Hermitianmatrix $H$ . If $X\in C_{0}$ , then$S_{0}(X)\leq S_{0}(\mathrm{Y})$2.

Trace inequalities related to Tsallis entropies

2.1

A variational expression for Tsallis relative entropy
In this subsection, we review of the derivation of a variational expression for the Tsallis relative entropy
as a parametric extension of that of the relative entropy in Lemma 1.2 of [29]. A variational expressionof the relative entropy has been studied in the general setting of von Neumann algebras $[41, 32]$ .

In the below, we sometimes relax the condition of the unital trace for the matrices in the deflnitionof the Tsallis relative entropy, since it is not essential in the mathematical studies of the entropic
functionals.
$D_{\lambda}(\cdot|\cdot)$

Theorem 2.1(1) If$A$

and$\langle$$\mathrm{Y}$$[15])$ For $\lambda\in(0,1]$, we have the following relations.

are nonnegative matrices, then$\ln_{\lambda}Tr[e_{\lambda}^{A+\ln_{\lambda}Y}]=\max\{Tr[X^{1-\lambda}A]-D_{\lambda}(X|\mathrm{Y}):X\geq 0,Tr[X]=1\}$.

(2) If $X$ is a positive matrix with $Tr[X]=1$ and $B$ is a Hermitian matrix, then$D_{\lambda}(X|e_{\lambda}^{B})= \max\{Tr[X^{1-\lambda}A]-\ln_{\lambda}Tr[e_{\lambda}^{4+B}] :A\geq 0\}$.

Taking the limit as, Theorem 2.1 recovers the similar form of Lemma 1.2 in [29] under theassumption of non-negativity of . If
and $B=0$ in (1) and (2) of Theorem 2.1, respectively, thenwe obtain the following corollary.
$\lambdaarrow 0$$A$Corollary 2.2(1) If$A$$\mathrm{Y}=I$

is a nonnegative matrix, then$\ln_{\lambda}Tr[e_{\lambda}^{A}]=\mathrm{m}\alpha\{Tr[X^{1-\lambda}A]+S_{\lambda}(X):X\geq 0, Tr[X]=1\}$.

(2) For a density matrix $X$ , we have$-S_{\lambda}(X)= \max\{Tr[X^{1-\lambda}A]-\ln_{\lambda}Tr[e_{\lambda}^{4}] :Taking the limit asof non-negativity of$\lambdaarrow 0$$A$.

A\geq 0\}$.

, Corollary 2.2 recovers the similar form of Theorem 1 in [6] under the assumption

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2.2

Generalized logarithmic and exponential trace inequalities

In this subsection, we derive some trace inequalities in terms of the results obtained in the previous
section. That is, the results of this subsection are derived by the application of a variational expression
for the Tsallis relative entropy. From (1) of Corollary 2.2, we have the generalized thermodynamic
inequality:,(9)
for a density matrix $D$ and a nonnegative matrix $H$ . Putting $D=\tau,A$ and $H=\ln_{\lambda}B$ in Eq.(9) for$A\geq 0$ and $B\geq I$ ,we have the generalized Peierls-Bogoliubov inequality (cf.Theorem 3.3 of [21]):$\ln_{\lambda}Tr[e_{\lambda}^{H}]\geq Tr[D^{1-\lambda}H]+S_{\lambda}(D)$$(Tr[A])^{1-\lambda}(\ln_{\lambda}Tr[A]-\ln_{\lambda}Tr[B])\leq Tr[A^{1-\lambda}(\ln_{\lambda}A-\ln_{\lambda}B)]$

for nonnegative matrices

$A$

,

(10)

and $B\geq I$ .

Lemma 2.3 The following statements are equivalent.

is convex in a Hermitian matrix(1)$F_{\lambda}(A)=\ln_{\lambda}Tr[e_{\lambda}^{A}]$(2)$f_{\lambda}(t)=\mathrm{i}_{\lambda}Tr[e_{\lambda}^{A+tB}]$is convex in.

R.

$t\in$

Corollary 2.4 For nonnegative matrices

$A$

$A$

and

$B$

, we have$\ln_{\lambda}Tr[e_{\lambda}^{A+B}]-\ln_{\lambda}Tr[e_{\lambda}^{A}]\geq\frac{Tr[B(e_{\lambda}^{A})^{1-\lambda}]}{(Tr[e_{\lambda}^{A}])^{1-\lambda}}$For nonnegative real numbers
and $0<\lambda\leq 1$ , the relations
relations naturally motivate u\S to consider the following inequalities in$x,$ $y$.

(11)$e_{\lambda}^{x+y}<e^{x+\mathrm{y}+\lambda xy}=eaee^{\mathrm{y}}$Proposition 2.5 ([15]) For non-negative matrices $X$ and$\mathrm{Y}$, and$\mathrm{t}^{-\lambda\lambda\lambda}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$$0<\lambda\leq 1$$Tr[e_{\lambda}^{X+Y}]\leq Tr[e_{\lambda}^{X+Y+\lambda Y^{1/\mathrm{a}_{XY^{1/2}}}}]$

hold. Thesecase.

