A New Quantum f-divergence for Trace Class Operators in Hilbert Spaces

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1 Entropy 04, 6, ; doi:0.3390/e65853 OPEN ACCESS entropy ISSN Article A New Quantum f-divergence for Trace Class Operators in Hilbert Spaces Silvestru Sever Dragomir, Mathematics, School of Engineering & Science, Victoria University, PO Box 448, Melbourne City, MC 800, Australia; sever.dragomir@vu.edu.au School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 050, South Africa External Editor: Kevin H. Knuth Received: 3 October 04; in revised form: 30 October 04 / Accepted: 3 November 04 / Published: 6 November 04 Abstract: A new quantum f-divergence for trace class operators in Hilbert Spaces is introduced. It is shown that for normalised convex functions it is nonnegative. Some upper bounds are provided. Applications for some classes of convex functions of interest are also given. Keywords: self-adjoint bounded linear operators; functions of operators; trace of operators; quantum divergence. Introduction Let X, A be a measurable space satisfying A > and µ be a σ-finite measure on X, A. Let P be the set of all probability measures on X, A which are absolutely continuous with respect to µ. For P, Q P, let p = dp dq and q = denote the Radon Nikodym derivatives of P and Q with respect to µ. dµ dµ Two probability measures P, Q P are said to be orthogonal and we denote this by Q P if: P {q = 0} = Q {p = 0} =. Let f : 0,, ] be a convex function that is continuous at zero, i.e., f 0 = lim u 0 f u. In 963, I. Csiszár ] introduced the concept of f-divergence as follows.

2 Entropy 04, Definition. Let P, Q P. Then: I f Q, P = X p x f is called the f-divergence of the probability distributions Q and P. ] q x dµ x, p x Remark. Observe that, the integrand in the formula is undefined when p x = 0. The way to overcome this problem is to postulate for f as above that: ] ] q x 0f = q x lim uf, x X. 0 u 0 u and: For f continuous convex on 0, we obtain the -conjugate function of f by: f u = uf, u 0, u f 0 = lim u 0 f u. It is also known that if f is continuous convex on 0,, then so is f. The following two theorems contain the most basic properties of f-divergences. For their proofs we refer the reader to Chapter of ] see also 3]. Theorem Uniqueness and symmetry theorem. Let f, f be continuous convex on 0,. We have: I f Q, P = I f Q, P, for all P, Q P if and only if there exists a constant c R, such that: for any u 0,. f u = f u + c u, Theorem Range of values theorem. Let f : 0, R be a continuous convex function on 0,. For any P, Q P, we have the double inequality: i If P = Q, then the equality holds in the first part of 3. f I f Q, P f 0 + f 0. 3 If f is strictly convex at one, then the equality holds in the first part of 3 if and only if P = Q; ii If Q P, then the equality holds in the second part of 3. If f 0 + f 0 <, then equality holds in the second part of 3 if and only if Q P. The following result is a refinement of the second inequality in Theorem see Theorem 3 in 3]. A function f defined on 0, is called normalised if f = 0.

3 Entropy 04, Theorem 3. Let f be a continuous convex and normalised function on 0, with f 0 + f 0 <. Then 0 I f Q, P f 0 + f 0] V Q, P 4 for any Q, P P. For other inequalities for f-divergence see 4 5]. We now give some examples of f-divergences that are well-known and often used in the literature see also 3]. The class of χ α -divergences. The f-divergences of this class, which is generated by the function χ α, α,, defined by: χ α u = u α, u 0, have the form: I f Q, P = X p q p α dµ = X p α q p α dµ. 5 From this class only the parameter α = provides a distance in the topological sense, namely the total variation distance V Q, P = q p dµ. The most prominent special case of this class is, however, X Karl Pearson s χ -divergence: χ q Q, P = X p dµ that is obtained for α =. Dichotomy class. From this class, generated by the function f α : 0, R: u ln u for α = 0; f α u = αu + α α α uα ] for α R\ {0, } ; u + u ln u for α = ; only the parameter α = f u = u provides a distance, namely, the Hellinger distance: H Q, P = X q ] p dµ. Another important divergence is the Kullback Leibler divergence obtained for α =, q KL Q, P = q ln dµ. p 3 Matsushita s divergences. The elements of this class, which is generated by the function ϕ α, α 0, ] given by: ϕ α u := u α α, u 0,, are prototypes of metric divergences, providing the distances I ϕα Q, P ] α. 4 Puri Vincze divergences. This class is generated by the functions Φ α, α, given by: X Φ α u := u α α, u 0,. u +

