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

Rastegin, Alexey E.

doi:10.1007/s11040-013-9128-z

Cited by 15 publications

(17 citation statements)

References 44 publications

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“…The Fano inequality in terms of q-entropies and some of its applications are considered in Refs. [41,43]. We will use the conditional q-entropy of order q ≥ 1 for expressing inequalities of Bell's type.…”

Section: Conditional Q-entropies and Their Propertiesmentioning

confidence: 99%

Fano type quantum inequalities in terms of q-entropies

Rastegin

2011

Quantum Inf Process

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The information-theoretic approach to Bell's theorem is developed with use of the conditional qentropies. The q-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional q-entropies for all q ≥ 1. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of n-cycle scenario is briefly mentioned. The new inequalities with conditional q-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The q-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the q-entropic formulation.

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Section: Conditional Q-entropies and Their Propertiesmentioning

confidence: 99%

Fano type quantum inequalities in terms of q-entropies

Rastegin

2011

Quantum Inf Process

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“…(Actually, [39] shows this for the Tsallis entropy, but via | log x − log y| ≤ |x − y| for x, y ≥ 1 one can easily see that the bound carries over to Rényi entropy.) Suppose the dimension of the catalysts was bounded, i.e.…”

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confidence: 94%

Stochastic Independence as a Resource in Small-Scale Thermodynamics

2015

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It is well known in thermodynamics that the creation of correlations costs work. It seems then a truism that if a thermodynamic transformation A→B is impossible, so will be any transformation that in sending A to B also correlates among them some auxiliary systems C. Surprisingly, we show that this is not the case for nonequilibrium thermodynamics of microscopic systems. On the contrary, the creation of correlations greatly extends the set of accessible states, to the point that we can perform on individual systems and in a single shot any transformation that would otherwise be possible only if the number of systems involved was very large. We also show that one only ever needs to create a vanishingly small amount of correlations (as measured by mutual information) among a small number of auxiliary systems (never more than three). The many, severe constraints of microscopic thermodynamics are reduced to the sole requirement that the nonequilibrium free energy decreases in the transformation. This shows that, in principle, reliable extraction of work equal to the free energy of a system can be performed by microscopic engines.

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“…It must be stressed that generalized entropic functions do not succeed all the standard properties. For entropic functions of the Tsallis type, some questions of such a kind were addressed in [42,43]. For this reason, we further refrain from a consideration of generalized conditional entropies.…”

Section: Some Applicationsmentioning

confidence: 99%

“…The papers [44,45] presented Rényi-entropy uncertainty relations and respective steering inequalities for POVMs assigned to quantum designs. A similar treatment can be developed on the base of (42). We refrain from presenting the details here.…”

Section: Some Applicationsmentioning

confidence: 99%

Entropic uncertainty relations from equiangular tight frames and their applications

Rastegin¹

2021

Preprint

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Finite tight frames are interesting in various respects, including potential applications in quantum information science. Indeed, each complex tight frame leads to a non-orthogonal resolution of the identity in the Hilbert space. In a certain sense, equiangular tight frames are very similar to the maximal sets that provide symmetric informationally complete measurements. Hence, applications of equiangular tight frames in quantum physics deserve more attention than they have obtained. We derive entropic uncertainty relations for a quantum measurement built of the states of an equiangular tight frame. First, the corresponding index of coincidence is estimated from above, whence desired results follow. State-dependent and state-independent formulations are both presented. We also discuss applications of the corresponding measurements to detect entanglement and steerability.

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

Cited by 15 publications

References 44 publications

Fano type quantum inequalities in terms of q-entropies

Fano type quantum inequalities in terms of q-entropies

Stochastic Independence as a Resource in Small-Scale Thermodynamics

Entropic uncertainty relations from equiangular tight frames and their applications

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