, we have.

Note that we have the matrix inequality:$e_{\lambda}^{X+\mathrm{Y}}\leq e_{\lambda}^{X+\mathrm{Y}+\lambda Y^{1/2}X\mathrm{Y}^{1/2}}$

for$\lambda\geq 1$

by the application of the L\"owner-Heinz inequality [34, 27, 38].

Proposition 2.6 ([15]) For nonnegative matrices $X,$ , and A $\in(0,1]$ , we have$\mathrm{Y}$$Tr[e_{\lambda}^{X+\mathrm{Y}+\lambda X\mathrm{Y}}]\leq Tr[e_{\lambda}^{X}e_{\lambda}^{Y}]$

.

(12)

Notice that Golden-Thompson inequality $[26, 47]$ ,$Tr[e^{X+\mathrm{Y}}]\leq Tr[e^{X}e^{\mathrm{Y}}]$

which holds for Hermitian matrices $X$ and , is recovered by taking the limit asin Proposition
2.6, in particular case of nonnegative matrices $X$ and Y.
Since $Tr[HZHZ]\leq Tr[H^{2}Z^{2}]$ for Hermitian matrices $H$ and $Z[41,10]$ , we have for nonnegative
matrices $X$ and ,$\mathrm{Y}$$\lambdaarrow 0$$Y$$Tr[(I+X+\mathrm{Y}+\mathrm{Y}^{1/2}X\mathrm{Y}^{1/2})^{2}]\leq Tr[(I+X+Y+X\mathrm{Y})^{2}]$by easy calculations. This implies the inequality$Tr[e_{1/2}^{X+Y+1/2\mathrm{Y}^{1/2}XY^{1/\iota}}]\leq Tr[e_{1/2}^{X+Y+1/2X\mathrm{Y}}]$.

Thus we have$Tr[e_{1/2}^{X+Y}]\leq Tr[e_{1/2}^{X}e_{1/2}^{Y}]$

(13)

101
ffom Proposition 2.5 and Proposition 2.6. Putting $B=\ln_{1/2}Y$ and $A=\ln_{1/2}\mathrm{Y}^{-1/2}XY^{-1/2}$ in (2) of
Theorem 2.1 under the assumption of $I\leq Y\leq X$ and using Eq.(13), we have$D_{1/2}(X|Y)$$=$$D_{1/2}(X|e_{\mathrm{l}/2}^{\ln_{1/2}Y})$2$Tr[x^{1/2}A]-\ln_{1/2}Tr[e_{1/2}^{A+B}]$$\geq$$Tr[\mathrm{x}^{1/2}A]-\ln_{1/2}Tr[e_{1/2}^{A}e_{1/2}^{B}]$$=$$Tr[X^{1/2}\ln_{1/2}Y^{-1/2}XY^{-1/2}]-\ln_{1/2}Tr[Y^{-1/2}X\mathrm{Y}^{-1/2}\mathrm{Y}]$$=$$Tr[X^{1/2}\ln_{1/2}Y^{-1/2}X\mathrm{Y}^{-1/2}]$,

which gives a lower bound of the Tsallis relative entropy in the case of $\lambda=1/2$ and(14)$I\leq \mathrm{Y}\leq X$.

2.3 Hiai-Petz type trace inequalities
In this subsection, we consider an extension of the following inequality:(15)$Tr[X( \log X+\log \mathrm{Y})]\leq\frac{1}{p}Tr[X\log X^{\mathrm{p}/2}\mathrm{Y}^{p}X^{p/2}]$

for nonnegative matrices $X$ and$\mathrm{Y}$, and $p>0$ .

Theorem 2.7 ([16])(1) For positive matrices $X$ and $Y,$ $p\geq 1$ and $0<\lambda\leq 1$ , we have
$Tr[X^{1-\lambda}(\ln_{\lambda}X-\ln_{\lambda}\mathrm{Y})]\leq-Tr[X\ln_{\lambda}(X^{-\mathrm{p}/2}Y^{\mathrm{p}}X^{-p/2})^{1/\mathrm{p}}]$

.

(16)

(2) For positive matrices $X$ and $Y,$ $0<p<1$ and $0<\lambda\leq 1$ , the following inequality dose not hold:$Tr[X^{1-\lambda}(\ln_{\lambda}X-\ln_{\lambda}\mathrm{Y})]\leq-Tr[X\ln_{\lambda}(X^{-p/2}\mathrm{Y}^{\mathrm{p}}X^{-\mathrm{p}/2})^{1/\mathrm{p}}]$Corollary 2.8(1) For positive matrices $X$ and$\mathrm{Y}$(17), the trace inequality$D_{\lambda}(X|\mathrm{Y})\leq-Tr[X\ln_{\lambda}(X^{-1/2}YX^{-1/2})]$(18)

holds.
(2) For positive matrices $X$ and2.4$\mathrm{Y}$, and $p\geq 1$ , we have the inequality (15).