4 Entropy 04, It has been shown in 6] that this class provides the distances I Φα Q, P ] α. 5 Divergences of Arimoto-type. This class is generated by the functions: ] α + u α α α α + u for α 0, \ {} ; Ψ α u := + u ln + u ln u + u ln + u for α = ; u for α =. It has been shown in 7] that this class provides the distances I Ψα Q, P ] minα, α for α 0, and V Q, P for α =. In order to introduce a quantum f-divergence for trace class operators in Hilbert spaces and study its properties we need some preliminary facts as follows.. Trace of Operators Let H,, be a complex Hilbert space and {e i } i I an orthonormal basis of H. We say that A B H is a Hilbert Schmidt operator if: Ae i <. 6 i I It is well know that, if {e i } i I and {f j } j J are orthonormal bases for H and A B H, then: Ae i = Af j = A f j 7 i I j I j I showing that the definition 6 is independent of the orthonormal basis and A is a Hilbert Schmidt operator iff A is a Hilbert Schmidt operator. Let B H the set of Hilbert Schmidt operators in B H. For A B H we define: / A := Ae i 8 i I for {e i } i I an orthonormal basis of H. This definition does not depend on the choice of the orthonormal basis. Using the triangle inequality in l I, one checks that B H is a vector space and that is a norm on B H, which is usually called in the literature as the Hilbert Schmidt norm. Denote the modulus of an operator A B H by A := A A /. Because A x = Ax for all x H, A is Hilbert Schmidt iff A is Hilbert Schmidt and A = A. From 7 we have that if A B H, then A B H and A = A. The following theorem collects some of the most important properties of Hilbert Schmidt operators: Theorem 4. We have: i B H, is a Hilbert space with inner product: A, B := i I Ae i, Be i = i I B Ae i, e i 9

5 Entropy 04, and the definition does not depend on the choice of the orthonormal basis {e i } i I ; ii We have the inequalities: A A 0 for any A B H and: for any A B H and T B H; iii B H is an operator ideal in B H, i.e., AT, T A T A B H B H B H B H ; iv B fin H, the space of operators of finite rank, is a dense subspace of B H; v B H K H, where K H denotes the algebra of compact operators on H. If {e i } i I an orthonormal basis of H, we say that A B H is trace class if: A := i I A e i, e i <. The definition of A does not depend on the choice of the orthonormal basis {e i } i I. We denote by B H the set of trace class operators in B H. The following proposition holds: Proposition. If A B H, then the following are equivalent: i A B H; ii A / B H; ii A or A is the product of two elements of B H. The following properties are also well known: Theorem 5. With the above notations: i We have A = A and A A 3 for any A B H; ii B H is an operator ideal in B H, i.e., iii We have: iv We have: B H B H B H B H ; B H B H = B H ; A = sup { A, B B B H, B } ; v B H, is a Banach space. iv We have the following isometric isomorphisms: B H = K H and B H = B H, where K H is the dual space of K H and B H is the dual space of B H.

