1 Introduction

The well-known concavity theorem by Lieb [12, Theorem 6] states that the map

$$\begin{aligned} A\rightarrow {\mathrm{Tr}}\exp (L+\log A), \end{aligned}$$
(1.1)

for a fixed self-adjoint matrix L,  is concave in positive definite matrices. This theorem is the basis for the proof of strong subadditivity of the quantum mechanical entropy [13], and it is also very important in random matrix theory [17].

Lieb’s concavity theorem is also closely related to the Golden–Thompson trace inequality. Recently, the second author [7, Theorem 3.1] studied (the known) concavity of the trace function

$$\begin{aligned} A \rightarrow {\mathrm{Tr}}\exp \bigl (H^*\log (A)H\bigr ) \end{aligned}$$

in positive definite matrices, if H is a contraction. This analysis led to multivariate generalizations of the Golden–Thompson trace inequality. Furthermore, the second author [8, Theorem 3.1] obtained convexity/concavity properties of the trace function

$$\begin{aligned} (A_1,\ldots ,A_k)\rightarrow {\mathrm{Tr}}\exp _q\left( \sum _{i=1}^k H^*_i\log _q(A_i)H_i\right) \end{aligned}$$

for \( H_1^*H_1+\cdots +H_k^*H_k=I, \) where \( \exp _q \) denotes the deformed exponential function, respectively, \( \log _q \) denotes the deformed logarithmic function, for the deformation parameter \( q\in [1,3]. \) This analysis led to a generalization of Golden–Thompson’s trace inequality for q-exponentials with \( q\in [1,3]. \)

There is furthermore a close relationship between Lieb’s concavity theorem (1.1) and entropies. In [16], Tropp formulated a variational representation:

$$\begin{aligned} {\mathrm{Tr}}\exp (L+\log A)= \max _{X> 0} \left\{ {\mathrm{Tr}}(L+I)X -D(X\mid A)\right\} , \end{aligned}$$
(1.2)

for \(A>0 \) and \( L=L^*, \) where \(D(X\mid A)={\mathrm{Tr}}(X\log X-X\log A)\) denotes the quantum relative entropy. This variational representation, together with convexity of the quantum relative entropy, enabled Tropp to give an elementary proof of Lieb’s concavity theorem [12, Theorem 6]. Tropp’s variational representation can easily be inverted to obtain a variational representation

$$\begin{aligned} D(X\mid A)= \max _{L=L^*}\left\{ {\mathrm{Tr}}(L+I)X-{\mathrm{Tr}}\exp (L+\log A)\right\} \end{aligned}$$
(1.3)

of the quantum relative entropy. The well-known Gibbs variational principle for the quantum entropy \(S(X)=-{\mathrm{Tr}}X\log X\) states that

$$\begin{aligned} \log {\mathrm{Tr}}\exp L=\max _{X>0, {\mathrm{Tr}}X=1} \left\{ {\mathrm{Tr}}X L - {\mathrm{Tr}}X \log X\right\} , \end{aligned}$$
(1.4)

and for \(X>0\) and \({\mathrm{Tr}}X=1,\)

$$\begin{aligned} -S(X)=\max _{L} \left\{ {\mathrm{Tr}}X L - \log {\mathrm{Tr}}\exp L\right\} , \end{aligned}$$
(1.5)

where L is self-adjoint. Other variational representations in terms of the quantum relative entropy were given by Hiai and Petz [11, Lemma 1.2]:

$$\begin{aligned} \log {\mathrm{Tr}}\exp (L+\log A)= \max _{X> 0, {\mathrm{Tr}}X=1} \left\{ {\mathrm{Tr}}XL-D(X\mid A)\right\} , \end{aligned}$$
(1.6)

and for \(X>0\) and \( {\mathrm{Tr}}X=1,\)

$$\begin{aligned} D(X\mid A)= \max _{L}\left\{ {\mathrm{Tr}}XL-\log {\mathrm{Tr}}\exp (L+\log A)\right\} , \end{aligned}$$
(1.7)

where \(A>0,\) and \(L=L^*.\) Note that the identities (1.6) and (1.7) reduce to (1.4) and (1.5) when \(A=I.\) See [20] for the various relations between the above variational representations. Furuichi [5] extended the two representations above to the deformed logarithmic and exponential functions with parameter \(q\in [1,2]\):

$$\begin{aligned} \log _q {\mathrm{Tr}}\exp _q(L+\log _q A)= \max _{X> 0,\, {\mathrm{Tr}}X=1} \left\{ {\mathrm{Tr}}LX^{2-q}-D_{2-q}(X\mid A)\right\} , \end{aligned}$$
(1.8)

and if \(X>0 \) and \( {\mathrm{Tr}}X=1,\)

$$\begin{aligned} D_{2-q}(X\mid A)= \max _{L}\left\{ {\mathrm{Tr}}LX^{2-q}-\log _q {\mathrm{Tr}}\exp _q(L+\log _q A)\right\} , \end{aligned}$$
(1.9)

where \(A>0,\)L is self-adjoint and \(D_{2-q}(X\mid A)\) denotes the Tsallis relative quantum entropy with parameter \( 2-q. \)