A generalized Fannes’s inequality

We give an upper bound of the Tsallis entropy. To do so, we state a few lemmas.

Lemma 2.9 For a density operator

on , we have$\mathcal{H}$$\rho$$S_{q}(\rho)\leq\ln_{q}d$,where $d=\dim \mathcal{H}<\infty$ .

Lemma 2.10 If$f$

is a concave function and $f(\mathrm{O})=f(1)=0$ , then we have$|f(t+s)-f(t)| \leq\max\{f(\epsilon), f(1-s)\}$for any $s\in[0,1/2]$ and $t\in[0,1]$ satisfying $0\leq s+t\leq 1$ .

Lemma 2.11 For any real number $u,v\in[0,1]$ and $q\in[0,2]$ , if $|u-v|\leq f1$ , then $|\eta_{q}(u)-\eta_{q}(v)|\leq$$\eta_{q}(|u-v|)$ , where.
$\eta_{q}(x)\equiv\frac{x^{q}-x}{1-q}$

102Theorem 2.12 ([23]) For two density operators$\rho_{1}$

then

and

on

$\rho_{2}$$\mathcal{H}$and $q\in[0,2]$ , if$||A||_{1}\equiv \mathrm{R}[(A^{*}A)^{1/2}]$

for a bounded linear operator

$A$

,

.

$|S_{q}(\rho_{1})-S_{q}(\rho_{2})|\leq||\rho_{1}-\rho_{2}||_{1}^{q}\ln_{q}d+\eta_{q}(||\rho_{1}-\rho_{2}||_{1})$Where we denote$||\rho_{1}-\rho_{2}||_{1}\leq q^{1/(1-q)}$.

By taking the limit as $qarrow 1$ , we have the following Fannes’s inequality (see pp.512 of [35], also
[8, 5, 37] as a corollary, since.
$)$$\lim_{qarrow 1q^{1/(1-q)}}=\frac{1}{\mathrm{e}}$Corollary 2.13 For two density operators$\rho_{1}$and$\rho_{2}$on$\mathcal{H}$, if$|| \rho_{1}-\rho_{2}||_{1}\leq\frac{1}{\mathrm{e}}$$|S_{1}(\rho_{1})-S_{1}(\rho_{2})|\leq||\rho_{1}-\rho_{2}||_{1}\ln d+\eta_{1}(||\rho_{1}-\rho_{2}||_{1})$where$S_{1}$

represents von Neumann entropy and $\eta_{1}(x)=-x\ln x$, then,.

3

Operator inequalities related to the Tsallis entropies

3.1

Properties of Tsallis relative operator entropy

In this subsection, we give several fundamental properties of Tsallis relative operator entropy

Section 1:

defined in.

$T_{\lambda}(A|B)=A^{1/2}\ln_{\lambda}(A^{-1/2}BA^{-1/2})A^{1/2}$

for two positive operators andand $\lambda\in(0,1]$ . along the line of the paper [12]. Note that moregeneral operator has been introduced in [13], it was named solidarity and then several properties
wereshown in [13, 11, 9].
$A$

$B$

Proposition 3.1 ([11, 9, 13])(1) (Homogeneity)$T_{\lambda}(\alpha A|\alpha B)=\alpha T_{\lambda}(A|B)$(2) (Monotonicity) If $B\leq C$ , thenfor any positive number a.

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

(3) (Superadditivity) $T_{\lambda}(A_{1}+A_{2}|B_{1}+B_{2})\geq T_{\lambda}(A_{1}|B_{1})+T_{\lambda}(A_{2}|B_{2})$ .
(4) (Joint concavity)$T_{\lambda}(\alpha A_{1}+\beta A_{2}|\alpha B_{1}+\beta B_{2})\geq\alpha T_{\lambda}(A_{1}|B_{1})+\beta T_{\lambda}(A_{2}|B_{2})$.

(5) (Unitary invariance)$T_{\lambda}(UAU^{\cdot}|UBU^{*})=UT_{\lambda}(A|B)U^{*}$

for any unitary operator$U$.

(6) (Monotonicity) For a unital positive linear map
Hilbert space to itself, we have$\Phi$

from the set of the bounded linear operators on.

$\Phi(T_{\lambda}(A|B))\leq T_{\lambda}(\Phi(A)|\Phi(B))$(7) (Bounds) For any invertible positive operator$T_{-\lambda}(A|B)\leq S(A|B)\leq T_{\lambda}(A|B)$$A$,

and $B,$$0<\lambda\leq 1$, we have$T_{\lambda}(A|B)\leq H_{\lambda}(A)+A^{1-\lambda}\ln_{\lambda}||B||$and$\mu A\leq B\Rightarrow T_{\lambda}(A|B)\geq(\ln_{\lambda}\mu)A$,.

Moreover we have the following bounds for Tsallis relative operator entropy.
Lemma 3.2 For any positive real number$x$and $0<\lambda\leq 1$ , the following inequalities hold

$1-\underline{1}\leq\ln_{\lambda}x\leq x-1$
$x$

.

