Some quantum f-divergence inequalities for convex functions of self-adjoint operators

: In this paper, some new inequalities for convex functions of self-adjoint operators are obtained. As applications, we present some inequalities for quantum f -divergence of trace class operators in Hilbert Spaces.


Introduction
Let (X, A) be a measurable space satisfying |A| > 2 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 dµ and q = dQ dµ denote the Radon-Nikodym derivatives of P and Q with respect to µ.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}) = 1.
In 1963, I. Csiszár [6] introduced the concept of f -divergence as follows.
Definition 1.1.Let P, Q ∈ P. Then is called the f -divergence of the probability distributions Q and P. Remark 1.2.Observe that, the integrand in the formula (1.1) is undefined when p (x) = 0.The way to overcome this problem is to postulate for f as above that 0f q (x) 0 = q (x) lim 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 ∈ [1, ∞), defined by have the form From this class only the parameter α = 1 provides a distance in the topological sense, namely the total variation distance V (Q, P ) = X |q − p| dµ.The most prominent special case of this class is, however, Karl Pearson's χ 2 -divergence that is obtained for α = 2.

Dichotomy Class
From this class, generated by the function 2 provides a distance, namely, the Hellinger distance .
Another important divergence is the Kullback-Leibler divergence obtained for α = 1, KL (Q, P ) = X q ln q p dµ.

Divergences of Arimoto-type
This class is generated by the functions It has been shown in [19] that this class provides the distances [I Ψα (Q, P )] The following two theorems contain the most basic properties of f -divergences.For their proofs we refer the reader to Chapter 1 of [17] (see also [3]).
Theorem 1.3 (Uniqueness and Symmetry Theorem).Let f, f 1 be continuous convex on [0, ∞).We have for all P, Q ∈ P if and only if there exists a constant c ∈ R such that For any P, Q ∈ P, we have the double inequality (i) If P = Q, then the equality holds in the first part of (1.4).
If f is strictly convex at 1, then the equality holds in the first part of (1.4) if and only if P = Q; (ii) If Q ⊥ P, then the equality holds in the second part of (1.4).
If f (0) + f * (0) < ∞, then equality holds in the second part of (1.4) if and only if Q ⊥ P.
The following result is a refinement of the second inequality in Theorem 1.4 (see [3,Theorem 3]).
For other inequalities for f -divergence see [2], [7]- [12] and [13] Motivated by the above results, in this paper we obtain some new inequalities for quantum fdivergence of trace class operators in Hilbert spaces.It is shown that for normalised convex functions it is nonnegative.Some upper bounds for quantum f -divergence in terms of variational and χ 2 -distance are provided.Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.
In what follows we recall some facts we need concerning the trace of operators and quantum f -divergence for trace class operators in infinite dimensional complex Hilbert spaces.

Some inequalities for convex functions of self-adjoint operators
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 , 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.
Let A be a self-adjoint operator on the complex Hilbert space (H, •, • ) with Sp(A) ⊆ [m, M ] for some real numbers m < M and let {E λ } λ be its spectral family.Then for any continuous function f : [m, M ] → R, it is known that we have the following spectral representation in terms of the Riemann-Stieltjes integral (see for instance [14, p. 257]): for any x, y ∈ H.
Proof.Without loosing the generality, we can assume that f is differentiable on I.By the convexity of f on I, we have for any s ∈ [m, M ] and x ∈ H. Using the spectral representation theorem for self-adjoint operators we get for any s ∈ [m, M ] and x ∈ H. Now if we integrate (2.2) over s on [m, M ] and divide by M − m, we get However, we know that Since f (t) = t −1 is convex function for t > 0, by above theorem we have which is equivalent to for any s ∈ [m, M ] and x ∈ H.
Integrating above inequality over s on [m, M ] with integraler F s x, x we then get which is equivalent to We know that On the other hand, we have for all x ∈ H.We obtain the desired result.✷ Remark 2.4.Let B = n j=1 α j X j in above theorem and α j ≥ 0, j ∈ {1, . . ., n} such that n j=1 α j = 1, then we have Multiply above inequality by α k and summing on k ∈ {1, . . ., n}, we get Equivalently, which is a Jensen type inequality.

