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Generalization of operator type Shannon inequality and its reverse one (Advanced Topics of Information Science and Functional Analysis)

Takayuki Furuta

Published Oct 1, 2004• 0 citations
OpenAlex

Abstract

We shall state the following generalization of operator type Shannoninequal- ity and its reverse one as a simple corollary of parametric extensions of Shannon inequality in Hilbert space operators.Let $\{A_{1}, A_{2}, \ldots, A_{n}\}$ andn $\{B_{1}, B_{2},\ldots, B_{n}\}n$ be two sequences of strictly positive operators on a Hilbert space H.If $E^{A_{j}=}L$ $B_{j}=I,$ thenwhere $S_{q}(A|B)=A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{q}(\log A^{\frac{-1}{2}}BA^{\frac{-1}{2}})A\mathrm{i}$ for $A>0$, $B>0$ and any real number $q$ and $S(A|B)=S_{0}(A|B)=A^{\frac{1}{2}}(\log A^{\frac{-1}{2}}BA^{\frac{-1}{2}})A^{\frac{1}{2}}$ which is the relative operator entropy of $A>0$ and $B>0.$Our results can be considered as parametric extensions of the following celebrated Shan- non inequality ( $[3],[5]$ and [233 $\mathrm{p},1]$ ) which is very useful and so famous in information theory.Let $\{a_{1}, a_{2}, \ldots, a_{n}\}$ and $\{b_{1}, b_{2}, \ldots, b_{n}\}$ be two probability vectors.Then $0 \geq\sum_{j=1}^{n}a_{j}\log b_{j}-\sum_{j=1}^{n}a_{j}\log a_{j}$ (see inequalities (2.4) of Corollary 2.4).

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We shall state the following generalization of operator type Shannoninequal- ity and its reverse one as a simple corollary of parametric extensions of Shannon inequality in Hilbert space operators.Let {A1,A2,…,An}\{A_{1}, A_{2}, \ldots, A_{n}\}{A1​,A2​,…,An​} andn {B1,B2,…,Bn}n\{B_{1}, B_{2},\ldots, B_{n}\}n{B1​,B2​,…,Bn​}n be two sequences of strictly positive operators on a Hilbert space H.If EAj=LE^{A_{j}=}LEAj​=L Bj=I,B_{j}=I,Bj​=I, thenwhere Sq(A∣B)=A12(A−12BA−12)q(log⁡A−12BA−12)AiS_{q}(A|B)=A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{q}(\log A^{\frac{-1}{2}}BA^{\frac{-1}{2}})A\mathrm{i}Sq​(A∣B)=A21​(A2−1​BA2−1​)q(logA2−1​BA2−1​)Ai for A>0A>0A>0, B>0B>0B>0 and any real number qqq and S(A∣B)=S0(A∣B)=A12(log⁡A−12BA−12)A12S(A|B)=S_{0}(A|B)=A^{\frac{1}{2}}(\log A^{\frac{-1}{2}}BA^{\frac{-1}{2}})A^{\frac{1}{2}}S(A∣B)=S0​(A∣B)=A21​(logA2−1​BA2−1​)A21​ which is the relative operator entropy of A>0A>0A>0 and B>0.B>0.B>0.Our results can be considered as parametric extensions of the following celebrated Shan- non inequality ( [3],[5][3],[5][3],[5] and [233 p,1]\mathrm{p},1]p,1] ) which is very useful and so famous in information theory.Let {a1,a2,…,an}\{a_{1}, a_{2}, \ldots, a_{n}\}{a1​,a2​,…,an​} and {b1,b2,…,bn}\{b_{1}, b_{2}, \ldots, b_{n}\}{b1​,b2​,…,bn​} be two probability vectors.Then 0≥∑j=1najlog⁡bj−∑j=1najlog⁡aj0 \geq\sum_{j=1}^{n}a_{j}\log b_{j}-\sum_{j=1}^{n}a_{j}\log a_{j}0≥∑j=1n​aj​logbj​−∑j=1n​aj​logaj​ (see inequalities (2.4) of Corollary 2.4).
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
One of the beautiful and very simple inequalities for a convex function is the Hermit-Hadamard inequality [S. Mehmood, et. al. Math. Methods Appl. Sci., 44 (2021) 3746], [S. Dragomir, et. al., Math. Methods Appl. Sci., in press]. The concept of log-convexity is a stronger property of convexity. Recently, the refined Hermit-Hadamard’s inequalities for log-convex functions were introduced by researchers [C. P. Niculescu, Nonlinear Anal. Theor., 75 (2012) 662]. In this paper, by the Hermit-Hadamard inequality, we introduce two parametric Tsallis quantum relative entropy, two parametric Tsallis-Lin quantum relative entropy and two parametric quantum Jensen-Shannon divergence in quantum information theory. Then some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices are investigated by this inequality. \newline \textbf{Keywords:} \textit{ Hermit-Hadamard’s inequality; log-convexity; Density matrices; Quantum relative entropy; Tsallis quantum relative entropy; quantum Jensen-Shannon divergence divergence.
In this article we propose a quantum version of Shannon's conditional entropy. Given two density matrices rho\\rhorho and sigma\\sigmasigma on a finite dimensional Hilbert space and with S(rho)=−trrholnrhoS(\\rho)=-\\tr\\rho\\ln\\rhoS(rho)=−trrholnrho being the usual von Neumann entropy, this quantity S(rho∣sigma)S(\\rho|\\sigma)S(rho∣sigma) is concave in rho\\rhorho and satisfies 0leS(rho∣sigma)leS(rho)0\\le S(\\rho|\\sigma)\\le S(\\rho)0leS(rho∣sigma)leS(rho), a quantum analogue of Shannon's famous inequality. Thus we view S(rho∣sigma)S(\\rho|\\sigma)S(rho∣sigma) as the entropy of rho\\rhorho conditioned by sigma\\sigmasigma.
Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.