Peierls-Bogolyubov's inequality for deformed exponentials

We study convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and obtain in this way new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls-Bogolyubov's inequality. We use these results to improve previously known lower bounds for the Tsallis relative entropy.


Introduction
In statistical mechanics and in quantum information theory the calculation of the partition function Tr exp H of the Hamiltonian H of a physical system is an important issue, but the computation is often difficult. However, it may be simplified by first computing a related quantity Tr exp A, where A is an easier to handle component of the Hamiltonian. Usually, the Hamiltonian is written as a sum H = A + B of two operators, and the Peierls-Bogolyubov inequality states that which then provides information about the difficult to calculate partition function. We give in this paper generalizations of Peierls-Bogolyubov's inequality in terms of the so-called deformed exponential and logarithmic functions. We formulate the results for operators on a finite dimensional Hilbert space H, but note that the results with proper modifications extend also to infinite dimensional spaces.
Main Theorem. Let A, B ∈ B(H) be self-adjoint operators, and let ϕ be a positive functional on B(H).
(i) If −∞ < q < 1 and r ≥ q and both A and A + B are bounded from above by −(q − 1) −1 , then log r Tr exp q (A + B) − log r Tr exp q A ≥ Tr exp q A r−2 Tr (exp q A) 2−q B.
(ii) If −∞ < q ≤ 0 and r ≥ q and both A and A + B are bounded from above by −(q − 1) −1 , then log r ϕ exp q (A+B) −log r ϕ exp q (A) ≥ ϕ exp q (A) r−2 ϕ d exp q (A)B .
(iii) If 1 < q ≤ 2 and r ≥ q and both A and A + B are bounded from below by −(q − 1) −1 , then log r Tr exp q (A + B) − log r Tr exp q A ≥ (Tr exp q A) r−2 Tr(exp q A) 2−q B.
(iv) If 3 2 ≤ q ≤ 2 and r ≥ q and both A and A + B are bounded from below by −(q − 1) −1 , then log r ϕ exp q (A + B) − log r ϕ(exp q A) ≥ ϕ exp q A r−2 ϕ d exp q (A)B .
(v) If q ≥ 2 and r ≤ q and both A and A + B are bounded from below by −(q − 1) −1 , then If in particular ϕ is the trace this inequality reduces to log r Tr exp q (A + B) − log r Tr exp q A ≤ (Tr exp q A) r−2 Tr(exp q A) 2−q B.
In subsection 5.2 we give explicit formulae for the Fréchet differential operators d exp q (A) in the parameter ranges q ≤ 0 and q ≥ 3/2. Note that the left-hand sides in the above theorem may be written as where ϕ in (i) is replaced by the trace. If we in (iii) let q tend to one, we obtain the inequality Tr exp A 2−r for r > 1 and arbitrary self-adjoint operators A and B. If we furthermore let r tend to one we recover Peierls-Bogolyubov's inequality (1.1).
Furuichi [4,Corollary 3.2] proved (iii) in the case r = q by very different methods. It may be instructive to compare the above results with the first author's study [7] of the deformed Golden-Thompson trace inequality.
We obtain, in Theorem 3.1, another variant Peierls-Bogolyubov type of inequality, and we improve, in Theorem 4.2, previously known lower bounds for the Tsallis relative entropy.
The Peierls-Bogolyubov inequality has been widely used in statistical mechanics and quantum information theory. Recently, Bikchentaev [1] proved that the Peierls-Bogolyubov inequality characterizes the tracial functionals among all positive functionals on a C * −algebra. Moreover, Carlen and Lieb in [2] combined this inequality with the Golden-Thompson inequality to discover sharp remainder terms in some quantum entropy inequalities.

Deformed exponentials
The deformed logarithm log q is defined by setting The deformed logarithm is also denoted the q-logarithm. The inverse function is called the q-exponential. It is denoted by exp q and is given by the formula Note also that If q tends to one then the q-logarithm and the q-exponential functions converge, respectively, toward the logarithmic and the exponential functions. (i) If f is convex and p > 0, then f 1/p is convex.
(ii) If f is convex and p < 0, then f 1/p is concave.
(iii) If f is convex and p < 0 and r > 0, then f r is convex.
(iv) If f is concave and p > 0, then f 1/p is concave.
(v) If f is concave and p < 0, then f 1/p is convex.
(vi) If f is concave and p > 0 and r < 0, then f r is convex.
Proof. Assume first that f is a convex function. The level set is then convex. Take x, y ∈ B(H) + and assume p > 0. Let c and d be any choice of positive numbers such that f (x) 1/p < c and f (y) 1/p < d. We note that c −1 x, d −1 y ∈ L and obtain Therefore f (x + y) 1/p ≤ f (x) 1/p + f (y) 1/p and by homogeneity, we conclude that f 1/p is convex. If p < 0 we choose c, d > 0 such that f (x) 1/p > c and f (y) 1/p > d. This is possible since f is assumed to be positive. Since the exponent is negative we obtain f (x) < c p and f (y) < d p , and therefore by homogeneity where we again used that the exponent is negative. Therefore f (x + y) 1/p ≥ f (x) 1/p + f (y) 1/p and by homogeneity we conclude that f 1/p is concave. This proves (i) and (ii). Under the assumptions in (iii) we proceed as under (ii) to obtain f (x + y) 1/p ≥ c + d.
By homogeneity and since the exponent rp is negative, we obtain the inequality implying convexity of f r . We obtain (iv), (v) and (vi) by a variation of the reasoning used to obtain (i), (ii) and (iii).