In Sect. 2, we consider variational representations related to the deformed exponential and logarithmic functions by making use of the tracial Young’s inequalities. In Sect. 3, we derive variational representations related to the Tsallis relative entropies, which may be considered extensions of equation (1.2). In Sect. 4, we consider the generalization of the Gibbs variational representations and then tackle the variational representations related to the Tsallis relative entropy under the conditions \(X>0\) and \({\mathrm{Tr}}X=1.\) Finally, in Sect. 5, we extend Golden–Thompson’s trace inequality to deformed exponentials with deformation parameter \( q\in [0,1).\)

Throughout this paper, the deformed logarithm denoted \(\log _q\) is defined by setting

$$\begin{aligned} \log _q x=\left\{ \begin{array}{ll} \displaystyle \frac{x^{q-1}-1}{q-1} &{}\quad q\ne 1,\\ \log x &{}\quad q=1, \end{array}\quad x>0. \right. \end{aligned}$$

The deformed logarithm is also denoted the q-logarithm. The deformed exponential function or the q-exponential is defined as the inverse function to the q-logarithm. It is denoted by \(\exp _q\) and is given by the formula

$$\begin{aligned} \exp _q x=\left\{ \begin{array}{lll} (x (q-1)+1)^{1/(q-1)},&{}\quad x> -1/(q-1), &{}\ q>1,\\ (x (q-1)+1)^{1/(q-1)}, &{}\quad x< -1/(q-1), &{}\ q<1,\\ \exp x, &{}\quad x\in {\mathbf {R}},&{}\ q=1. \end{array} \right. \end{aligned}$$

The Tsallis relative entropy \( D_{p}(X\mid Y) \) is for positive definite matrices X, Y and \( p\in [0,1) \) defined, see [18], by setting

$$\begin{aligned} D_{p}(X\mid Y)= \frac{{\mathrm{Tr}}(X-X^{p}Y^{1-p})}{1-p}={\mathrm{Tr}}X^{p}(\log _{2-p}X-\log _{2-p}Y), \end{aligned}$$

and the definition may be extended to positive semi-definite matrices. The expression converges for \( p\rightarrow 1 \) to the relative quantum entropy \(D(X\mid Y)\) introduced by Umegaki [19]. It is known that the Tsallis relative entropy is nonnegative for states [6, Proposition 2.4]; see also [9, Lemma 1] for a direct proof of the nonnegativity.

2 Variational representations for some trace functions

We consider variational representations related to the deformed logarithm functions.

Lemma 2.1

For positive definite operators X and Y, we have

$$\begin{aligned} {\mathrm{Tr}}Y=\left\{ \begin{array}{ll} \displaystyle \max _{X>0}\bigl \{{\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-\log _q Y\right) \bigr \}, &{}\quad q\le 2,\\ \displaystyle \min _{X>0}\bigl \{{\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-\log _q Y\right) \bigr \}, &{}\quad q>2. \end{array} \right. \end{aligned}$$

Proof

For positive definite operators X and Y,  the tracial Young inequality states that

$$\begin{aligned} {\mathrm{Tr}}X^{p}Y^{1-p}\le & {} p {\mathrm{Tr}}X+(1-p){\mathrm{Tr}}Y, \quad p\in [0,1]. \end{aligned}$$

As for the reverse tracial Young inequalities, we refer the readers to the proof of [3, Lemma 12] from which we extracted the inequality

$$\begin{aligned} {\mathrm{Tr}}X^s\le s {\mathrm{Tr}}XY + (1-s){\mathrm{Tr}}Y^{-s/(1-s)}, \quad 0<s<1. \end{aligned}$$

Replacing s by 1/p and then replacing X by \(X^p\) and Y by \(Y^{1-p},\) it follows for \(p>1\) that

$$\begin{aligned} {\mathrm{Tr}}X^pY^{1-p}\ge p {\mathrm{Tr}}X + (1-p) {\mathrm{Tr}}Y. \end{aligned}$$

It is also easy to see that the above inequality holds for \(p<0.\) Thus, it follows that

$$\begin{aligned} {\mathrm{Tr}}Y\ge {\mathrm{Tr}}X-\frac{{\mathrm{Tr}}X-{\mathrm{Tr}}X^pY^{1-p}}{1-p}, \quad 0\le p<1\,\text {or}\, 1<p<\infty , \end{aligned}$$

and

$$\begin{aligned} {\mathrm{Tr}}Y\le {\mathrm{Tr}}X-\frac{{\mathrm{Tr}}X-{\mathrm{Tr}}X^pY^{1-p}}{1-p}, \quad p\in (-\infty ,0). \end{aligned}$$

For \(X=Y\), the above inequalities become equalities, and hence

$$\begin{aligned} {\mathrm{Tr}}Y=\left\{ \begin{array}{ll} \displaystyle \max _{X>0}\Bigl \{{\mathrm{Tr}}X-\frac{{\mathrm{Tr}}X-{\mathrm{Tr}}X^pY^{1-p}}{1-p}\Bigr \}, &{}\quad 0\le p<1\,\text {or}\, 1<p<\infty ,\\ \displaystyle \min _{X>0}\Bigl \{{\mathrm{Tr}}X-\frac{{\mathrm{Tr}}X-{\mathrm{Tr}}X^pY^{1-p}}{1-p}\Bigr \}, &{}\quad p\in (-\infty ,0). \end{array} \right. \end{aligned}$$

