Some applications of matrix inequalities in Rényi entropy

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1 Some applications of matrix inequalities in Rényi entropy arxiv: v [math-ph] Aug 206 Hadi Reisizadeh, and S. Mahmoud Manjegani 2, Department of Electrical and Computer Engineering, 2 Department of Mathematical Sciences, Isfahan University of Technology Abstract The Rényi entropy is one of the important information measures that generalizes Shannon s entropy. The quantum Rényi entropy has a fundamental role in quantum information theory, therefore, bounding this quantity is of vital importance. Another important quantity is Rényi relative entropy on which Rényi generalization of the conditional entropy, and mutual information are defined based. Thus, finding lower bound for Rényi relative entropy is our goal in this paper. We use matrix inequalities to prove new bounds on the entropy of type β, Rényi entropy. Introduction There are several entropic quantities belonging to the family of β-entropies entropies that have been shown to be useful. The entropies of type β are defined by means of information functions [3]. Recall that a real function f defined on [0,] is an information function if it satisfies the boundary conditions and the functional equation hadi.reisizadeh@gmail.com manjgani@cc.iut.ac.ir f0 = f; f =,. 2 y fx+ xf x = fy+ yf x y,

2 2 for all x,y D, where D = {x,y; 0 x,0 y,and x+y }. Let f be an information function and p,p 2,,p n be an finite discrete probability distribution. Then in [3] the entropy of the distribution p,p 2,,p n with respect to f is defined by H f n p,p 2,,p n = i=2 s i f p i s i, s i = p + +p i ; i = 2,,n. Definition.. [3] Let β be a positive number. We call the real function defined in [0,] an information function of type β if it satisfies the boundary condition. and the function equation for all x,y D. fx+ x β y f x = x fy+ yβ f y, The entropy of type β of a probability distribution p,p 2,,p n is defined by H β np,p 2,,p n = i=2 where f is an information function of type β. s β i fp i s i, s i = p + +p i ; i = 2,,n. Theorem.2. [3] Let f be an information function of type β with β. Then fx = 2 β [x β + x β ], for all x [0,]. Theorem.3. [3] Let β be a positive number β with β. Then we have for the entropy of type β of a probability distribution p,p 2,,p n, Hnp β,p 2,,p n = 2 β p β i..2 The Shannon s entropy H n p,p 2,,p n is the limit function of Hn βp,p 2,,p n, when β. For a positive number β with β, Rényi [7] has extended the concept of Shannon s entropy by defining the entropy of order β of a probability distribution p,p 2,,p n as βh n p,p 2,...,p n = β log 2 p β i..3

3 3 Let H be a complex Hilbert space with dimension n. The set of linear operators on H is denoted by M n C, where M n C is the set the C -algebra of all n by n complex matrices. The conjugate transpose of matrix A M n C is denoted by A or A. If A = A, then Ais called Hermitian matrix. A Hermitian or self adjoint matrix A M n C is called positive semi definite resp. positive definite if Ax,x 0 resp. Ax,x > 0 for each x C n. The set M n + C of all positive semi definite matrices is then a closed convex cone in M n C and makes the set of all Hermitian matrices partially ordered: for Hermitian matrices A and B, A B if and only if B A M n + C [2]. Lemma.4. If A 0, then for all r R;A r 0. Proof. It follows from definition of positive semidefinite matrix. Lemma.5. If A and B are positive semidefinite matrices, then 0 trab tratrb. Proof. This is an immediate result of the Von Neumann s trace inequality [5]. Lemma.6. Let A and B be n by n positive semidefinite matrices, then Equality holds if and only if B 2AB 2 = ci ; c R +. ndetadetb n trab..4 Proof. B is a positive semidefinite matrix, therefore it has a unique square root. Also, for square matrices X and Y, the trace of XY is equal to the trace of YX. It follows that the inequality.4 can be rewritten as detb trb 2AB 2 2 AB 2 n. n Assumingthatλ,λ 2,,λ n R + 0 aretheeigenvaluesofb 2AB 2,byarithmetic-geometric mean inequality we have detb 2 AB 2 n = n λ λ 2 λ n n λ i n = trb 2AB 2 n Moreover, equality holds in the arithmetic-geometric mean inequality if and only if λ = λ 2 = = λ n which implies equality holds in.4, if B 2AB 2 = λ I. It is clear that B 2AB 2 = ci for some c R +, then ndetadetb n = trab..

