Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Abstract

Pinsker's and Fannes' type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum $f$-divergence is used for its estimating from below. For order $\alpha \in (0,1)$, a family of lower bounds of Pinsker type is obtained. For $\alpha >1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi's entropies. The Fano inequality is extended to Tsallis' entropies for all $\alpha >0$. The deduced bounds on the Tsallis conditional entropy are used for obtaining inequalities of Fannes' type.

12 pages, no figures. The title has been changed. The paper is substantially revised. To appear in Math. Phys. Anal. Geom

References in corpus (10)

• Quantum f-divergences and error correction
• Some extensions of the uncertainty principle
• Some general properties of unified entropies
• Subadditivity of q-entropies for q>1
• Continuity bounds on the quantum relative entropy - II
• Entropic uncertainty relations for extremal unravelings of super-operators
• The Wigner-Yanase entropy is not subadditive
• Stochastic Isometries in Quantum Mechanics
• Upper continuity bounds on the relative $q$-entropy for $q>1$
• On the failure of subadditivity of the Wigner-Yanase entropy

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• Emergence of a second law of thermodynamics in isolated quantum systems
• Relating relative Rényi entropies and Wigner-Yanase-Dyson skew information to generalized multiple quantum coherences
• On quantum conditional entropies defined in terms of the $f$-divergences