103
Proposition 3.3 ([13]) For any invertible positive operator$A$and $B,$ $0<\lambda\leq 1$ ,$A-AB^{-1}A\leq T_{\lambda}(A|B)\leq-A+B$

.

(19)

Moreover, $T_{\lambda}(A|B)=0$ if and only if $A=B$ .
Finally we prove the further bounds of Tsallis relative operator entropy with the -power mean$\lambda$

Lemma 3.4 For any positive real number$\alpha$and$x$$\#\lambda$.

and $0<\lambda\leq 1$ , the inequalities hold$x^{\lambda}(1- \frac{1}{\alpha x})+\ln_{\lambda}\frac{1}{\alpha}\leq\ln_{\lambda}x\leq\frac{x}{\alpha}-1-x^{\lambda}\ln_{\lambda}\frac{1}{\alpha}$

.

(20)

The equality of the right hand side of the above inequalities hold if and only if $x=\alpha$ . The equality of
the left hand side of the above inequalities hold if and only if.
$x= \frac{1}{a}$Theorem 3.5 ([22]) For any invertible positive operators
the following inequality holds

$A$

and

$B$

, and any positive real number ,$\alpha$$A \#\lambda B-\frac{1}{\alpha}A\#\lambda-1B+(\ln_{\lambda}\frac{1}{\alpha})A\leq T_{\lambda}(A|B)\leq\frac{1}{\alpha}B-A-(\ln_{\lambda}\frac{1}{\alpha})A\#\lambda B$

.

(21)

The equality of the right hand side of the above inequalities holds if and only if $B=\alpha A$ . The equalityof the left hand side of the above inequalities holds if and only if $A=\alpha B$ . We have that $T_{\lambda}(A|B)=0$ isequivalent to $A=B$ .

Remark 3.6 We note that Eq.(21) recovers the inequalities shown in [24] :$(1- \log\alpha)A-\frac{1}{\alpha}AB^{-1}A\leq S(A|B)\leq(\log\alpha-1)A+\frac{1}{\alpha}B$as$\lambdaarrow 0$. Moreover, if we put$\alpha=1$, then we have

$A-AB^{-1}A\leq S(A|B)\leq B-A$

which recover the inequalities of Corollary 5 in [13], cf. Eq.(19).
Taking account for the non-additivity (Eq.(2) and) which are the typical features of Tsallis
entropies, we consider the Tsallis relative operator entropy of two positive operator of the tensor productand. To show our theorem, we state the following lemma for the convenience of the$\mathrm{E}\mathrm{q}.(3)$$A_{1}\otimes A_{2}$$B_{1}\otimes B_{2}$readers.

$\mathrm{T}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}3.7([22])$

For any $0<\lambda\leq 1$ and any strictly positive operators$A_{1},$ $A_{2},$ $B_{1}$and$B_{2}$, we have$T_{\lambda}(A_{1}\otimes A_{2}|B_{1}\otimes B_{2})=T_{\lambda}(A_{1}|B_{1})\otimes A_{2}+A_{1}\emptyset T_{\lambda}(A_{2}|B_{2})+\lambda T_{\lambda}(A_{1}|B_{1})\otimes T_{\lambda}(A_{2}|B_{2})$Taking the limit as$\lambdaarrow 0$

.

(22)

in Theorem 3.7, we have the following corollary.

Corollary 3.8 For any strictly positive operators$A_{1},$$A_{2},B_{1}$and$B_{2}$, we have$S(A_{1}\otimes A_{2}|B_{1}\otimes B_{2})=S(A_{1}|B_{1})\otimes A_{2}+A_{1}\otimes S(A_{2}|B_{2})$.

Since we havefor any $B\geq A,$
for any $B\leq A$ and we have
any $X\geq 0$ and $Y\geq 0$ , Theorem 3.7 implies the following corollary.
$T_{\lambda}(A|B)\geq 0$$T_{\lambda}(A|B)\leq 0$$X\otimes \mathrm{Y}\geq 0$

forCorollary 3.9(1) Hbr any $0<\lambda\leq 1$ and $0<A_{i}\leq B_{:},$ $(i=1,2)$ , we have the following inequalities.

(a)
(b)

.

$T_{\lambda}(A_{1} @A_{2}|B_{1} @B_{2})\geq\lambda T_{\lambda}(A_{1}|B_{1})\otimes T_{\lambda}(A_{2}|B_{2})$$T_{\lambda}(A_{1}\otimes A_{2}|B_{1}\otimes B_{2})\geq T_{\lambda}(A_{1}|B_{1})QA_{2}+A_{1}\otimes T_{\lambda}(A_{2}|B_{2})$.

(2) For any $0<\lambda\leq 1$ and $0<B_{:}\leq A_{i},$ $(i=1,2)$ , we have the following inequalities.
(c)(d)$T_{\lambda}(A_{1}\otimes A_{2}|B_{1} @B_{2})\leq\lambda T_{\lambda}(A_{1}|B_{1})\otimes T_{\lambda}(A_{2}|B_{2})$.