4 4 Lemma.7. If A is a positive definite matrix, then Equality holds if and only if A = I. tri A logdeta tra I..5 Proof. Let λ,λ 2,,λ n R + 0 be the eigenvalues of A. Then by functional calculus theorem, the inequality.5 can be rewritten as or, by setting λ i = e u i, λ i logλ i e u i u i λ i, e u i, that follows from the convexity of the exponential function. Equality holds if and only if u i = 0 for every i n implying λ i = for every i n. Thus, equality holds in.5 if and only if A = I. Definition.8. An operator ρ on a finite dimensional Hilbert space H, is called density operator if it satisfies the following three requirements: i ρ is Hermitian, ii trρ =, and iii ρ is a positive semi-definite operator. 2 Entropy of type β and Rényi entropy First, we prove some results about the Rényi entropy of order β. For n, we denote { } n = p,p 2,...,p n : p i 0, p i =. Theorem 2.. For all p,p 2,...,p n n and positive real number β,. if 0 < β <, then βh n p,p 2,...,p n β log 2 + β n n 0 p i.

5 5 2. if β >, then βh n p,p 2,...,p n β log 2 + β n n 0 p i, where p i {p i n p i > 0} and n 0 is the number of p i that p i = 0. Proof. We have βh n p,p 2,..,p n = β log 2 n pβ i = β log 2 Therefore, for 0 < β <, we get p β i = β log 2 +log 2 n n0 βh n p,p 2,..,p n β log 2 + and for β >, we get p β i. β log 2 p i = β log 2 + β n n0 log 2 p i, βh n p,p 2,..,p n β log 2 + β log 2 p i = β log 2 + β n n0 log 2 p i. From.2 and.3 we have the following relations between the entropy of order β and the entropy of type β [3]: ] βh n = β log 2 [2 β Hn β + Theorem 2.2. If 0 < β <, then for all p,p 2,...,p n n we have n n 0 Hn β p 0 ],p 2,...,p n 2 β [n n 0 p i β, where p i {p i n p i > 0} and n 0 is the number of p i that p i = 0. 2.

6 6 Proof. For 0 < β < from.3, we have: [ ] β log 2 2 β Hn β + β log 2 + β n n 0 log n n 2 p i 0 ] [ n n 0 log 2 [2 β Hn β + log 2 p i β ], which implies Thus, n n 0 2 β Hn β + p i β. n n 0 ] Hnp β,p 2,...,p n 2 β [n n 0 p i β. Product of non-zero probabilities of distribution identifies upper bound, so it is an important criteria for limiting value of entropy of type β. 3 Some bounds on the quantum Rényi entropy In this section, we re going to extend Theorem 2. to quantum setting. In [] the Rényi entropy in quantum setting of order α 0,, is given as H α ρ = α logtrρα. Theorem 3.. Let ρ S with rankρ = d. Then,. if 0 < α <, H α ρ log 0 + α 0 log λ i. α 0 2. if α >, H α ρ log 0 + α 0 log λ i, α 0 where λ i {λ i σρ λ i > 0} and d 0 is the number of λ i that λ i = 0.

7 7 Proof. For any density matrix ρ, there exist unitary matrix U and diagonal matrix D such that D = UρU. Also, D S. By property of quantum entropy, we have where and λ i σρ. By Theorem 2., we get Thus, α log α log H α D = H α UρU = H α ρ, H α D = α logtrdα = α log λ λ D = λ d λ i α, λ α i log 0 + α 0 logλ i α 0 λ α i log 0 + α 0 α 0 H α D log 0 + α 0 logλ i α 0 H α D log 0 + α 0 logλ i α 0 Also, H α D = H α ρ. Theorem 3.2. Let ρ S with rankρ = d. Then, H α ρ logd. Proof. By the convexity of t α α >, we obtain Thus, for α >, λ α i = d d λ α i d d H α ρ = H α D = α log logλ i λ i α = d α. if 0 < α <, if α >. if 0 < α <, if α >. λ i α α logd α = logd

