Geometric properties for a class of deformed trace functions

We investigate geometric properties of a class of trace functions expressed in terms of the deformed logarithmic and exponential functions. These trace functions and their properties may be of independent interest, but we use them in this paper to extend earlier results of Beklan, Carlen and Lieb.


Preliminaries
Tsallis [7] generalised in 1988 the standard Bolzmann-Gibbs entropy to a nonextensive quantity S q depending on a parameter q.In the quantum version it is given by where ρ is a density matrix.It has the property that S q (ρ) → S(ρ) for q → 1, where S(ρ) = −Tr ρ log ρ is the von Neumann entropy.

The deformed logarithm and exponential
The Tsallis entropy may be written on a similar form S q (ρ) = −Tr ρ log q (ρ), where the deformed logarithm log q defined for positive x is given by The deformed logarithm is also denoted the q-logarithm.The range of the q-logarithm is given by the intervals The inverse function exp q (denoted the q-exponential) is always positive and given by exp q (x) = (x(q − 1) + 1) 1/(q−1) for q > 1 and x > −(1 − q) −1 (x(q − 1) + 1) 1/(q−1) for q < 1 and x < −(1 − q) −1 exp x for q = 1 and x ∈ R.
The q-logarithm and the q-exponential functions converge, respectively, to the logarithmic and the exponential functions for q → 1.We note that (1) d dx log q (x) = x q−2 and d dx exp q (x) = exp q (x) 2−q .
We will also need the following lemma.
Lemma 1.1.Take arbitrary q ∈ R. Independent of x > 0, we have where α = 1+p(q−1).Furthermore, take arbitrary q and set β = 1+(q−1)/p for p = 0.For any x ∈ R in the domain of exp q , we obtain that px is in the domain of exp β and that (exp q x) p = exp β (px).
Proof.We substitute u = t 1/p (thus t = u p ) in log q x p = x p 1 t q−2 dt and note that du = p −1 t (1−p)/p dt.Therefore, dt = pt (p−1)/p du and thus The definition of β implies p/(1 − q) = 1/(1 − β).There are now four cases depending on p > 0, p < 0, q > 1 and q < 1.In all four cases it follows that px is in the domain of exp β .We finally obtain from the first result in the lemma that log β (exp q x) p = p log q (exp q x) = px and therefore (exp q x) p = exp β (px).

Convexity and min-max theorems
The following results are well-know in convexity theory and very elementary.Since they play a pivotal role in this paper we provide the proofs as a convenience to the reader.
Lemma 1.2.Let f : X × Y → R be a function of two variable and set Proof.Take ε > 0 and elements Then

The Young tracial inequalities
The following inequalities are known as the tracial Young inequalities.We prefer to prove them as below.
Proposition 1.3.Let A and B be positive definite matrices.Then Proof.Let first 0 ≤ p ≤ 1.We may write where f (t) = t p for t > 0, and L A and R B are the left and right multiplication operators.The first equality above in terms of the quasi-entropy S I f (A, B) follows since L A and R B commute, and the first inequality in the proposition then follows from the geometric-arithmetic mean inequality.Since Jensen's inequality reverses for the extensions of a chord (corresponding to the cases p ≤ 0 or p ≥ 1), the second inequality of the proposition follows.

Variational expressions
We take the following variational representations from our paper [6, Lemma 2.1] with a slightly simplified proof.
Proposition 2.1.For positive definite operators X and Y we have Proof.We learned in Proposition 1.3 that By combining the first inequality for 0 ≤ p < 1 with the case p > 1 in the second, we obtain For X = Y the above inequalities become equalities.Setting q = 2 − p, the first range (p ≥ 0, p = 1) is transformed to the range (q ≤ 2, q = 1), while the second range (p ≤ 0) is transformed to the range (q ≥ 2).Since p = 2 − q and 1 − p = q − 1 we obtain By using the definition of the deformed logarithm we note that and by inserting this in the expressions above, we obtain the desired statements of the proposition, except for q = 1.We may finally let q tend to one in the first inequality and obtain the variational expression by continuity.This completes the proof.
Note that the last statement in the above proof entails the inequality for the relative quantum entropy S(X | Y ).

Main results
Let H be a contraction and A positive definite.Take q = 1 and set β = 1 + (p − 1)/q .Since A is positive definite and H is a contraction, it follows that We may therefore in both cases apply exp p to H * log p (A)H.Thus is well-defined and positive, and we set (2) ϕ p,q (A) = Tr Y q = Tr exp p H * log p (A)H q .
Since (exp p x) q = exp β (qx) we obtain We define and calculate for positive definite X and A the function where we used that q/(p − 1) = 1/(β − 1) .By replacing q with β in Proposition 2.1 we then obtain We next determine parameter values p and q such that F (X, A) is either (jointly) concave/convex or concave/convex in the second variable.
Since H = I is a possibility, we realise that concavity of ϕ p,q requires 0 ≤ q ≤ 1, while convexity of ϕ p,q requires q ≤ 0 or q ≥ 1.
Since we intend to eventually use operator convexity/concavity of the function t → t p , Lieb's concavity theorem, or Ando's convexity theorem, we are restricted to the cases Likewise, we must impose the condition If β = 1, then p = 1.We next determine parameter values p and q that impart convexity or concavity of ϕ p,q .
Convexity (Necessarily q ≤ 0 or q ≥ 1) To impart convexity on ϕ p,q for β ≤ 2, it is sufficient to show that F (X, A) is convex in the second variable.Since β < 1, it is equivalent to 0 This is satisfied only for q < 0. In conclusion, ϕ p,q is convex for 1 ≤ p ≤ 2 and q ≤ 0. (ϕ is constant for q = 0).
Convexity of F (X, A) in the second variable is equivalent to In conclusion, ϕ p,q is convex for 0 ≤ p ≤ 1 and q ≤ p − 1, or for 2 ≤ p ≤ 3 and p − 1 ≤ q.