Setting \(q=2-p\), we obtain

$$\begin{aligned}&{\mathrm{Tr}}Y\\&\quad =\left\{ \begin{array}{ll} \displaystyle \max _{X>0}\Bigl \{{\mathrm{Tr}}X-\frac{{\mathrm{Tr}}X^{2-q}\left( X^{q-1}-Y^{q-1}\right) }{q-1}\Bigr \}, &{}\quad -\infty<q<1\,\text {or}\, 1<q\le 2,\\ \displaystyle \min _{X>0}\Bigl \{{\mathrm{Tr}}X-\frac{{\mathrm{Tr}}X^{2-q}\left( X^{q-1}-Y^{q-1}\right) }{q-1}\Bigr \}, &{}\quad q\in (2,\infty ). \end{array} \right. \end{aligned}$$

This proves the assertions except for the case \( q=1. \) For \( q=1 \), the assertion reduces to the identity

$$\begin{aligned} {\mathrm{Tr}}Y=\max _{X>0}\Bigl \{{\mathrm{Tr}}X-{{\mathrm{Tr}}X\left( \log X-\log Y\right) }\Bigr \}, \end{aligned}$$

which follows from the well-known inequality \( D(X\mid Y)\ge {\mathrm{Tr}}(X-Y). \)\(\square \)

Theorem 2.2

Let H be a contraction. For a positive definite operator A, we have the variational representations

$$\begin{aligned}&{\mathrm{Tr}}\exp _q\left( H^*\log _q (A) H\right) \\&\quad =\left\{ \begin{array}{ll} \displaystyle \max _{X>0}\bigl \{{\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \bigr \}, &{}\quad q\le 2,\\ \displaystyle \min _{X>0}\bigl \{{\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \bigr \}, &{}\quad q>2. \end{array} \right. \end{aligned}$$

Proof

Since H is a contraction, it follows for \( q>1 \) that

$$\begin{aligned} H^*\log _q(A)H> \frac{-1}{q-1}. \end{aligned}$$

By setting \(Y= \exp _q\left( H^*\log _q (A) H\right) \) in Lemma 2.1, we obtain the conclusions in the case \( q\ge 1. \) For \( q<1 \), we have

$$\begin{aligned} H^*\log _q(A)H< \frac{-1}{q-1}. \end{aligned}$$

Setting \(Y= \exp _q\left( H^*\log _q (A) H\right) \) in Lemma 2.1, we obtain the conclusions for \( q< 1. \)\(\square \)

Corollary 2.3

Let H be a contraction and consider the map

$$\begin{aligned} \varphi (A)= {\mathrm{Tr}}\exp _q\left( H^*\log _q (A) H\right) \end{aligned}$$

defined in positive definite operators. The following assertions are valid:

  1. (i)

    \( \varphi (A) \) is concave for \( 0\le q<1, \)

  2. (ii)

    \( \varphi (A) \) is concave for \( 1\le q\le 2, \)

  3. (iii)

    \( \varphi (A) \) is convex for \( 2< q\le 3. \)

Proof

By calculation, we obtain

$$\begin{aligned}&{\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \\&\quad = \left( 1-\frac{1}{q-1}\right) {\mathrm{Tr}}X + \frac{1}{q-1}\left[ {\mathrm{Tr}}X^{2-q}(I-H^*H)+{\mathrm{Tr}}X^{2-q}H^*A^{q-1}H\right] . \end{aligned}$$

Under the assumption in (i),  we have

$$\begin{aligned} 1<2-q\le 2,\quad -1\le q-1<0,\quad (2-q)+(q-1)=1. \end{aligned}$$

By Ando’s convexity theorem [1], the trace function \({\mathrm{Tr}}X^{2-q}H^*A^{q-1}H\) is thus jointly convex in (X, A). We also realize that \({\mathrm{Tr}}X^{2-q}(I-H^*H)\) is convex in X. Therefore,

$$\begin{aligned} {\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is jointly concave in (X, A). Hence, by Theorem 2.2 and [4, Lemma 2.3], we obtain that \( \varphi (A) \) is concave for \( 0\le q<1. \) Under the assumption in (ii),  we have

$$\begin{aligned} 0\le 2-q\le 1,\quad 0\le q-1\le 1,\quad (2-q)+(q-1)=1. \end{aligned}$$

By Lieb’s concavity theorem [12], the trace function \({\mathrm{Tr}}X^{2-q}H^*A^{q-1}H\) is jointly concave in (X, A). The expression

$$\begin{aligned} {\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is therefore also jointly concave in (X, A). By Theorem 2.2 and [4, Lemma 2.3] we obtain that \( \varphi (A) \) is concave for \( 1\le q\le 2. \) Under the assumption in (iii),  we have

$$\begin{aligned} -1\le 2-q< 0,\qquad 1< q-1\le 2,\qquad (q-1)+(2-q)=1. \end{aligned}$$

By Ando’s convexity theorem [1], the trace function \({\mathrm{Tr}}X^{2-q}H^*A^{q-1}H\) is jointly convex in (X, A). Since obviously \({\mathrm{Tr}}X^{2-q}(I-H^*H)\) is convex in X,  we obtain that

$$\begin{aligned} {\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is jointly convex in (X, A). Hence, \( \varphi (A) \) is convex for \( 2< q\le 3, \) by Theorem 2.2 and [4, Lemma 2.3]. \(\square \)

Remark 2.4

The second author [8] proved the cases \(1\le q\le 2\) and \(2\le q\le 3\) in the above corollary by another method. The case \(0\le q< 1\) may be similarly proved by using that the trace function

$$\begin{aligned} A\rightarrow {\mathrm{Tr}}(H^* A^{q-1} H)^{1/(q-1)} \end{aligned}$$

is concave for \(0\le q< 1\) (see [2]).