8 8 From concavity of t α 0 < α <, we obtain Thus, for 0 < α <, λ α i = d d λ α i d d H α ρ = H α D = α log λ i α = d α λ i α α logd α = logd. Corollary 3.3. Let ρ S with rankρ = d and 0 < α <. Then, log 0 + α 0 log λ i H α ρ logd, α 0 where λ i {λ i σρ λ i > 0} and d 0 is the number of λ i that λ i = 0. This theorem provides a lower bound which does not depend on the off-diagonal elements called as coherences in[4], and it also shows that product of non-zero probabilities of finding the system in the respective states gives us certain least value of quantum Rényi entropy. 4 The Rényi relative entropy, conditional entropy, and mutual information Definition 4.. [6] The Rényi relative entropy of α 0 is defined as D α ρ σ = α logtr [ ρ α σ α]. Theorem 4.2. For Rényi relative entropy of α >, we have D α ρ σ α where d is the dimension of H. logd+ α α logdetρ+ d d log detσ, Proof. Since ρ and σ are positive definite matrices, by Lemma.4, ρ α and σ α are also positive definite. trρ α σ α d detρ α detσ α d logtrρ α σ α logd+ d logdetρα + d logdetσ α.

9 9 Since α >, α logtrρα σ α logd+ α d logdetρα + d logdetσ α. Thus, D α ρ σ logd+ α α logdetρ+ log detσ. α d d Definition 4.3. [] From Rényi relative entropy, we define Rényi generalization of entropy, conditional entropy, and mutual information in analogy respectively with the below formulations; H α A ρ = D α ρ A I A, where µ A = dimh A I A. H α A B ρ := logdimh A min σ B D α ρ AB µ A σ B, I α A;B ρ := min σ B D α ρ AB ρ A σ B, Theorem 4.4. For Rényi generalization of conditional entropy and mutual information α >, if there are c,c 2 R + and σ B H B that µ α A ρ α A σ α B = c 2 ρ α AB, then we have H α A B ρ = logd A α I α A;B ρ = α σ α B = c ρ α AB, logd A d B + 2α d A d B logdetρ AB +logc logd A d B + 2α d A d B logdetρ AB +logc 2 Proof. From Theorem 4.2, D α ρ σ logd+ α α logdetρ+ log detσ. α d d Equality holds if and only if σ α = cρ α ; c R +. So, if there are c,c 2 R + and σ B H B that µ α A σ α B = c ρ α AB, ρ α A σ α B = c 2 ρ α AB for α >, then min σ B D α ρ AB µ A σ B = Therefore, α = α logd A d B + α logdetρ AB + logdetcρ α AB d A d B d A d B logd A d B + 2α logdetρ AB +logc. d A d B H α A B ρ = logd A logd A d B + 2α logdetρ AB +logc. α d A d B

10 0 Similarly, we have these conclusion for I α A;B ρ : I α A;B ρ = logd A d B + 2α logdetρ AB +logc 2, α d A d B where ρ α A σ α B = c 2 ρ α AB, c 2 R +, and α >. Now, we are going to survey previous theorem in a wider sense. Theorem 4.5. For Rényi generalization of mutual information α >, we have Proof. From Lemma.7 and for α >, From Theorem 4.2, I α A;B ρ α logd A d B + logdetρ AB α d A d B α d detσ = d d detdσ. logdetσ α trdσ I dlogd d = α logd D α ρ σ logd+ α α logdetρ+ log detσ α d d logd+α logd+ α α d logdetρ = α logd+. α d logdetρ Hence, I α A;B ρ α logd A d B + logdetρ AB α d A d B Therefore, we know the least Rényi generalization of mutual information for α >, before finding optimum σ B. For instance, we suppose ρ AB = d A d B I AB, so c = c 2 = 2α, ρ A = tr B ρ AB = I A d A d B d A

11 and optimum σ B is d B I B, we have below conclusions : H α A B ρ = logd A logd A d B + 2α logdet I AB +log 2α α d A d B d A d B d A d B = logd A logd A d B + 2α log d Ad B +log 2α α d A d B d A d B d A d B = logd A logd A d B +2αlog + 2αlog α d A d B d A d B = logd A logd A d B +log α d A d B = logd A = H α A ρ and also, I α A;B ρ = 0. The last theorem shows relationship between Renyi relative entropy of ρ, σ and identity for α >. Theorem 4.6. For Rényi relative entropy of α >, Proof. Using Lemma.6, we get D α ρ σ D α ρ I+D α I σ. So, trρ α σ α trρ α trσ α logtrρ α σ α logtrρ α +logtrσ α α logtrρα σ α α logtrρα + α logtrσ α for α >. D α ρ σ D α ρ I+D α I σ. References [] M. Berta, K. Seshadreesan, M. M Wilde, Renyi generalizations of the conditional quantum mutual information. J. Math. Phys. 56, arxiv:

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