(iii) 2 ≤ β
To impart convexity on ϕ p,q we have to show that F (X, A) is (jointly This excludes q < 0 but is satisfied for 1 ≤ q ≤ p − 1.In conclusion, ϕ p,q is convex for 2 ≤ p ≤ 3 and 1 ≤ q ≤ p − 1.
To impart concavity on ϕ p,q for β ≤ 2, we must show that F (X, A) is (jointly which is satisfied for all 0 < q ≤ 1 and 0 ≤ p < 1.In conclusion, ϕ p,q is concave for 0 ≤ p ≤ 1 and 0 ≤ q ≤ 1.
(vi) 2 < β Concavity on ϕ p,q only requires that F (X, A) is concave in the second variable, and since β > 1 this is equivalent to 0 and this is satisfied for 0 < q < p − 1.In conclusion, ϕ p,q is concave for 1 ≤ p ≤ 2 and 0 ≤ q ≤ p − 1.
Note that we under (i), (ii), (iv) and (v) in the preceding analysis used that ϕ p,q is the partial maximisation of F (X, A) over positive definite X, while we in (iii) and (vi) used that ϕ p,q is the partial minimisation of F (X, A).We have thus proved the following theorem.
Theorem 3.1.The trace function ϕ p,q defined in (2) has the following geometric properties depending on the parameters p and q.
(i) ϕ p,q (A) is concave in positive definite A for the parameter values (ii) ϕ p,q (A) is convex in positive definite A for the parameter values The special case q = 1 was proved in [6, Corollary 2.3 ].
4 Extensions to several variables Theorem 4.1.Let H 1 , . . ., H k be matrices with is well-defined in k-tuples of positive definite matrices, and it has the same convexity/concavity properties (in terms of p and q) as it has for one variable.
Proof.Note that if p > 1, then log p (A i ) ≥ −(p − 1) −1 for i = 1, . . ., k and therefore k i=1 since the sum is a convex combination of k terms, possibly including zero, that all are majorising −(p − 1) −1 • I.The sum is thus in the domain of exp p so ϕ p,q is well-defined.A similar argument shows that ϕ p,q is well-defined also for p < 1.We introduce the k × k block matrices with zero matrices of suitable orders inserted and note that Ĥ is a contraction.Furthermore, and the statement follows.
5 Comparison with Υ p,q Carlen and Lieb considered the trace functions (4) Υ p,q (A) = Tr H * A p H q/p for arbitrary parameters p and q, where H is an arbitrary matrix and A is positive definite.Note that ϕ p,q is defined only for contractions H, while Υ p,q is well-defined without any restrictions on H.We may extend Υ p,q to several variables in the same way as we just extended ϕ p,q to several variables and obtain that Υ p,q (A 1 , . . ., A k ) has the same geometric properties as Υ p,q (A).
Theorem 5.1.Let H 1 , . . ., H k be matrices such that and consider the functions ϕ p,q (A 1 , . . ., A k ) and Υ p,q (A 1 , . . ., A k ) defined in positive definite matrices A 1 , . . ., A k .Then ϕ p,q has the same convexity/concavity properties (with respect to p and q) as have Υ p−1,q .
Proof.Choose positive definite matrices A 1 , . . ., A k .For q = 1 we obtain where we used Since the concavity/convexity properties of ϕ p,q and Υ p,q do not depend of the number of variables, we conclude that the concavity/convexity properties of ϕ p,q (A) and Υ p−1,q (A) are the same.We have thus obtained the following result.
Theorem 5.2.The trace function Υ p,q defined in (4) has the following geometric properties depending on the parameters p and q.
(i) Υ p,q (A) is concave in positive definite A for the parameter values Carlen and Lieb [3, Theorem 1.1] proved that the trace functions Υ p,q (A) are concave for 0 ≤ p ≤ 1 and p ≤ q ≤ 1, and convex for 1 ≤ p ≤ 2 and p ≤ q.By the very different method developed in this paper we recovered the previous results of Carlen and Lieb and added a number of more cases possibly exhausting all possible cases.

Variant trace functions
We may also consider trace functions of the form A → (H * A p H) 1/p q = Tr (H * A p H) q/p 1/q = Υ p,q (A) 1/q for arbitrary H, where A is positive definite.Note that • q is a norm for q > 0, and that Υ p,q (A) is homogeneous of degree q.Because of homogeneity it follows from the general theory of convexity that if Υ p,q (A) is convex or concave, so is Υ p,q (A) 1/q .By a general amplification argument we then obtain that the function Φ p,q (A 1 , . . ., A k ) = with zero matrices of suitable orders inserted.Since by calculation Υ p,q ( Â) 1/q = Tr Ĥ * Âp Ĥ q/p 1/q =   Tr k i=1 A p i q/p   1/q , the statement follows.Bekjan [2, Lemma 1] proved that Φ p,q (A 1 , . . ., A k ) is concave for −1 ≤ p ≤ 0 and q = 1.

Declaration of conflict of interest
The author has no conflict of interest.
No datasets were generated or analysed during the current study.
concave for the same parameter values p, q as Υ p,q .Indeed, consider k × k block matrices To do this we use that the functions t → t p are operator concave, if and only if 0 ≤ p ≤ 1, and operator convex, if and only if−1 ≤ p ≤ 0 or 1 ≤ p ≤ 2.It may be of interest to note that the same parameter conditions apply, if we only require matrix convexity or matrix concavity of order two, cf.[4, Proposition 3.1].We also make use of Lieb's concavity theorem and Ando's convexity theorem.Together they state that the trace functions