Proposition 2.5

Let H be a contraction.

  1. (i)

    If \(1< q\le 2,\) then for positive definite A and self-adjoint L such that

    $$\begin{aligned} L+H^*\log _q (A) H>-\frac{1}{q-1}, \end{aligned}$$

    we have the equality

    $$\begin{aligned}&{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H)\\&\quad = \max _{X>0}\bigl \{{\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}L - {\mathrm{Tr}}X^{2-q} \left( \log _q X-H^*\log _q (A) H\right) \bigr \}. \end{aligned}$$
  2. (ii)

    If \( q>2, \) then for positive definite A and self-adjoint L such that

    $$\begin{aligned} L+H^*\log _q (A) H>-\frac{1}{q-1}, \end{aligned}$$

    we have the equality

    $$\begin{aligned}&{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H)\\&\quad = \min _{X>0}\bigl \{{\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}L -{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \bigr \}. \end{aligned}$$
  3. (iii)

    If \( q< 1, \) then for positive definite A and self-adjoint L such that

    $$\begin{aligned} L+H^*\log _q (A) H<-\frac{1}{q-1}, \end{aligned}$$

    we have the equality

    $$\begin{aligned}&{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H)\\&\quad =\max _{X>0}\bigl \{{\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \bigr \}. \end{aligned}$$

Proof

Under the assumptions of (i), (ii) and (iii),  the expression \( \exp _q(L+H^*\log _q(A)H) \) is well defined and positive definite. By setting \( Y=\exp _q(L+H^*\log _q(A)H) \) in Lemma 2.1, we obtain (i), (ii) and (iii). \(\square \)

Corollary 2.6

Let H be a contraction, and let L be positive definite. The map

$$\begin{aligned} A\rightarrow {\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H), \end{aligned}$$

defined in positive definite operators, is concave for \(1<q\le 2\) and convex for \( 2<q\le 3. \) The map

$$\begin{aligned} A\rightarrow {\mathrm{Tr}}\exp _q (-L+H^*\log _q (A) H), \end{aligned}$$

defined in positive definite operators, is concave for \( 0\le q< 1.\)

Proof

If \(1< q\le 2,\) the map \( X\rightarrow {\mathrm{Tr}}X^{2-q}L \) is concave. By an argument similar to the proof of Corollary 2.3(ii), we obtain that the expression

$$\begin{aligned} {\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is jointly concave in (X, A). Then, obviously

$$\begin{aligned} {\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is jointly concave in (X, A). By Proposition 2.5(i) and [4, Lemma 2.3], we then conclude that

$$\begin{aligned} {\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H) \end{aligned}$$

is concave in A for \(1< q\le 2.\) The case for \(2<q\le 3\) can be proved by a similar argument as above. If \(0\le q<1,\) then the map \( X\rightarrow {\mathrm{Tr}}X^{2-q}L \) is convex. By an argument similar to the proof of Corollary 2.3(i), we obtain that the expression

$$\begin{aligned} {\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is jointly concave in (X, A). Thus,

$$\begin{aligned} {\mathrm{Tr}}X-{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (A) H\right) \end{aligned}$$

is jointly concave in (X, A). By Proposition 2.5(iii) and [4, Lemma 2.3], we then obtain that

$$\begin{aligned} {\mathrm{Tr}}\exp _q (-L+H^*\log _q (A) H) \end{aligned}$$

is concave in A for \(0\le q<1.\)\(\square \)

Letting \(q\rightarrow 1\) in Corollary 2.6, we obtain:

Corollary 2.7

Let H be a contraction, and let L be self-adjoint. The map

$$\begin{aligned} A\rightarrow {\mathrm{Tr}}\exp (L+H^*\log (A) H) \end{aligned}$$

is concave in positive definite operators.

The corollary was shown in a more general setting in Theorem 3 of Lieb and Seiringer [14] and in Theorem 2.5 of an earlier paper of Hiai [10].

Proposition 2.8

Let H be a contraction, and let L be positive definite. The map

$$\begin{aligned} A\rightarrow {\mathrm{Tr}}\exp _q\left( L+H^*\log _r (A) H\right) \end{aligned}$$

is convex in positive definite operators for \(q, r \in [2,3]\) with \(r\ge q.\)

Proof

Since H is a contraction and \(r\ge q\ge 2\), we obtain the inequalities

$$\begin{aligned} H^*\log _r (A)H> \frac{-1}{r-1}\ge \frac{-1}{q-1}. \end{aligned}$$

We may thus apply the deformed exponential and set \(Y=\exp _q\left( L+H^*\log _r (A) H\right) \) in Lemma 2.1 to obtain

$$\begin{aligned}&{\mathrm{Tr}}\exp _q (L+H^*\log _r (A) H)\\&\quad = \min _{X>0}\bigl \{{\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}L - {\mathrm{Tr}}X^{2-q} \left( \log _q X-H^*\log _r (A) H\right) \bigr \}\\&\quad = \min _{X>0}\Bigl \{\Bigl (1-\frac{1}{q-1}\Bigr ){\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}\Bigl (L+\frac{1}{q-1}-\frac{H^* H}{r-1}\Bigr )\\&\qquad +\frac{1}{r-1}{\mathrm{Tr}}X^{2-q}H^* A^{r-1} H \Bigr \}, \end{aligned}$$

where by the assumptions

$$\begin{aligned} 2-q\in [-1,0],\quad r-1\in [1,2],\quad (r-1)+(2-q)\ge 1. \end{aligned}$$