6 Entropy 04, We define the trace of a trace class operator A B H to be: tr A := i I Ae i, e i, 4 where {e i } i I an orthonormal basis of H. Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series 4 converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace: Theorem 6. We have: i If A B H, then A B H and: ii If A B H and T B H, then AT, T A B H and: tr A = tr A; 5 tr AT = tr T A and tr AT A T ; 6 iii tr is a bounded linear functional on B H with tr = ; iv If A, B B H, then AB, BA B H and tr AB = tr BA; v B fin H is a dense subspace of B H. Utilising the trace notation, we obviously have that: A, B = tr B A = tr AB and A = tr A A = tr A for any A, B B H. The following Hölder s type inequality has been obtained by Ruskai in 8]: tr AB tr AB tr A /α] α tr B / α] α 7 where α 0, and A, B B H with A /α, B / α B H. In particular, for α = we get the Schwarz inequality: tr AB tr AB tr A ] / tr B ] / 8 with A, B B H. If A 0 and P B H with P 0, then: 0 tr P A A. 9 Indeed, since A 0, then Ax, x 0 for any x H. If {e i } i I an orthonormal basis of H, then: 0 AP / e i, P / e i A P / e i = A P ei, e i for any i I. Summing over i I, we get: 0 AP / e i, P / e i A P e i, e i = A i I i I

7 Entropy 04, and since: AP / e i, P / e i = P / AP / e i, e i = tr P / AP / = tr P A i I i I we obtain the desired result 9. This obviously imply the fact that, if A and B are self-adjoint operators with A B and P B H with P 0, then: tr P A tr P B. 0 Now, if A is a self-adjoint operator, then we know that: Ax, x A x, x for any x H. This inequality follows by Jensen s inequality for the convex function f t = t defined on a closed interval containing the spectrum of A. If {e i } i I is an orthonormal basis of H, then: tr P A = AP / e i, P / e i AP / e i, P / e i i I i I A P / e i, P / e i = tr P A, i I for any A a self-adjoint operator and P B + H := {P B H with P 0}. For the theory of trace functionals and their applications the reader is referred to 9]. For some classical trace inequalities see 0 4], which are continuations of the work of Bellman 5]. For related works the reader can refer to 0,4,6 3]. 3. Classical Quantum f-divergence On complex Hilbert space B H,,, where the Hilbert Schmidt inner product is defined by: U, V := tr V U for A, B B + H consider the operators L A : B H B H and R B : B H B H defined by: L A T := AT and R B T := T B. We observe that they are well defined and since: L A T, T = AT, T = tr T AT = tr T A 0 and: R B T, T = T B, T = tr T T B = tr T B 0 for any T B H, they are also positive in the operator order of B B H, the Banach algebra of all bounded operators on B H with the norm where T = tr T, T B H.

8 Entropy 04, Since tr X = tr X for any X B H, then also: tr T AT = tr T A / A / T A = tr / T A / T A = tr / T = tr A / T T = tr A / for A 0 and T B H. We observe that L A and R B are commutative, therefore the product L A R B is a self-adjoint positive operator on B H for any positive operators A, B B H. For A, B B + H with B invertible, we define the Araki transform A A,B : B H B H by A A,B := L A R B. We observe that for T B H, and we have A A,B T = AT B and: Observe also, by the properties of trace, that: A A,B T, T = AT B, T = tr T AT B. tr T AT B = tr B / T A / A / T B / A = tr / T B / A / T B / A = tr / T B / giving that: A A,B T, T = tr A / T B / 0 for any T B H. Let U be a self-adjoint linear operator on a complex Hilbert space K;,. The Gelfand map establishes a -isometrically isomorphism Φ between the set C Sp U of all continuous functions defined on the spectrum of U, denoted Sp U, and the C -algebra C U generated by U and the identity operator K on K as follows: For any f, g C Sp U and any α, β C we have i Φ αf + βg = αφ f + βφ g; ii Φ fg = Φ f Φ g and Φ f = Φ f ; iii Φ f = f := sup t SpU f t ; iv Φ f 0 = K and Φ f = U, where f 0 t = and f t = t, for t Sp U. With this notation we define: f U := Φ f for all f C Sp U and we call it the continuous functional calculus for a self-adjoint operator U. If U is a self-adjoint operator and f is a real valued continuous function on Sp U, then f t 0 for any t Sp U implies that f U 0, i.e., f U is a positive operator on K. Moreover, if both f and g are real valued functions on Sp U, then the following important property holds: in the operator order of B K. f t g t for any t Sp U implies that f U g U P