$T_{\lambda}(A_{1}\otimes A_{2}|B_{1}\otimes B_{2})\geq T_{\lambda}(A_{1}|B_{1})\otimes A_{2}+A_{1}QT_{\lambda}(A_{2}|B_{2})$

.

104
3.2

Generalized Shannon inequalities based on Tsallis relative operator en-

tropy

The fundamental properties ofare shown in the previous subsection. In this subsection we give
the Shannon type operator inequality and its reverse one satisfied by Tsallis relative operator entropy.
To do so, we need the following lemma.
$T_{\lambda}(A|B)$

Theorem 3.10 ([53]) Let$\{A_{1}, A_{2}, \ldots, A_{n}\}$operators on a Hilbert space $H$ . If

nand$\{B_{1}, B_{2}, \ldots, B_{n}\}$be two sequences of strictly positive, then$\sum_{j=1}^{\mathrm{n}}A_{j}=\sum_{j=1}B_{j}=I$.

$0 \geq\sum_{j=1}^{n}T_{\lambda}(A_{\mathrm{j}}|B_{j})\geq\frac{(\sum_{j=1}^{n}A_{j}B_{j}^{-1}A_{j})^{-\lambda}-I}{\lambda}$

We also obtain the operator version of the Shannon inequality and reverse one given by FUruta [25]as a corollary of Theorem 3.10 in the following.
Corollary 3.11 ([25]) Let$\{A_{1}, A_{2}, \ldots, A_{n}\}$.

operators on a Hilbert space $H$ If

and$\{B_{1}, B_{2}, \ldots, B_{n}\}$n$\sum_{j=1}^{n}A_{\dot{f}}=\sum_{j=1}B_{j}=I$

be two sequences of strictly positive, then$0 \geq\sum_{\mathrm{j}=1}" A_{j}^{1/2}(\log A_{j}^{-1/2}B_{j}A_{j}^{-1/2})A_{j}^{1/2}\geq-\log(\sum_{j=1}^{n}A_{j}B_{j}^{-1}A_{f})$.

Actually the above Corollary 3.11 is a part of the Corollary 2.4 in [25].

4Applications of Tsallis entropies

Finally we review the application of the Tsallis relative entropy as a measure of entanglements. The
concept of entanglement has been important in quantum information theory, especially quantum teleportation and quantum computing and so on. Therefore it is important to quantify the degree of entanglement, in order to scientifically treat the concept of entanglement. Here we give the definition of aseparable (disentangled) state and an entangled state in the following [45, 39, 31,30].

Deflnition 4.1 A state acting on the composite system
state if it is represented by$\kappa$$\mathcal{H}_{1}\emptyset \mathcal{H}_{2}$is called a separable (disentangled).

$\otimes\kappa_{2}^{1},$$\kappa=\sum_{:}p:\kappa_{1}^{1}$for states
acting on the subsystems
andstate if a state is not a separable state.
$\kappa_{1}^{i}$$\kappa_{2}^{i}$$\mathcal{H}_{1}$$p:\geq 0,$

and$\mathcal{H}_{2}$$\sum_{i}p_{1}=1$,, respectively. It is also called an entangled

When we quantify the entanglement, we should pay the attention whether the considering entangledstate is pure or mixed. For pure entangled states, it is easily calculated by the von Neumann entropy of
the reduced states. For mixed entangled states, we have several measure of entanglements.
Definition 4.2 ([7]) For mixed entangled states, where $\sum_{:}p_{i}=1,$ .
andare pure entangled states on, the entanglernent of formation is defined by$\sigma=\sum_{:^{p:}}\sigma^{(\dot{*})}$$|\emptyset:\rangle\langle\phi:|$$p_{*}$$\geq 0$$\sigma^{(:)}=$$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$E^{F}( \sigma)\equiv\min\sum_{:}p_{\dot{*}}S(\sigma_{1}^{(:)})$

as a minimum of the average of the von Neumann entropyof the reduced states
pure entangled states, where the minimum is taken over all the possible states$S(\sigma_{1}^{(*)})$$\sigma^{(:)}$$\sigma_{1}^{(:)}=tr_{\mathcal{H}\mathrm{a}}\sigma^{(:)}$.

$\sigma_{1}^{(:)}$

for thewith$\sigma=\sum_{i}p_{\dot{*}}\sigma^{(*)}$

105
Deflnition 4.3 ([52]) For mixed entangled states
is defined by$\sigma$on$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$E^{R}( \sigma)\equiv\min_{\rho\in D}U(\sigma|\rho)$

where the minimum is taken for all $\rho\in D$ , wherestates on.