By Ando’s convexity theorem and [4, Lemma 2.3], we then get the desired conclusions. \(\square \)

3 Variational expressions related to Tsallis relative entropy

Theorem 3.1

Let H be a contraction. For positive definite operators X and A, the following assertions hold:

  1. (i)

    For \(1< q\le 2\), we have the equality

    $$\begin{aligned}&{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) \\&\quad = \max _{L>-H^*\log _q (A) H-(q-1)^{-1}}\left\{ {\mathrm{Tr}}X +{\mathrm{Tr}}X^{2-q}L -{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H)\right\} . \end{aligned}$$
  2. (ii)

    For \(q>2\), we have the equality

    $$\begin{aligned}&{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) \\&\quad = \min _{L>-H^*\log _q (A) H-(q-1)^{-1}}\left\{ {\mathrm{Tr}}X +{\mathrm{Tr}}X^{2-q}L -{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H)\right\} . \end{aligned}$$
  3. (iii)

    For \(q<1\), we have the equality

    $$\begin{aligned}&{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) \\&\quad = \max _{L<-H^*\log _q (A) H-(q-1)^{-1}}\left\{ {\mathrm{Tr}}X +{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H)\right\} . \end{aligned}$$

Proof

Under the assumptions in (i) and the natural condition

$$\begin{aligned} L+H^*\log _q (A) H>-\frac{1}{q-1}, \end{aligned}$$

ensuring that \(\exp _q (L+H^*\log _q (A) H)\) makes sense, we set

$$\begin{aligned} G(L)={\mathrm{Tr}}X +{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H) \end{aligned}$$

and obtain that G(L) is concave. By Proposition 2.5(i), we then obtain the inequality

$$\begin{aligned} G(L)\le {\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) . \end{aligned}$$

Inserting \(L_0=\log _q X-H^* \log _q (A) H \) yields

$$\begin{aligned} G(L_0)={\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) , \end{aligned}$$

such that G(L) attains its maximum in \(L_0.\) Thus, we obtain

$$\begin{aligned} \max _{L>-H^*\log _q (A) H-(q-1)^{-1}}G(L)= {\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) , \end{aligned}$$

which proves (i). The case (ii) can be proved by a similar argument. Under the assumptions in (iii),  and the condition

$$\begin{aligned} L+H^*\log _q (A) H<-\frac{1}{q-1}, \end{aligned}$$

we set

$$\begin{aligned} G(L)={\mathrm{Tr}}X+{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}\exp _q (L+H^*\log _q (A) H) \end{aligned}$$

and obtain that G(L) is concave. By Proposition 2.5(iii), we then obtain the inequality

$$\begin{aligned} G(L)\le {\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) . \end{aligned}$$

Inserting \(L_0=\log _q X-H^* \log _q (A) H \) yields

$$\begin{aligned} G(L_0)={\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) \end{aligned}$$

such that G(L) attains its maximum in \(L_0.\) Hence,

$$\begin{aligned} \max _{L<-H^*\log _q (A) H-(q-1)^{-1}}G(L)= {\mathrm{Tr}}X^{2-q}\left( \log _q X-H^* \log _q(A)H\right) , \end{aligned}$$

which proves (iii). \(\square \)

Setting \(H=I\), we obtain in particular

Corollary 3.2

The equality

$$\begin{aligned} D_{2-q}(X\mid A)= & {} {\mathrm{Tr}}X^{2-q}\left( \log _q X- \log _q A \right) \nonumber \\= & {} \max _{L+\log _q A>-(q-1)^{-1}}\bigl \{{\mathrm{Tr}}X +{\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}\exp _q (L+\log _q A )\bigr \}\nonumber \\ \end{aligned}$$
(3.1)

holds for \(q\in (1,2].\)

Corollary 3.2 may be considered as a variational representation of the Tsallis relative entropy. Letting \(q\rightarrow 1\), we recover the well-known representation:

$$\begin{aligned} D(X\mid A)= & {} {\mathrm{Tr}}X\left( \log X- \log A \right) \nonumber \\= & {} \max _{L}\left\{ {\mathrm{Tr}}X +{\mathrm{Tr}}X L-{\mathrm{Tr}}\exp (L+\log A )\right\} , \end{aligned}$$

where the supremum is taken over self-adjoint L. This variational expression of \(D(X\mid A)\) was essentially given by Tropp [16] as mentioned in Introduction. However, this was formerly given by Petz [15] in the von Neumann algebra setting.

4 Variant representations related to Tsallis relative entropy

In this section, we generalize the Gibbs variational representations and the variational representations in terms of the quantum relative entropy obtained by Hiai and Petz [11]. We recall the Peierls–Bogolyubov-type inequalities for deformed exponentials and quote from [9, Theorem 7].