9 Entropy 04, Let f : 0, R be a continuous function. Utilising the continuous functional calculus for the Araki self-adjoint operator A Q,P B B H we can define the quantum f-divergence for Q, P S H := {P B H, P 0 with = } and P invertible, by: S f Q, P := f A Q,P P /, P / = tr P / f A Q,P P /. If we consider the continuous convex function f : 0, R, with f 0 := 0 and f t = t ln t for t > 0, then for Q, P S H and Q, P invertible, we have: S f Q, P = tr Q ln Q ln P ] =: U Q, P, which is the Umegaki relative entropy. If we take the continuous convex function f : 0, R, f t = t for t 0, then for Q, P S H with P invertible, we have: S f Q, P = tr P / A Q,P B H P / = tr Q P =: V Q, P, where V Q, P is the variational distance. If we take f : 0, R, f t = t for t 0, then for Q, P S H with P invertible, we have: S f Q, P = tr Q P =: χ Q, P, 3 which is called the χ -distance. Let q 0, and define the convex function f q : 0, R by f q t = tq q. Then: S fq Q, P = tr Qq P q, q which is Tsallis relative entropy., If we consider the convex function f : 0, R by f t = t then: S f Q, P = tr Q / P / =: h Q, P, which is known as Hellinger discrimination. If we take f : 0, R, f t = ln t, then for Q, P S H and Q, P invertible, we have: S f Q, P = tr P ln P ln Q] = U P, Q. In the important case of finite dimensional space H and the generalized inverse P, numerous properties of the quantum f-divergence have been obtained in the recent papers 33 36] and the references therein. We omit the details.

10 Entropy 04, A New Quantum f-divergence In order to simplify the writing, we denote by S H the set of all density operators which are elements of B + H having unit trace. We observe that, if P, Q are self-adjoint with P, Q 0 and P is invertible, then P QP 0. Let f : 0, R be a continuous convex function on 0,. We can define the following new quantum f-divergence functional: ] D f Q, P := tr P f P QP D for Q, P S H with P invertible. The definition can be extended for any continuous function. If we take the convex function f t = t, t 0, then we get: ]] D f Q, P := tr P P QP = tr Q P =: χ Q, P, for Q, P S H with P invertible, which is the Karl Pearson s χ -divergence version for trace class operators. This divergence is the same as the one generated by the classical f-divergence, see 3. More general, if we take the convex function f t = t n, t 0 and n a natural number with n, then we get: n ]] D f Q, P := tr P P QP = tr Q QP n =: D χ n Q, P for Q, P S H with P invertible. If we take the convex function f t = t ln t for t > 0 and f 0 := 0, then we get: ]] D f Q, P = tr P P QP ln P QP ] = tr P QP ln P QP =: D KL Q, P for Q, P S H with P and Q invertible. We observe that this is not the same as Umegaki relative entropy introduced above. If we take the convex function f t = ln t for t > 0, then we get: ] D f Q, P = tr P ln P QP ] = tr P ln P Q P = tr ] P ln P QP =: D KL Q, P for Q, P S H with P and Q invertible. If we take the convex function f t = t, t 0, then we get: ] D f Q, P = tr P P QP H =: D V Q, P for Q, P S H with P invertible.

11 Entropy 04, If we consider the convex function f t =, t > 0, then: t ]] D f Q, P = tr P P QP H = tr P ] P Q P = χ P, Q for Q, P S H with P and Q invertible. If we take the convex function f t = f q t = tq, q 0,, then we get: q D fq Q, P = q ] tr P P QP, q which is different, in general, from the Tsallis relative entropy introduced above. Other examples may be considered by taking the convex functions from the introduction. The details are omitted. Suppose that I is an interval of real numbers with interior I and f : I R is a convex function on I. Then, f is continuous on I and has finite left and right derivatives at each point of I. Moreover, if x, y I and x < y,, then f x f + x f y f + y which shows that both f and f + are nondecreasing function on I. It is also known that a convex function must be differentiable except for at most countably many points. For a convex function f : I R, the subdifferential of f denoted by f is the set of all functions ϕ : I, ], such that ϕ I R and: f x f a + x a ϕ a for any x, a I. 4 It is also well known that if f is convex on I, then f is nonempty, f, f + f and if ϕ f, then: f x ϕ x f + x for any x I. In particular, ϕ is a nondecreasing function. If f is differentiable and convex on I, then f = {f }. Theorem 7. Let f be a continuous convex function on 0, with f = 0. Then, we have: for any Q, P S H with P invertible. If f is continuously differentiable on 0,, then we also have: 0 D f Q, P 5 D f Q, P D f Q, P D f Q, P. 6 Proof. For any x 0, we have from the gradient inequality 4 that: f x f + x f + and since f is normalised, then: f x x f +. 7

