<i>α</i>-<i>z</i>-Rényi relative entropies

Audenaert, Katrien; Datta, Nilanjana

doi:10.1063/1.4906367

Cited by 128 publications

(147 citation statements)

References 45 publications

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“…For example, a two-parameter family of Rényi divergences proposed by Jaksic et al [95] and further investigated by Audenaert and Datta [9] (see also [84] and [35]) captures both the minimal and Petz quantum Rényi divergence.…”

Section: Background and Further Readingmentioning

confidence: 99%

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Quantum Information Processing with Finite Resources

Tomamichel¹

2016

SpringerBriefs in Mathematical Physics

284

398

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Section: Background and Further Readingmentioning

confidence: 99%

“…2.6.4) from H A to H B . Let E † be that map, then 9) where the second maximization is over all E ∈ CPTP(B, A ), i.e. all maps whose adjoint is completely positive and unital from A to B.…”

Section: The Min-entropymentioning

confidence: 99%

Quantum Information Processing with Finite Resources

Tomamichel¹

2016

SpringerBriefs in Mathematical Physics

284

398

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“…See [25], p. 58, and also [14], [26]. The former two sorts of quantum Rényi relative entropy are particular cases of the so-called α − z-Rényi relative entropies which were introduced by Audenaert and Datta in [1].…”

Section: Introduction and Statements Of The Resultsmentioning

confidence: 99%

Quantum Rényi relative entropies on density spaces of C⁎-algebras: Their symmetries and their essential difference

Molnár

2019

Journal of Functional Analysis

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We extend the definitions of different types of quantum Rényi relative entropy from the finite dimensional setting of density matrices to density spaces of C * -algebras. We show that those quantities (which trivially coincide in the classical commutative case) are essentially different on non-commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces. Besides, we determine the symmetry groups of density spaces corresponding to each of those quantum Rényi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented.2010 Mathematics Subject Classification. Primary: 46L40, 47B49, 81P45. We next collect some useful properties of Jordan *-isomorphisms that we will use in what follows. First, any Jordan *-isomorphism J :holds for every nonnegative integer n, see 6.3.2 Lemma in [16]. In particular, J is unital meaning that J sends the identity to the identity. Since J is clearly positive (in fact, it preserves the order between self-adjoint elements in both directions), it is bounded. Indeed, more is true: J is an isometry with respect to the C * -norm. By Proposition 1.3 in [22], J preserves invertibility, namely we have J (X −1 ) = J (X ) −1 for every invertible element X ∈ A . It follows that J preserves the spectrum and, using continuous function calculus, from (18) we deduce that J ( f (X )) = f (J (X )) holds for any self-adjoint element X ∈ A s and continuous real function f on the spectrum of X .

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“…PpX |Bq ψ " 2´H min pX |Bq ψ (16) Its dual, the max-entropy, is related to the quality of X as a secret key relative to B, as the fidelity measures how close the CQ state is to one in which X is uniform and completely independent of B. This is a form of "decoupling" of X from B.…”

Section: Preliminaries 21 Mathematical Setupmentioning

confidence: 99%

“…In all these examples, the dual entropy is itself known to be a divergence-based entropy for an appropriate choice of divergence. But we could also choose different variants of the Rényi divergence, for which the duals of the associated conditional entropies are not known to themselves come from a divergence, such as the maximal [15] or the reversed sandwiched relative entropy [16] …”

Section: Preliminaries 21 Mathematical Setupmentioning

confidence: 99%

Duality of channels and codes

Renes

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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For any given channel W with classical inputs and possibly quantum outputs, a dual classical-input channel W K can be defined by embedding the original into a channel N with quantum inputs and outputs. Here we give new uncertainty relations for a general class of entropies that lead to very close relationships between the original channel and its dual. Moreover, we show that channel duality can be combined with duality of linear codes, whereupon the uncertainty relations imply that the performance of a given code over a given channel is entirely characterized by the performance of the dual code on the dual channel. This has several applications. In the context of polar codes, it implies that the rates of polarization to ideal and useless channels must be identical. Duality also relates the tasks of channel coding and privacy amplification, implying that the finite blocklength performance of extractors and codes is precisely linked, and that optimal rate extractors can be transformed into capacity-achieving codes, and vice versa. Finally, duality also extends to the EXIT function of any channel and code. Here it implies that for any channel family, if the EXIT function for a fixed code has a sharp transition, then it must be such that the rate of the code equals the capacity at the transition. This gives a different route to proving a code family achieves capacity by establishing sharp EXIT function transitions. IntroductionDuality is an important concept in many branches of mathematics, often enabling given problems to be transformed into dual versions that are simpler to solve. Recently, the author and collaborators have introduced a dual channel in the context of quantum information processing and polar coding [1][2][3][4]. The dual construction applies to channels with classical inputs and classical or quantum outputs and is designed so that the original channel and its dual can both be embedded into the same quantum channel. Constraints on the form of quantum channels then lead to nontrivial constraints on the behavior of the channel and its dual.Here we investigate the notion of duality more comprehensively. We find that it is entirely compatible with the duality of linear codes generally, as well as the notion of channel convolution appearing in belief propagation decoding and polar coding more specifically. Entropic uncertainty relations imply constraints between a wide variety of entropic functions of the channel and code, including EXIT functions. As the class of entropies is quite large, including Rényi entropies for instance, this essentially means that the behavior of a code over a channel is determined by that of the dual code over the dual channel.Channel duality has several applications, which we briefly describe here by way of outlining the structure of the paper. In the next section we set the mathematical stage and define the class of entropies under consideration. Section 3 is then concerned with the definition and properties of dual channels themselves. In particular, duals of simple classica...

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α-z-Rényi relative entropies

Cited by 128 publications

References 45 publications

Quantum Information Processing with Finite Resources

Quantum Information Processing with Finite Resources

Quantum Rényi relative entropies on density spaces of C⁎-algebras: Their symmetries and their essential difference

Duality of channels and codes

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