Lemma 4.1

Let A and B be self-adjoint \( n\times n \) matrices. The following assertions hold:

  1. (i)

    If \(q<1,\) and both A and \(A+B\) are bounded from above by \(-(q-1)^{-1},\) then

    $$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q(A+B)-\log _q{\mathrm{Tr}}\exp _q A \ge \bigl ({\mathrm{Tr}}\exp _q A\bigr )^{q-2}{\mathrm{Tr}}(\exp _q A)^{2-q}B. \end{aligned}$$
  2. (ii)

    If \(1<q\le 2,\) and both A and \(A+B\) are bounded from below by \(-(q-1)^{-1},\) then

    $$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q(A+B)-\log _q{\mathrm{Tr}}\exp _q A \ge \bigl ({\mathrm{Tr}}\exp _q A\bigr )^{q-2}{\mathrm{Tr}}(\exp _q A)^{2-q}B. \end{aligned}$$
  3. (iii)

    If \(q\ge 2,\) and both A and \(A+B\) are bounded from below by \(-(q-1)^{-1},\) then

    $$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q(A+B)-\log _q{\mathrm{Tr}}\exp _q A \le \bigl ({\mathrm{Tr}}\exp _q A\bigr )^{q-2}{\mathrm{Tr}}(\exp _q A)^{2-q}B. \end{aligned}$$

Using these Peierls–Bogolyubov-type inequalities, we obtain:

Theorem 4.2

The following variational representations hold:

  1. (i)

    If \(q<1,\) then for \(L< -(q-1)^{-1},\)

    $$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q L=\max _{X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\log _q X \right\} , \end{aligned}$$

    and for \(X>0\) with \({\mathrm{Tr}}X=1,\)

    $$\begin{aligned} {\mathrm{Tr}}X^{2-q}\log _q X=\max _{L< -(q-1)^{-1}}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q L \right\} . \end{aligned}$$
  2. (ii)

    If \(1<q\le 2,\) then for \(L> -(q-1)^{-1},\)

    $$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q L=\max _{ X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\log _q X \right\} , \end{aligned}$$

    and for \(X>0\) with \({\mathrm{Tr}}X=1,\)

    $$\begin{aligned} {\mathrm{Tr}}X^{2-q}\log _q X=\max _{ L> -(q-1)^{-1}}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q L \right\} . \end{aligned}$$
  3. (iii)

    If \(q>2,\) then for \(L> -(q-1)^{-1},\)

    $$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q L=\min _{X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\log _q X \right\} , \end{aligned}$$

    and for \(X>0\) with \({\mathrm{Tr}}X=1,\)

    $$\begin{aligned} {\mathrm{Tr}}X^{2-q}\log _q X=\min _{L> -(q-1)^{-1}}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q L \right\} . \end{aligned}$$

Proof

We just prove the case of \(1<q\le 2.\) For \(X>0\) with \( {\mathrm{Tr}}X=1\) and setting \(A=\log _q X\), we have \({\mathrm{Tr}}\exp _q A=1.\) By (ii) of Lemma 4.1, we thus obtain

$$\begin{aligned} {\mathrm{Tr}}X^{2-q}B\le \log _q{\mathrm{Tr}}\exp _q(\log _q X+B), \end{aligned}$$

which holds for \(X>0\) with \({\mathrm{Tr}}X=1\) and B with \(\log _q X+B>-(q-1)^{-1}.\) Replacing B with \(L-\log _q X\) yields

$$\begin{aligned} {\mathrm{Tr}}X^{2-q}L\le \log _q{\mathrm{Tr}}\exp _q L+{\mathrm{Tr}}X^{2-q}\log _q X, \end{aligned}$$
(4.1)

which is valid for \(X>0\) with \({\mathrm{Tr}}X=1\) and \(L>-(q-1)^{-1}.\) It is easy to see that for a fixed X,  there is equality in (4.1) for \(L=\log _q X.\) We thus obtain

$$\begin{aligned} {\mathrm{Tr}}X^{2-q}\log _q X=\max _{ L>-(q-1)^{-1}}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q L \right\} . \end{aligned}$$

By an elementary calculation, we obtain the equalities

$$\begin{aligned} \log _q{\frac{y}{x}}=\log _q y + y^{q-1}\log _q{\frac{1}{x}} \end{aligned}$$

and

$$\begin{aligned} \log _q{\frac{1}{x}}=-x^{1-q}\log _q x \end{aligned}$$

for \(q\in {\mathbf {R}}.\) Therefore,

$$\begin{aligned} \log _q\frac{\exp _q L}{{\mathrm{Tr}}\exp _qL}= & {} \log _q\exp _qL+(\exp _q L)^{q-1}\log _q\frac{1}{{\mathrm{Tr}}\exp _qL}\\= & {} L-(\exp _q L)^{q-1}({\mathrm{Tr}}\exp _q L)^{1-q}\log _q{\mathrm{Tr}}\exp _q L. \end{aligned}$$

It follows that

$$\begin{aligned}&\log _q {\mathrm{Tr}}\exp _q L+{\mathrm{Tr}}\left[ \left( \frac{\exp _q L}{{\mathrm{Tr}}\exp _q L}\right) ^{2-q}\log _q \left( \frac{\exp _q L}{{\mathrm{Tr}}\exp _q L}\right) \right] \\&\quad = \log _q{\mathrm{Tr}}\exp _q L+\frac{{\mathrm{Tr}}(\exp _q L)^{2-q}L}{({\mathrm{Tr}}\exp _q L)^{2-q}}-\frac{{\mathrm{Tr}}\exp _q L ({\mathrm{Tr}}\exp _q L)^{1-q}\log _q {\mathrm{Tr}}\exp _q L}{({\mathrm{Tr}}\exp _q L)^{2-q}} \\&\quad = {\mathrm{Tr}}\left( \frac{\exp _q L}{{\mathrm{Tr}}\exp _q L}\right) ^{2-q} L. \end{aligned}$$