12 Entropy 04, Utilising the property P for the positive operator A = P QP where Q, P S H with P invertible, then we have the inequality in the operator order: f P QP P QP f +. 8 Utilising the property 0 for the inequality 8, we have: ] ] tr P f P QP tr P P QP f + ] = f + tr P P QP = f + tr Q ] = 0 and the inequality 5 is proven. From the gradient inequality, we also have for any x 0: x f x + f f x and since f is normalised, then: x f x f x which, as above, implies that: P QP H f P QP f P QP. 9 Making use of the property 0 for the inequality 9, then we get: ] tr P P QP H f P QP tr which is the required inequality 6. P f Remark. If we take f t = ln t, t > 0 in Theorem 7, then we get: ] P QP, 30 0 D KL Q, P χ P, Q 3 for any Q, P S H with P and Q invertible. If we take the convex function ε t = e t, then: ] D ε Q, P = tr P exp P QP H, where Q, P S H with P invertible. By Theorem 7, we get: ] 0 D ε Q, P tr P P QP H exp P QP H, 3 where Q, P S H with P invertible. The inequality in 3 is equivalent to: 0 D ε Q, P ] tr P QP exp P QP H where Q, P S H with P invertible. ] +,

13 Entropy 04, The following lemma is of interest in itself: Lemma. Let S be a self-adjoint operator such that γ H S Γ H for some real constants Γ γ. Then, for any P B + H \ {0}, we have: 0 tr P S Γ γ tr tr P S 33 P tr P S S H ] / tr P S 4 Γ γ. Γ γ tr P S Proof. Observe that: tr P = tr Γ + γ = tr P S S Γ + γ tr P S H S H tr P S P S S H tr tr P S P S H tr P S 34 since, obviously: Now, since γ H S Γ H, then: tr P S tr P S H = 0. S Γ + γ H Γ γ. Taking the modulus in 34 and using the properties of trace, we have: tr P S = tr which proves the first part of 33. tr P S 35 P tr S Γ + γ tr P S H S H P S Γ + γ tr P S H S H P tr P S S H, Γ γ tr

14 Entropy 04, By Schwarz inequality for trace, we also have: tr P tr P S S H ] / tr tr P S P S H ] tr P S / tr P S =. 36 From 35 and 36, we get: tr P S tr P S Γ γ tr P S ] / tr P S, which implies that: By 36, we then obtain: tr P S tr P tr P S ] / tr P S Γ γ. tr P S S H tr P S ] / Γ γ that proves the last part of 33. Corollary. Let Q, P S H with P invertible and such that there exists 0 < r R satisfying the condition 38. Then: 0 χ Q, P D V Q, P χ Q, P Proof. Utilising the inequality 33 for S = P QP, we have: 0 χ Q, P P tr P QP H and the inequality 37 is proved. χ Q, P 4, We observe that if Q, P S H with P invertible and there exists r, R > 0 with: r H P QP RH, 38

