Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators

Abstract

We shall state the following parametric extensions of Shannon inequality and its reverse one in Hilbert space operators. Let p∈[0,1] and also let { A 1, A 2,…, A n } and { B 1, B 2,…, B n } be two sequences of strictly positive operators on a Hilbert space H such that ∑ j=1 n A j ♯ p B j ⩽ I . Then ∑ j=1 nS p+1(A j|B j)⩾ ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j × log ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j ⩾ log ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j ⩾∑ j=1 nS p(A j|B j) ⩾− log ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j ⩾− ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j × log ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j ⩾∑ j=1 nS p−1(A j|B j), where S q(A|B)=A 1 2 A −1 2 BA −1 2 q logA −1 2 BA −1 2 A 1 2 for A>0 , B>0 and any real number q and A♮ qB=A 1 2 A −1 2 BA −1 2 qA 1 2 for A>0 , B>0 and any real number q . In particular, if ∑ j=1 n A j =∑ j=1 n B j = I , then ∑ j=1 nS 2(A j|B j)⩾ ∑ j=1 nB jA j −1B j log ∑ j=1 nB jA j −1B j ⩾ log ∑ j=1 nB jA j −1B j ⩾∑ j=1 nS 1(A j|B j)⩾0 ⩾∑ j=1 nS(A j|B j)⩾− log ∑ j=1 nA jB j −1A j ⩾− ∑ j=1 nA jB j −1A j log ∑ j=1 nA jB j −1A j ⩾∑ j=1 nS −1(A j|B j), where S(A|B)=S 0(A|B)=A 1 2 logA −1 2 BA −1 2 A 1 2 which is the relative operator entropy of A>0 and B>0 . Our results can be considered as parametric extensions of the following celebrated Shannon inequality [Ann. Math. Statistics 22 (1951) 79; Bull. Syst. Tech. J. 27 (1948) 379; Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman, 1998, p. 233] which is very useful and so famous in information theory. Let { a 1, a 2,…, a n } and { b 1, b 2,…, b n } be two probability vectors. Then 0⩾∑ j=1 n a j log b j −∑ j=1 n a j log a j (see inequalities (2.4) of Corollary 2.4).