$D$, the relative entropy of entanglement,

represents the set of all separable (disentangled)$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$This measure is a kind of distance (difference) between the entangled states and the separable(disentangled) states . They also proposed the conditions that the entanglement-measure
for any
entangled states on the total system. It is given in [52] byshould$\sigma$$E(\sigma)$$\rho$$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$\sigma$(E1)$E(\sigma)=0\Leftrightarrow\sigma$(E2)$E(\sigma)$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\eta$is separable.

is invariant under the local unitary operations:$E(\sigma)=E(U_{1}\otimes U_{2}\sigma U_{1}^{*}\otimes U_{2}^{*})$

where$U_{1},$$(i=1,2)$ represent the unitary operators acting on(E3) The measure of entanglement
positive map given by Q. That is,$E(\sigma)$,$\mathcal{H}_{:},$$(i=1,2)$ .

can not be increased under the tracepreserving completely$E(\Phi\sigma)\leq E(\sigma)$.

As a measure satisfying the above conditions, a special case of V.Vedral’s definition, we introduced
the entanglement degree due to the mutual entropy and then applied it to the analysis of the JaynesCummings model in [20, 18, 19].

Deflnition 4.4 ([20, 18, 19]) For mixed entangled statesto the mutual entropy is given by$\sigma$on$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$E^{M}(\sigma)\equiv Tr[\sigma(\log\sigma-\log\rho_{1}\otimes\rho_{2})]$

where$\rho_{1}\equiv tr_{\mathcal{H}_{2}}\sigma$

and$\rho_{2}\equiv tr_{\mathcal{H}_{1}}\sigma$, the entanglement degree due,.

In the above definition, we fixed the separable (disentangled) state such as, because it
was difficult to find the separable (disentangled) state attaining the minimum value of the relative entropy$\rho=tr_{\mathcal{H}_{2}}\sigma\emptyset tr_{\mathcal{H}_{1}}\sigma$

of entanglement. The separable (disentangled) state chosen by
is nontrivial state but
our measure contains both quantum and classical entanglement. That is, our measure takes greater value
than V.Vedral’s one. That is, from the definitions, we easily find
. For example, for pureentangled states, by the above Araki-Lieb’s triangle inequality, we easily find that our measure is equal
to the twice of von Neumann entropy, namely $E^{M}(\sigma)=2E^{R}(\sigma)$ for pure entangled states , since
becomes von Neumann entropy for pure entangled states [51]. However it was sufficient to get the roughdegree of entanglement for the analysis of the time development of the Janeys-Cummings model.
In the previous paper, we adopted a parametrically extended entanglement-measure due to the Tsallis
relative entropy which is a generalization of our previous entanglement-measure.
$\rho=tr_{\mathcal{H}_{2}}\sigma\emptyset tr_{\mathcal{H}_{1}}\sigma$$E^{R}(\sigma)\leq E^{M}(\sigma)$$\sigma$Deflnition 4.5 ([3, 17]) For mixed entangled states a on
the Tsallis relative entropy is given by$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$E_{\lambda}^{T}( \sigma)\equiv\frac{Tr[\sigma-\sigma^{1-\lambda}(\rho_{1}\otimes\rho_{2})^{\lambda}]}{\lambda}$

where $0<\lambda\leq 1,$$\rho_{1}\equiv tr_{\mathcal{H}_{2}}\sigma$

and$\rho_{2}\equiv tr_{\mathcal{H}_{1}}\sigma$$E^{R}(\sigma)$, the entanglement-measure due to,.

Note that the above entanglement-measure is a special version of the generalized Kullback-Leibler measureof quantum entanglement introduced in [3]. In addition, the above entanglement-measure for a non-trivial
example was studied in [3]. From the definition, we easily flnd that.
In the below, we show the equality condition of the inequality ((1) of Proposition 1.5) in the propertiesof the Tsallis relative entropy.
$\lim_{qarrow 1}E_{q}^{T}(\sigma)=E^{M}(\sigma)$

106
Lemma 4.6 For $\lambda\in[-1,0)\cup(0,1]$ and density operators$D_{\lambda}(\rho|\sigma)\geq 0$with equality if and only if$\rho=\sigma$$\rho$

and , we have$\sigma$,.

Proposition 4.7 ([17]) For $0<\lambda\leq 1,$$E_{q}^{T}$

satisfies the conditions (E1), (E2) and (E3)

We also have the following proposition.
Proposition 4.8 ([17])(1) For any entangled states(2) There exists$\lambda$$\sigma$on$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$in $(0,1]$ such that

(3) For any entangled states$\sigma$and, we have$E_{\lambda}^{T}(\sigma)=E^{R}(\sigma)$on$\sigma’$$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$$E_{0}^{T}(\sigma)=0$.

for any entangled states

We note that we have the additivity
takes when$0$$\lambda=1$$\mathcal{H}_{1}\otimes \mathcal{H}_{2}$

.

.

as
. In addition, we should
is not separable (disentangled) state.