15 Entropy 04, then by the property 0, we get: r tr P P QP R showing that r R. The following result provides a simple upper bound for the quantum f-divergence D f Q, P. Theorem 8. Let f be a continuous convex function on 0, with f = 0. Then, we have: 0 D f Q, P f R f + r ] D V Q, P 39 f R f + r ] χ Q, P 4 f R f + r ] for any Q, P S H with P invertible and satisfying the condition 38. Proof. Without loosing the generality, we prove the inequality in the case when f is continuously differentiable on 0,. We have: ]] tr P P QP H f P QP λ H ] = tr P P QP H f P QP for any λ R and for any Q, P S H with P invertible. Since f is monotonic nondecreasing on r, R], then: f + r f x f R for any x r, R]. This implies in the operator order that: f + r H f P QP f R H, therefore: f P QP 40 f R + f + r H f R f + r ] H. 4 From 30 and 40, we have: ] ] 0 tr P f P QP tr P P QP H f P QP = tr P P QP H f P QP f ]] R + f + r H = P P tr QP H f P QP f ]] R + f + r H tr P P QP H f P QP f ] ] R + f + r H = tr P P QP H f P QP f R f + r ] tr P = f R f + r ] V Q, P, P QP H ] f ] R + f + r H

16 Entropy 04, which proves the first inequality in 39. The rest follows by 37. Example. If we take f t = ln t, t > 0 in Theorem 8, then we get: 0 D KL Q, P rr D V Q, P 4 χ Q, P rr 4rR for any Q, P S H with P, Q invertible and satisfying the condition: r H P QP RH, 43 with r > 0. With the same conditions as in for Q, P and if we take f t = t ln t, t > 0 in Theorem 8, then we get: 0 D KL Q, P R ln D V Q, P 44 r R ln χ Q, P R r 4 ln. r 3 If we take in 39 f t = f q t = tq, then we get: q q R q r q 0 D fq Q, P q R q r q q R q r q χ Q, P q R q r q q R q r q 4 q R q r q provided that Q, P S H, with P, Q invertible and satisfying the condition 43. We have the following upper bound, as well: V Q, P 45 Theorem 9. Let f : 0, R be a continuous convex function that is normalized. If Q, P S H, with P invertible, and there exists R r 0 such that the condition 38 is satisfied, then: 0 D f Q, P Proof. By the convexity of f, we have: R t r + t r R f t = f R f r + r f R. 46 R t f r + t r f R for any t r, R]. This inequality implies the following inequality in the operator order of B H: R H P f P QP QP f r + P QP r H f R, 47

17 Entropy 04, for Q, P S H, with P invertible, and R r 0 such that the condition 38 is satisfied. Utilising the property 0, we get from 47 that: ] tr P f P QP f r ] tr P R H P QP + f R ] tr P P QP rh R f r + r f R =, and the inequality 46 is thus proven. 48 Remark 3. If we take in 46 f t = t, then we get: 0 χ Q, P R r R + r + for Q, P S H, with P invertible and satisfying the condition 38. If we take in 46 f t = t ln t, then we get the inequality: ] 0 D KL Q, P ln r R r R r R r R R r provided that Q, P S H, with P, Q invertible and satisfying the condition 38. With the same assumptions for P, Q, if we take in 46 f t = ln t, then we get the inequality: 0 D ] KL Q, P ln r R r R r R R r Further Upper Bounds We also have: Theorem 0. Let f : 0, R be a continuous convex function that is normalized. If Q, P S H, with P invertible, and there exists R > > r 0 such that the condition 38 is satisfied, then: 0 D f Q, P R r where Ψ f ; r, R : r, R R is defined by: R r Ψ f ; r, R 5 sup Ψ f t; r, R t r,r R r f R f + r 4 f R f + r ] Ψ f t; r, R = f R f t R t f t f r. 53 t r

18 Entropy 04, We also have: 0 S f Q, P 4 Ψ f ; r, R R r Ψ f ; r, R 54 4 sup Ψ f t; r, R t r,r 4 f R f + r ]. Proof. By denoting: we have: f t; r, R := t r f R + R t f r f t, t r, R] t r f R + R t f r f t f t; r, R = t r f R + R t f r T t + t r f t = t r f R f t] R t f t f r] = M m R t t r = Ψ f t; r, R for any t r, R. From the proof of Theorem 9 and since f = 0, we have: ] tr P f P QP R f r + r f R f R r = f ; r, R = Ψ f ; r, R for any Q, P S H, with P invertible, and R > > r 0 such that the condition 38 is valid. Since: 55 Ψ f ; r, R sup t r,r Ψ f t; r, R 56 ] f R f t f t f r = sup t r,r R t t r ] ] f R f t f t f r sup + sup t r,r R t t r,r t r ] ] f R f t f t f r = sup inf t r,r R t t r,r t r = f R f + r, and, obviously: R r, 57 4 then by 55 57, we have the desired result 5. The rest is obvious.