Proposition 2.2. Consider the function
(v) G is convex for 0 < p ≤ 1 and r < 0.
Proof. Since the real function t → t p is convex in positive numbers for p ≤ 0 and p ≥ 2 and concave for 0 ≤ p ≤ 1, it is well known that the trace function A → Tr A p retains the same properties. A historic account of this result may be found in [8,Introduction]. By (ii) and (i) in Proposition 2.1 we thus obtain that the function A → TrA p 1/p is concave for p < 0 and convex for p ≥ 2. Furthermore, since the real function t → t p/r is concave and increasing for r ≤ p < 0, we derive (i) in the assertion. Part (ii) then follows by Proposition 2.1(iii), and Part (iii) follows from Proposition 2.1(iv) by noting that 0 < p/r ≤ 1. Part (iv) follows from Proposition 2.1(i) by noting that p/r ≥ 1, and part (v) finally follows from Proposition 2.1(vi).
Note that (Tr A p ) 1/p for p ≥ 1 is the Schatten p-norm of the positive definite matrix A. The convexity in this case may also be derived by noting that a norm satisfies the triangle inequality and is positively homogeneous.
Proof. By continuity we may assume BB * invertible. Since the function t → t p is operator convex for −1 ≤ p ≤ 0 or 1 ≤ p ≤ 2, it follows that the trace function A → Tr B * A p B is convex for these parameter values. It then follows by (ii) and (i) in Proposition 2.1 that the function is concave for −1 ≤ p < 0 and convex for 1 ≤ p ≤ 2. Furthermore, since the real function t → t p/r is concave and increasing for r ≤ p < 0 we derive part (i) of the assertion. Part (ii) then follows by Proposition 2.1(iii). Parts (iii) to (vi) now follow by minor variations of the reasoning in the preceding proposition.

Some deformed trace functions
Theorem 2.4. Consider the function Proof. Note that the conditions on A ensure that A(q − 1) + 1 > 0 for both q < 1 and q > 1. By calculation we obtain Under the assumptions in (i) we obtain for q ≤ r < 1. By Proposition 2.2(i) and since the factor (r −1) −1 is negative, it follows that G is convex. If r > 1 then (1 − r) −1 > 0 and the convexity of G follows by Proposition 2.2(ii). This proves the first statement. Under the assumptions in (ii) we obtain thus G is convex by Proposition 2.2(iv). Under the assumptions in (iii) we first consider the case r > 1 and obtain thus G is concave by Proposition 2.2(iii). If r < 1 then we use Proposition 2.2(v) to obtain that (r − 1)G is convex. Since r − 1 < 0 we conclude that G is concave also in this case. (i) If −∞ < q ≤ 0 and r ≥ q, then F is convex, (ii) If 3 2 ≤ q ≤ 2 and r ≥ q, then F is convex, (iii) If q ≥ 2 and r ≤ q, then F is concave.
Proof. By calculation we obtain Under the assumptions in (i) we obtain for q ≤ r < 1. By Proposition 2.3(i) and since the factor (r −1) −1 is negative, it follows that F is convex. If r > 1 then (1 − r) −1 > 0 and the convexity of F follows by Proposition 2.3(ii). This proves the first statement. Under the assumptions in (ii) we obtain thus F is convex by Proposition 2.3(iv). The last case is argued as in the preceding theorem by considering the cases r > 1 and r < 1 separately.
Note in the above theorem there is a gap between 0 and 3/2 for the possible values of q.

