Equality conditions of Data Processing Inequality for $α$-$z$ Rényi relative entropies

Recovery and the Data Processing Inequality for quasi-entropies

We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies defined in terms of an operator monotone decreasing functions. Included in our results is a bound in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras where the results are already non-trivial and of interest in quantum information theory.

Remarks on on Kim's Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions

We describe recent work of Kim in arXiv:1210.5190 to show that operator convex functions associated with quasi-entropies can be used to prove a large class of new matrix inequalities in the tri-partite and bi-partite setting by taking a judiciously chosen partial trace over all but one of the spaces. We give some additional examples in both settings. Furthermore, we observe that the equality conditions for all the new inequalities are essentially the same as those for strong subadditivity.

Tight uniform continuity bound for a family of entropies

We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $α$-Rényi entropy, $S_α$, for $α\in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state …

Rényi relative entropies and noncommutative $L_p$-spaces II

We study an extension of the sandwiched Rényi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values $α\in [1/2,1)$. This work is intended as a continuation of [A. Jenčová, Ann. Henri Poincaré 19, 2513-2542, 2018], where the values $α>1$ were studied. We use the Araki-Masuda divergences of [Berta et al., Ann. Henri Poincaré 9, 1843-1867, 2018] and treat them in the framework of Kosaki's noncommutative $L_p$-spaces. Using the variational formula, recently obtained by F. Hiai, for $α\in [1/2,1)$, we prove the data processing inequality with respect to p…

Rényi and Tsallis entropies: three analytic examples

A comparative study of one-dimensional quantum structures which allow analytic expressions for the position and momentum Rényi $R(α)$ and Tsallis $T(α)$ entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms a special status of the Gaussian distribution: at any parameter $α$ it converts into the equality both Rényi and Tsallis uncertainty relations removing for the latter an additional requirement $1/2\leqα\leq1$ that is a necessary condition for all other geometries. It is shown that th…

Petz条件熵与夹层Rényi条件熵的严格连续性

Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

该研究确定了阶数在[1/2,1)区间的优化Petz与夹层Rényi条件熵的迹距离严格连续模，给出可达上界，验证其极限与已有量子条件熵连续界一致，通过线性化、Schmidt秩支配等方法完成证明。