Some quantum f -divergence inequalities for convex functions
Let (H, •, • ) be a complex Hilbert space and {e i } i∈I an orthonormal basis of H.We say that It is well know that, if {e i } i∈I and {f j } j∈J are orthonormal bases for H and showing that the definition (3.1) is independent of the orthonormal basis and A is a Hilbert-Schmidt operator iff A * is a Hilbert-Schmidt operator.Let B 2 (H) the set of Hilbert-Schmidt operators in B (H) .For A ∈ B 2 (H) we define 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 2 (I) , one checks that B 2 (H) is a vector space and that • 2 is a norm on B 2 (H) , which is usually called in the literature as the Hilbert-Schmidt norm.
Denote the modulus of an operator A ∈ B (H) by We define the trace of a trace class operator A ∈ B 1 (H) to be 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 (3.4) converges absolutely and it is independent from the choice of basis.
Utilising the trace notation we obviously have that we obtain the desired result (3.5).
This obviously imply the fact that, if A and B are self-adjoint operators with A ≤ B and P ∈ B 1 (H) with P ≥ 0, then Tr (P A) ≤ Tr (P B) .
(3.6) for any A a self-adjoint operator and P ∈ B + 1 (H) := {P ∈ B 1 (H) with P ≥ 0} .For the theory of trace functionals and their applications the reader is referred to [22].
On complex Hilbert space (B 2 (H) , •, • 2 ) , where the Hilbert-Schmidt inner product is defined by We observe that they are well defined and since for any T ∈ B 2 (H) , they are also positive in the operator order of B (B 2 (H)) , the Banach algebra of all bounded operators on B 2 (H) with the norm • 2 where for A ≥ 0 and T ∈ B 2 (H) .We observe that L A and R B are commutative, therefore the product L A R B is a self-adjoint positive operator in B (B 2 (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 : Observe also, by the properties of trace, that for any T ∈ B 2 (H) .We observe that, by the definition of operator order and by (3.8) we have r1 B2(H) ≤ A A,B ≤ R1 B2(H) for some R ≥ r ≥ 0 if and only if for any T ∈ B 2 (H) .
We also notice that a sufficient condition for (3.9) to hold is that the following inequality in the operator order of B (H) is satisfied for any T ∈ B 2 (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 1 K on K as follows: For any f, g ∈ C (Sp (U )) and any α, β ∈ C we have (iv) Φ (f 0 ) = 1 K and Φ (f 1 ) = U, where f 0 (t) = 1 and f 1 (t) = t, for t ∈ Sp (U ) .
With this notation we define 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) .
Let f : [0, ∞) → R be a continuous function.Utilising the continuous functional calculus for the Araki self-adjoint operator A Q,P ∈ B (B 2 (H)) we can define the quantum f -divergence for Q, P ∈ D 1 (H) := {P ∈ B 1 (H) , P ≥ 0 with Tr (P ) = 1 } and P invertible, by 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 ∈ D 1 (H) and Q, P invertible we have which is the Umegaki relative entropy.
If we take the continuous convex function f : [0, ∞) → R, f (t) = |t − 1| for t ≥ 0 then for Q, P ∈ D 1 (H) with P invertible we have Let q ∈ (0, 1) and define the convex function which is Tsallis relative entropy.

If we consider the convex function
which is known as Hellinger discrimination.
Quantum f -divergence Inequalities for Self-adjoint Operators

137
If we take f : (0, ∞) → R, f (t) = − ln t then for Q, P ∈ D 1 (H) and Q, P invertible we have The reader can obtain other particular quantum f -divergence measures by utilizing the normalized convex functions from Introduction, namely the convex functions defining the dichotomy class, Matsushita's divergences, Puri-Vincze divergences or divergences of Arimoto-type.We omit the details.
In the important case of finite dimensional space H and the generalized inverse P −1 , numerous properties of the quantum f -divergence, mostly in the case when f is operator convex, have been obtained in the recent papers [15], [16], [20], [21] and the references therein.
In the following theorem we apply the same proof of Theorem 2.3.
for T ∈ B 2 (H).In the following theorem we apply the same proof of Theorem 2.1.From above inequality we have where P, Q, U, V ∈ D 1 (H) and P, U are invertible.Equivalently, we have

Theorem 2 . 1 .
Let A be a self-adjoint operator on the complex Hilbert space (H, •, • ) and Sp((A) ⊆ [m, M ] ⊂ I where I is an interval.If the function f :

Theorem 2 . 3 .
Let A and B be two self-adjoint operators on the Hilbert space (H, •, • ) with Sp(A), Sp(B) ⊆ [m, M ] ⊂ I and f : I → R a continuously differentiable convex function on I. Then

A, B 2 = 2 =
Tr (B * A) = Tr (AB * ) and A 2 2 = Tr (A * A) = Tr |A| 2 for any A, B ∈ B 2 (H) .If A ≥ 0 and P ∈ B 1 (H) with P ≥ 0, then 0 ≤ Tr (P A) ≤ A Tr (P ) .(3.5)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 1/2 e i , P 1/2 e i ≤ A P 1/2 e i A P e i , e i for any i ∈ I. Summing over i ∈ I we get 0 ≤ i∈I AP 1/2 e i , P 1/2 e i ≤ A i∈I P e i , e i = A Tr (P ) and since i∈I AP 1/2 e i , P 1/2 e i = i∈I P 1/2 AP 1/2 e i , e i = Tr P 1/2 AP 1/2 = Tr (P A) for A, B ∈ B + (H) consider the operators L A : B 2 (H) → B 2 (H) and R B : B 2 (H) → B 2 (H) defined by L A T := AT and R B T := T B.

Theorem 3 . 3 . 2 ≥ 1 2 and S = U 1 2
Let A Q,P and B V,U be two Araki self-adjoint operators on B(B 2 (H)) with Sp(A Q,P ), Sp(B V,U ) ⊆ [m, M ] ⊂ I and f : I → R a continuously differentiable convex function on I. Thenf (A Q,P )T, T 2 S 2 − f (B V,U )S, S 2 T A Q,P T, T 2 f ′ (B V,U ) − T 2 f ′ (B V,U )B V,U S, S2for any x, y ∈ H.Remark 3.4.Let T = P in above theorem we getf (A Q,P )P ′ (B V,U ) − f ′ (B V,U )B V,U U 13) since Q ∈ D 1 (H).Put f (t) = t 2 − 1 for t ≥ 0 in above inequality (3.13) we getχ 2 (Q, P ) − χ 2 (V, U ) ≥ 2 Tr (B V,U U − B V,U V ) 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 i∈I|A| P 1/2 e i , P 1/2 e i = Tr (P |A|) ,