For a fixed L, we therefore have equality in (4.1) for \(X=({\mathrm{Tr}}\exp _q L)^{-1} \exp _q L.\) Hence,

$$\begin{aligned} \log _q{\mathrm{Tr}}\exp _q L=\max _{ X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\log _q X \right\} \end{aligned}$$

for \(L> -(q-1)^{-1}.\) The cases for \(q<1\) and \(q>2\) are proved by similar reasoning. \(\square \)

By replacing L with \(L+H^*\log _q(Y)H\) in Theorem 4.2, we obtain:

Theorem 4.3

Assume \(H^*H\le I.\) For any positive definite operator Y, the following assertions hold:

  1. (i)

    If \(q<1,\) then for \(L\le 0\) we have the equality

    $$\begin{aligned}&\log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q (Y)H)\\&\quad =\max _{X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H\right) \right\} , \end{aligned}$$

    and for \(X>0\) with \({\mathrm{Tr}}X=1\) the equality

    $$\begin{aligned}&{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y) H\right) \\&\quad =\max _{L\le 0}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q (Y)H) \right\} . \end{aligned}$$
  2. (ii)

    If \(1<q\le 2,\) then for \(L\ge 0\) we have the equality

    $$\begin{aligned}&\log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q (Y)H)\\&\quad =\max _{ X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H\right) \right\} , \end{aligned}$$

    and for \(X>0\) with \({\mathrm{Tr}}X=1\) the equality

    $$\begin{aligned}&{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H\right) \\&\quad =\max _{L\ge 0}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q(Y)H)\right\} . \end{aligned}$$
  3. (iii)

    If \(q>2,\) then for \(L\ge 0\) we have the equality

    $$\begin{aligned}&\log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q (Y)H)\\&\quad =\min _{X>0, \,{\mathrm{Tr}}X=1}\left\{ {\mathrm{Tr}}X^{2-q}L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H \right) \right\} , \end{aligned}$$

    and for \(X>0,\)\({\mathrm{Tr}}X=1\) the equality

    $$\begin{aligned}&{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H\right) \\&\quad =\min _{L\ge 0}\left\{ {\mathrm{Tr}}X^{2-q}L-\log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q (Y)H)\right\} . \end{aligned}$$

Remark 4.4

We note that the cases \(1< q\le 2\) in Theorem 4.2 and \(1< q\le 2\) with \(H=I\) in Theorem 4.3 were first obtained by Furuichi in [5], who gave a different proof. Note also that when \(q\rightarrow 1,\) we recover Gibbs’ variational principle for the von Neumann entropy \(S(X)=-{\mathrm{Tr}}X \log X \) together with the variational representations related to the quantum relative entropy obtained by Hiai and Petz, when L is self-adjoint. Moreover, we can derive convexity or concavity of the map

$$\begin{aligned} Y\rightarrow \log _q{\mathrm{Tr}}\exp _q (L+H^*\log _q (Y)H) \end{aligned}$$

by using the joint convexity or concavity of the Tsallis entropy-type functionals:

$$\begin{aligned} (X,Y)\rightarrow {\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H\right) , \end{aligned}$$

which in turn recovers the Peierls–Bogolyubov-type inequalities for deformed exponentials. Note that the joint convexity or concavity for the Tsallis entropy-type functionals can be traced back to Lieb’s concavity theorem and Ando’s convexity theorem, as demonstrated in Corollary 2.3.

Now, we consider two types of variational expressions with and without the restriction \({\mathrm{Tr}}X=1.\) A special case of Theorem 2.2 states that

$$\begin{aligned} \exp _q s=\max _{\lambda >0}\left\{ \lambda -\lambda ^{2-q}(\log _q \lambda -s)\right\} , \quad q \le 2, \end{aligned}$$
(4.2)

and

$$\begin{aligned} \exp _q s=\min _{\lambda>0}\left\{ \lambda -\lambda ^{2-q}(\log _q \lambda -s)\right\} , \quad q>2, \end{aligned}$$
(4.3)

for numbers s such that \(\exp _q s\) is defined. The above formulas may be viewed as Legendre–Fenchel-type dualities for deformed exponentials. The inequalities (4.2) and (4.3) may also easily be obtained from the scalar Young’s inequality and its reverse inequality. We now recover Theorem 2.2 from Theorem 4.3 and the above scalar Legendre–Fenchel dualities. For \(1< q\le 2\) and by using Theorem 4.3 (ii),  we obtain

$$\begin{aligned}&\max _{X>0}\left\{ {\mathrm{Tr}}X +{\mathrm{Tr}}X^{2-q} L-{\mathrm{Tr}}X^{2-q}\left( \log _q X-H^*\log _q (Y)H\right) \right\} \\&\quad = \max _{\lambda>0}\max _{{\bar{X}}>0, \,{\mathrm{Tr}}{\bar{X}}=1}\left\{ {\mathrm{Tr}}\lambda {\bar{X}} +{\mathrm{Tr}}(\lambda {\bar{X}})^{2-q} L-{\mathrm{Tr}}(\lambda {\bar{X}})^{2-q}\left( \log _q (\lambda {\bar{X}})\right. \right. \\&\qquad \left. \left. -H^*\log _q (Y)H\right) \right\} \\&\quad = \max _{\lambda>0}\max _{{\bar{X}}>0, {\mathrm{Tr}}\,{\bar{X}}=1}\left\{ \lambda ^{2-q}\left( {\mathrm{Tr}}{\bar{X}}^{2-q}L-{\mathrm{Tr}}{\bar{X}}^{2-q}(\log _q {\bar{X}}- H^*\log _q (Y)H)\right) \right. \\&\qquad \left. +\lambda - \lambda ^{2-q}\log _q \lambda \right\} \\&\quad = \max _{\lambda >0}\left\{ \lambda ^{2-q}\log _q{\mathrm{Tr}}\exp _q (L+H^* \log _q(Y) H)+\lambda - \lambda ^{2-q}\log _q \lambda \right\} \\&\quad = {\mathrm{Tr}}\exp _q(L+H^*\log _q (Y)H), \end{aligned}$$