19 Entropy 04, Remark 4. If we consider the convex normalized function f t = t, then: Ψ f t; r, R = R t R t and we get from 5 the simple inequality: t r t r =, t r, R 0 χ Q, P R r 58 for Q, P S H, with P invertible and satisfying the condition 38, which is better than 49. If we take the convex normalized function f t = t, then we have: Furthermore: Ψ f t; r, R = R t R t t r t r D f Q, P = χ P, Q. =, t r, R]. rrt Using 5, we get: 0 χ R r P, Q Rr for Q, P S H, with Q invertible and satisfying the condition 38. If we consider the convex function f t = ln t defined on r, R] 0,, then: Then, by 5, we have: ln R + ln t ln t + ln r Ψ f t; r, R = R t t r ln t R t ln r t r ln R = M t t m t R r R tt r = ln, t r, R. r R t M t r 0 D ] KL Q, P ln r R r R r R R r R r rr for Q, P S H, with P, Q invertible and satisfying the condition 38. If we consider the convex function f t = t ln t defined on r, R] 0,, then: which gives that: Using 5, we get: Ψ f t; r, R = R ln R t ln t R t Ψ f ; r, R = R ln R R r ln r r. 0 D KL Q, P ln R rr R r t ln t r ln r, t r, R, t r ] Rr r R r R r ln R r ] R r for Q, P S H, with P, Q invertible and satisfying the condition 38.

20 Entropy 04, We also have: Theorem. Let f : 0, R be a continuous convex function that is normalized. If Q, P S H, with P invertible, and there exists R > > r 0 such that the condition 38 is satisfied, then: 0 D f Q, P { R max, r } ] f r + f R r + R f ] f r + f R r + R f. 6 Proof. We recall the following result see for instance 37] that provides a refinement and a reverse for the weighted Jensen s discrete inequality: ] n min {p n n i} f x i f x i 63 i {,...,n} n n i= i= n n p i f x i f p i x i P n P i= n i= ] n max {p n n i} f x i f x i, i {,...,n} n n i= where f : C R is a convex function defined on the convex subset C of the linear space X, {x i } i {,...n} C are vectors and {p i } i {,...n} are nonnegative numbers with P n := n i= p i > 0. For n =, we deduce from 63 that: ] f x + f y x + y min {s, s} f 64 i= sf x + s f y f sx + s y ] f x + f y x + y max {s, s} f for any x, y C and s 0, ]. Now, if we use the second inequality in 64 for x = r, y = R, s = R t with t r, R], then we have: R r for any t r, R]. R t f r + t r f R f t 65 { R t max, t r } ] f r + f R r + R f ] f r + f R r + R f

21 Entropy 04, This implies that: and the proof is completed. ] tr P f P QP R f r + r f R { R max, r } ] f r + f R r + R f ] f r + f R r + R f Remark 5. If we take in 6 f t = t, then we have: 0 χ P, Q max {R, r} rr r + R for Q, P S H, with P invertible and satisfying the condition 38. If we take in 6 f t = ln t, then we have: { 0 D R KL Q, P max, r } R + r ln 4rR R + r ln 4rR for Q, P S H, with P invertible and satisfying the condition 38. From 4, we have the following absolute upper bound: 0 D KL Q, P 4rR for Q, P S H, with P invertible and satisfying the condition 38. Utilising the elementary inequality ln x x, x > 0, we have that: R + r ln 4rR 4rR, 68 which shows that 67 is better than Conclusions In this paper we have introduced a new quantum f-divergence for trace class operators in Hilbert Spaces. It is shown that for normalised convex functions it is nonnegative. Some upper bounds are provided. Applications for some classes of convex functions of interest are also given. Acknowledgments The author would like to thank the anonymous referees for valuable comments that have been implemented in the final version of the paper.

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