Fundamental properties of Tsallis relative entropy

Furuichi, Shigeru; Yanagi, Kenjiro; Kuriyama, Ken

doi:10.1063/1.1805729

Cited by 150 publications

(146 citation statements)

References 25 publications

(34 reference statements)

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“…where X 1 = Tr 2 X and I 2 is the identity in M n (see e. g. [BP13,FU04]), hence from the joint convexity it follows that the Bregman divergence is monotone in the following sense:…”

Section: An Application -The Tsallis Entropymentioning

confidence: 99%

On the Joint Convexity of the Bregman Divergence of Matrices

Pitrik

Virosztek

2015

Lett Math Phys

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We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantum Tsallis entropy of a tripartite state, which can be considered as a generalization of the strong subadditivity of the von Neumann entropy. (In general, the strong subadditivity of the Tsallis entropy fails for quantum states, but it holds for classical states.) Furthermore, we show that the joint convexity of the Bregman divergence does not imply the monotonicity under stochastic maps, but every monotone Bregman divergence is jointly convex.

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“…where X 1 = Tr 2 X and I 2 is the identity in M n (see e. g. [BP13,FU04]), hence from the joint convexity it follows that the Bregman divergence is monotone in the following sense:…”

Section: An Application -The Tsallis Entropymentioning

confidence: 99%

On the Joint Convexity of the Bregman Divergence of Matrices

Pitrik

Virosztek

2015

Lett Math Phys

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“…We again discuss this question from our perspective. For α → 1, Tsallis α-relative entropies reduce to the von Neumann relative entropy [42] …”

Section: For Single-qubit Mixed Statesmentioning

confidence: 99%

“…The upper bound is attained for the maximally coherent state [42,43] for the density matrices ρ and δ, denoted by D α (ρ δ), is defined as…”

Section: Preliminariesmentioning

confidence: 99%

Ordering states with Tsallis relative $$\alpha $$ α -entropies of coherence

Zhang

Shao

Luo

et al. 2016

Quantum Inf Process

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In this paper, we study the ordering states with Tsallis relative α-entropies of coherence and l1 norm of coherence for single-qubit states. We show that any Tsallis relative α-entropies of coherence and l1 norm of coherence give the same ordering for single-qubit pure states. However, they don't generate the same ordering for some high dimensional pure states, even though these states are pure. We also consider three special Tsallis relative α-entropies of coherence, such as C1 ,C2 and C 1 2 , and show any one of these three measures and C l 1 will not generate the same ordering for single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states.

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“…For these entropies, we have the formula 17) and its classical variety. For α > 1 and A, B ∈ L + (H), we also introduce…”

Section: Definitions and Notationmentioning

confidence: 99%

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

Rastegin

2013

Math Phys Anal Geom

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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 α ∈ (0, 1), a family of lower bounds of Pinsker type is obtained. For α > 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 α > 0. The deduced bounds on the Tsallis conditional entropy are used for obtaining inequalities of Fannes' type.

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Fundamental properties of Tsallis relative entropy

Cited by 150 publications

References 25 publications

On the Joint Convexity of the Bregman Divergence of Matrices

On the Joint Convexity of the Bregman Divergence of Matrices

Ordering states with Tsallis relative $$\alpha $$ α -entropies of coherence

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

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