$E^{M}(\sigma\otimes\sigma’)=E^{M}(\sigma)+E^{M}(\sigma’)$$E_{\lambda}^{T}(\sigma)$on, and any $0<\lambda\leq 1$ , we have the subadditivity:$E_{\lambda}^{T}(\sigma Q\sigma’)\leq E_{\lambda}^{T}(\sigma)+E_{\lambda}^{T}(\sigma’)$note that our measure$\sigma$, although$\sigma$$\lambdaarrow 0$

References[1] S.Abe, Nonadditive generalization of the quantum Kullback-Leibler divergence for measuring
thedegree of purification, Phys.Rev.A,Vo1.68,032302 (2003).
[2] S.Abe, Monotonic decrease of the quantum nonadditive divergence by projective
measurements
Phys. Lett. A, Vo1.312, pp. 336-338
its Corrigendum, Vol.
.
[3] S.Abe and A.K.Rajagopal, Quantum entanglement inferred by the principle of maximum
Tsallisentropy, Phys.Rev.
-3466(1999).
$(2003),\mathrm{a}\mathrm{n}\mathrm{d}$$324,\mathrm{p}\mathrm{p}.507(2004)$$\mathrm{A},\mathrm{V}\mathrm{o}\mathrm{l}.60,\mathrm{p}\mathrm{p}.3461$[4] S.Abe and A.K.Rajagopal, Validity of the second law in nonextensive quantum thermodynamics,
Phys.Rev. Lett.,Vol. 91,No. 12,120601 (2003).

[5] R.Alicki and M.Fhnnes, Quantum Dynamical Systems, Oxford University Press, 2001.
[6] N.Bebiano, J.da Provid\^encia Jr. and R.Lemos, Matrix inequalities in statistical mechanics,
LinearAlg. Appl., Vol.376, pp.265-273 (2004).

[7] C.H.Bennett, D.P.DiVincenzo, J.A.Smolin and W.K.Wootters, Mixed-state entanglement and quantum error correction, Phys.Rev.A,Vo1.54,pp.382 $-3851(1996)$ .
[8] M.Fannes,

A

continuity

propertyofCommun.Math.Phys.,Vol.31,pp.29l-294 (1973).

entropydensityforspinlatticesystems,[9] J.I.Fujii, Operator meao and the relative operator entropy,Operator Theory: Advancesand Applications, Vol.59,pp. 161-172(1992).

[10] J.I.Ftijii, A trace inequality arising &om quantum information theory, Linear Alg. Appl., Vol.400,$\mathrm{p}\mathrm{p}.141- 146(2\mathrm{N}5)$.

[11] J.I.Fujii, M.Fujii and Y.Seo, An extension of the Kubo-Ando theory: Solidarities, Math.Japonica,
Vol.35,-396(1990).
$\mathrm{p}\mathrm{p}.387$[12] J.I.Fujii and E.Kamei, Relative operator entropy in noncommutative information theory, Math.
Japonica, Vol.34,-348(1989).
$\mathrm{p}\mathrm{p}.341$[13] J.I.Fujii and E.Kamei, Uhlmann’s interpolational method for operator means, Math.Japonica,

Vol.

$34,\mathrm{p}\mathrm{p}.541- 547(1989)$.

107[14] S.Furuichi, On uniqueness theorems for Tsallis entropy and Tsallis relative entropy, IEEE Rao.
Information Theory, Vol.
-3645(2005).
$51,\mathrm{p}\mathrm{p}.3638$[15] S.Furuichi, Race inequalities in nonextensive statistical mechanics, Linear Alg. Appl., Vol.418,
pp.821-827(2006).
[16] S.Furuichi, Further results on Tsallis relative entropy, preprint.
[17] S.Furuichi, A note on a parametrically extended entanglement-measure due to Tsallis relative entropy, to appear in INFORMATION.
[18] S.Furuichi and M.Abdel-Aty, Entanglement in squeezed two.level atom, J.Phys.
6857(2001).

$\mathrm{A},\mathrm{V}\mathrm{o}\mathrm{l}.34,\mathrm{p}\mathrm{p}.6851-$[19] S.Furuichi and S.Nakamura, Rigorous derivation of quasi-mutual entropy in Jaynes-Cummings
model, J.Phys.A.Vol.35,pp.544 -5452(2002).

[20] S.Furuichi and M.Ohya,Entanglement degree in the time development of the Jaynes-Cummings
model,Lett. Math. Phys.,Vol.49,pp.279-285(1999).
[21] S.Furuichi, K.Yanagi and K.Kuriyama, IFXindamental properties of Tsallis relative entropy,
J.Math.Phys, Vol.45, No.12, pp. 4868-4877 (2004).

[22] S.Furuichi, K.Yanagi and K.Kuriyama, A note on operator inequalities of Tsallis relative opeartor
entropy, Linear Alg. Appl.,Vol.407,pp.l9-3l (2005).
[23] S.Furuichi, K.Yanagi and K.Kuriyama, A generalized Fannes’s inequality, preprint.
[24] T.Furuta,

to Linear Operators: fi}Space, CRC Pr I Llc, 2002.
$In\dot{m}ta\hslash on$$\mathfrak{v}m$

Matrix to Bounded Linear Operators on a Hilbert[25] T.Furuta, Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators,
Linear Alg. Appl., Vol.381,pp.21 -235(2004).