where the last equality follows from (4.2). The cases for \(q<1\) and \(q>2\) are proved by similar reasoning.

5 Golden–Thompson’s inequality for deformed exponentials

The second author generalized Golden–Thompson’s trace inequality to q-exponentials with deformation parameter \(q\in [1,3].\) We will now address the same question for parameter values \( q\in [0,1). \) The following result is an easy consequence of Corollary 2.3.

Corollary 5.1

Let \(H_1, \ldots , H_k\) be matrices with \(H_1^*H_1+\cdots +H_k^*H_k=I.\) The function

$$\begin{aligned} \varphi (A_1,\ldots , A_k)={\mathrm{Tr}}\exp _q\left( \sum _{i=1}^{k}H_i^*\log _q(A_i)H_i\right) , \end{aligned}$$
(5.1)

defined in k-tuples of positive definite matrices, is concave for \(q\in [0,1).\)

The second author [8, Theorem 3.1] proved that \(\varphi \) is positively homogeneous of degree one. Since \(\varphi \) is concave for \(q\in [0,1)\) and by appealing to [8, Lemma 2.1], we may reason as in [8, Corollary 3.4] to obtain:

Corollary 5.2

The function \(\varphi \) defined in (5.1) satisfies the inequality

$$\begin{aligned} \varphi (B_1,\ldots ,B_k)\le {\mathrm{Tr}}\exp _q \left( \sum _{i=1}^kH^*\log _q(A_i)H_i\right) ^{2-q}\sum _{j=1}^kH_j^*\bigl (d{}\log _q(A_j)B_j\bigr )H_j \end{aligned}$$

for \(0\le q<1,\) where \(d{}\log _q(A)\) denotes the Fréchet differential of \(\log _q(A).\)

Theorem 5.3

Let A and B be negative definite matrices. The inequality

$$\begin{aligned} {\mathrm{Tr}}\exp _q(A+B)\le {\mathrm{Tr}}\exp _q(A)^{2-q}\left( A(q-1)+\exp _qB\right) \end{aligned}$$

holds for \(0\le q<1.\)

Proof

In Corollary 5.2, we set \(k=2,\)\(A_1=B_1\) and \(A_2=I.\) We then obtain the inequality

$$\begin{aligned} \varphi (B_1,B_2)\le {\mathrm{Tr}}\exp _q(H_1^*\log _q(B_1)H_1)^{2-q}(H_1^*B_1^{q-1}H_1+H_2^*B_2H_2) \end{aligned}$$
(5.2)

for \(0\le q<1.\) Furthermore, we set \( H_1=\varepsilon ^{1/2} \) for \( 0< \varepsilon <1. \) To fixed negative definite matrices \(L_1\) and \(L_2\), we may choose \(B_1\) and \(B_2\) such that

$$\begin{aligned} \begin{array}{rl} L_1&{}=H_1^*\log _q(B_1)H_1=\varepsilon \log _q(B_1),\\ L_2&{}=H_2^*\log _q(B_1)H_2=(1-\varepsilon ) \log _q(B_2). \end{array} \end{aligned}$$

By inserting these operators in inequality (5.2), we obtain

$$\begin{aligned}&{\mathrm{Tr}}\exp _q(L_1+L_2) \\&\quad \le {\mathrm{Tr}}\exp _q(L_1)^{2-q}\bigl (\varepsilon \exp _q(\varepsilon ^{-1}L_1)^{q-1}+(1-\varepsilon )\exp _q((1-\varepsilon )^{-1}L_2)\bigr ) \\&\quad = {\mathrm{Tr}}\exp _q(L_1)^{2-q}\bigl (L_1(q-1)+\varepsilon +(1-\varepsilon )\exp _q((1-\varepsilon )^{-1}L_2)\bigr ). \end{aligned}$$

Since \( \lim _{\varepsilon \rightarrow 0}(1-\varepsilon )\exp _q((1-\varepsilon )^{-1}L_2)=\exp _q(L_2), \) we obtain

$$\begin{aligned} {\mathrm{Tr}}\exp _q(L_1+L_2) \le {\mathrm{Tr}}\exp _q(L_1)^{2-q}\bigl (L_1(q-1)+\exp _q(L_2)\bigr ). \end{aligned}$$

Finally, by replacing \(L_1\) and \(L_2\) with A and B,  the assertion follows. \(\square \)

Letting \(q\rightarrow 1\), we recover the Golden–Thompson inequality

$$\begin{aligned} {\mathrm{Tr}}\exp (A+B)\le {\mathrm{Tr}}\exp (A)\exp (B), \end{aligned}$$

firstly only for negative definite operators. However, by adding suitable constants to A and B,  we obtain the trace inequality for arbitrary self-adjoint operators.