Peierls-Bogolyubov type inequalities
We first obtain a variant Peierls-Bogolyubov type inequality as a consequence of Proposition 2.2. Take positive definite operators A, B ∈ B(H) and define the function Since g(t) is convex for p ≥ 1 and 0 < r ≤ p we obtain the inequality, for these parameter values. By concavity we obtain the opposite inequality for the parameter values 0 < p ≤ 1 and r ≥ p, and for the parameter values for p < 0 and r ≤ p < 0. (ii) If 0 < p ≤ 1 and r ≥ p or if p < 0 and r ≤ p < 0 then Proof. With the parameter values in (i) we may let t tend to zero in (3.1) and obtain the inequality g(1) − g(0) ≥ g ′ (0). We note that g(1) − g(0) is the left hand side in the desired inequality. Furthermore, where we used the chain rule for Fréchet differentiation, the linearity of the trace, and the formula in [5, Theorem 2.2]. This proves case (i). Case (ii) follows by virtually the same argument using the opposite inequality in (3.1).
We then explore consequences of Theorem 2.4. If −∞ < q < 1 we take self-adjoint operators A, B ∈ B(H) such that both A and A + B are bounded from above by − is thus well-defined and convex for −∞ < q < 1 and r ≥ q. Therefore, for these parameter values.
For q > 1 we take self-adjoint operators A, B ∈ B(H) such that both A and A + B are bounded from below by −(q − 1) −1 . For t ∈ [0, 1] we note that The function defined in (3.2) is thus well-defined. It is convex for 1 < q ≤ 2 and r ≥ q, and it is concave for q ≥ 2 and r ≤ q. In the first case we thus retain the inequality in (3.3), while the inequality is reversed in the latter case.   (i) If −∞ < q ≤ 0 and r ≥ q and both A and A + B are bounded from above by −(q − 1) −1 , then (ii) If 3 2 ≤ q ≤ 2 and r ≥ q and both A and A + B are bounded from below by −(q − 1) −1 then (iii) If q ≥ 2 and r ≤ q and both A and A + B are bounded from below by −(q − 1) −1 then Proof. We follow a similar path as in the proof of Theorem 3.2 and consider the function which by Theorem 2.5 is convex for the parameter values in (i). We obtain by an argument similar to the one given in the proof of Theorem 3.2 that h(1) − h(0) ≥ h ′ (0), and we note that h(1) − h(0) is the left hand side in the desired inequality. Furthermore, where we used the chain rule for Fréchet differentiation, the derivative of the deformed logarithmic function, and the linearity of the trace. This proves case (i). Since the function h in (3.4) is convex for the parameter values in (ii) and concave for the parameter values in (iii) these cases follow by virtually the same line of arguments as in (i).
Note that (iii) in Theorem 3.3 is a generalization of (iii) in Theorem 3.2. Since C is arbitrary in the above theorem we may replace the trace by any other positive functional on B(H). The main theorem now follows from Theorem 3.2 and Theorem 3.3.

The Tsallis relative entropy
In this section we study lower bounds for the (generalized) Tsallis relative entropy. For basic information about the Tsallis entropy and the Tsallis relative entropy we refer the reader to references [9,10].
Theorem 4.2. Let q ∈ (0, 1] and take p ≤ q. Then, for positive definite operators X, Y ∈ B(H), the inequality Proof. Let X, Y ∈ B(H) be positive definite operators and take 1 < q ≤ 2 and r ≥ q. By setting A = log q X and B = log q Y − log q X we obtain self-adjoint A, B such that both A and A + B are bounded from below by −(q − 1) −1 . We may thus apply (i) of Theorem 3.2 and obtain after a little calculation the inequality By setting p = 2 − r and renaming q by 2 − q we obtain the stated inequality for q ∈ (0, 1] and p ≤ q. The lower bound of the Tsallis relative entropy D q (X | Y ) in Theorem 4.2 was obtained in [3,Theorem 3.3] in the special case p = q. The family of lower bounds given above is in general not an increasing function in the parameter p and may therefore, depending on Tr X and Tr Y, provide better lower bounds.

Various Fréchet differentials
In order to obtain a more detailed understanding of the bounds obtained in the Main Theorem we need to provide explicit formulae for the Fréchet differential operator d exp q in the parameter range q ≥ 3/2. The integral representation valid for 0 < p < 1 is well-known. Since t → t p is operator monotone the representation may be quite easily derived by calculating the representing measure, see for example [6,Theorem 5.5]. Furthermore, since by an elementary calculation we obtain the integral representation valid for positive definite x. Since by (5.1) we have for 0 < p < 1 and we obtain the integral representation valid for positive definite x. By using the rule for the Fréchet differential of a product, or by an elementary direct calculation, we obtain the general identity which combined with (5.2) provides a formula for the Fréchet differential of x p for 1 < p < 2. If h is self-adjoint the formula in (5.5) may be written on the form d(x p+1 )h = hx p + x p h 2 + d(x p ) xh + hx 2 which is then manifestly self-adjoint. valid for 1 < q < 2. Since by an elementary calculation we derive the formula (5.7) d log q (x)h = sin(q − 1)π (q − 1)π ∞ 0 (x + λ) −1 h(x + λ) −1 λ q−1 dλ valid for positive definite x and 1 < q < 2. Note that for all q > 1 by the definition of the deformed logarithm. If we in formula (5.7) let q tend to 1 we obtain as expected. If we instead set h = 1, we recover the classical integral t q−2 = sin(q − 1)π (q − 1)π ∞ 0 λ q−1 (t + λ) 2 dλ valid for t > 0 and 1 < q < 2.
Note that we for any q > 1 and x > −(q − 1) −1 have the identity Tr d exp q (x)h = Tr exp q (x) 2−q h.
Likewise, d exp q (x)h = exp q (x) 2−q h for commuting x and h.