[26] S.Golden, Lower bounds for the Helmholtz function, Phys. Rev., Vol.137, pp.B1127-B1128 (1965).
[27] E.Heinz, Beitr\"age zur St\"orungsthmrie der Spektralzerlegung, Math. Ann., Vol.123, pp.415-438(1951).
[28] F.Hiai and D.Petz, The proper formula for relative entropy in asymptotics in quantum probability,
Comm. Math. Phys.,Vol.143,pp.99–114(1991).
[29] F.Hiai and D.Petz, The Golden-Thompson trace inequality is complemented, Linear Alg. Appl.,
Vol. 181,pp. 153-185(1993).

[30] R.Horodecki and M.Horodecki, Information-theoretic aspects of inseparability of mixed states,
Phys. Rev. A,Vo1.54,pp. 1838-1843(1996).
[31] M.Horodecki, P.Horodecki and R.Horodecki, Separability of mixed states: necessary and sufficientconditions, Phys.Lett.
-8(1996).
$\mathrm{A},\mathrm{V}\mathrm{o}\mathrm{l}.25,\mathrm{p}\mathrm{p}.1$[32] H.Kosaki, Relative entropy for states: a variational expression, J.Operator Theory, Vol.16, pp.335348 (1986).

[33] G.Lindblad, Completely positive maps and entropy inequalities, Comm.Math.Phys.,Vol.40,pp.147151(1975).
[34] K.L\"owner, \"Uber monotone Matrixfunktionen, Math.Z., Vol.38, pp.177-216 (1934).
[35] M.A.Nielsen and I.Chuang, Quantum Computation and Quantum Information,Cambridge
Press,2000.

[36] J.von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer-Berlin,1932.

108[37] M.Ohya and D.Petz, Quantum Entropy and its Use, Springer-Verlag,1993.
[38] G.K.Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc., Vol.36, pp.309-310(1972).

[39] A.Peres, Separabiliy criterion for density matrices, Phys.Rev.Lett.,Vol.77,pp. 1413-1415(1996).
[40] D.Petz, Quasi-entropies for finite quantum system, Rep.Math.Phys.,Vol.23,-65(1986).

$\mathrm{p}\mathrm{p}.57$[41] D.Petz, A variational expression for the relative entropy of states of a von Neumann algebra,
Commun.Math. Phys.,Vol.114,pp.345-349 (1988).

[42] D.Petz, A survey of certain trace inequalities, in Functional Analysis and Operator Theory, pp.287298,Banach Center Publications, 30 (Warszawa 1994).

[43] M.B.Ruskai and F.M.Stillinger, Convexity inequalities for estimating kee enery and relative entropy,
J.Phys.
-2437(1990).
$\mathrm{A},\mathrm{V}\mathrm{o}\mathrm{l}.23,\mathrm{p}\mathrm{p}.2421$[44] M.B.Ruskai, Inequalities for quantum entropy:
J. Math. Phys., Vol.43, pp.4358-4375 (2002).

a review with condition for equality,[45] R.F.Wemer, Quantum states with Binstein-Podolsky-Roaen correlations admitting a hidden-variable
model, Phys.Rev.A, Vo1.40,.
$\mathrm{p}\mathrm{p}.4277- 4281(1989)$[46] W.Thirring, Quantum mechanics of large systems, Springer-Verlag, 1980.
[47] C.J.Thompson, Inequality with applications in statistical mechanics, J.Math.Phys., Vol.6, pp.18121813(1965).
[48] C.Tsallis, Possible generalization of Bolzmann-Gibbs statistics,
487(1988).

J.Stat.Phys.,Vol.52,pp.479-[49] C. Tsallis et al., Nonextensive Statistical Mechanics and Its Applications, edited by S. Abe andY. Okamoto, Springer-Verlag, Heidelberg,2001; see also the comprehensive list of references at
http:. cat. cbpf.
.
$//\mathrm{t}\mathrm{s}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{s}$$\mathrm{b}\mathrm{r}/\mathrm{b}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{o}.\mathrm{h}\mathrm{t}\mathrm{m}$[50] H.Umegaki, Conditional expectation in an operator algebra, IV (entropy and information),Kodai
Math.Sem.Rep., Vol.14,pp.5 -85(1962).
[51] V.Vedral and M.B.Plenio, Bntanglement measure and purification procedures, Phys.Rev.A,Vo1.57,
pp.1619-1633(1998).
[52] V.Vedral,

M.B.Plenio,M.A.Rippin
andPhys.Rev.Lett.,Vol.78,pp.227 -2279(1997).

P.L.Night,

Quantifying

entanglement,[53] K.Yanagi, K.Kuriyama and S.Furuichi, Generalized Shannon inequalities based on Tsallis relative
operator entropy, Linear Alg. Appl.,Vol.394,-118(2005).
$\mathrm{p}\mathrm{p}.109$