Monotonicity of a relative Rényi entropy

Frank, Rupert L.; Lieb, Élliott H.

doi:10.1063/1.4838835

Cited by 216 publications

(253 citation statements)

References 16 publications

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“…However, the data-processing inequality holds more generally for all α ≥ 1 2 , as was shown by Frank and Lieb [57]. It thus remains to show data-processing for α ∈ [ for all H ≥ 0 with H ρ and equality can be achieved.…”

Section: Data-processing Via Joint Concavitymentioning

confidence: 99%

Quantum Information Processing with Finite Resources

Tomamichel¹

2016

SpringerBriefs in Mathematical Physics

256

398

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Section: Data-processing Via Joint Concavitymentioning

confidence: 99%

Quantum Information Processing with Finite Resources

Tomamichel¹

2016

SpringerBriefs in Mathematical Physics

256

398

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“…Generalisations of the relative entropy could also be used to define geometric measures of QCs. Two recently suggested generalisations are the sandwiched relative Rényi entropies [136,137] (see also [138])…”

Section: Hierarchy Of Geometric Measuresmentioning

confidence: 99%

Measures and applications of quantum correlations

Adesso¹,

Bromley²,

Cianciaruso³

2016

J. Phys. A: Math. Theor.

370

416

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Abstract. Quantum information theory is built upon the realisation that quantum resources like coherence and entanglement can be exploited for novel or enhanced ways of transmitting and manipulating information, such as quantum cryptography, teleportation, and quantum computing. We now know that there is potentially much more than entanglement behind the power of quantum information processing. There exist more general forms of non-classical correlations, stemming from fundamental principles such as the necessary disturbance induced by a local measurement, or the persistence of quantum coherence in all possible local bases. These signatures can be identified and are resilient in almost all quantum states, and have been linked to the enhanced performance of certain quantum protocols over classical ones in noisy conditions. Their presence represents, among other things, one of the most essential manifestations of quantumness in cooperative systems, from the subatomic to the macroscopic domain. In this work we give an overview of the current quest for a proper understanding and characterisation of the frontier between classical and quantum correlations in composite states. We focus on various approaches to define and quantify general quantum correlations, based on different yet interlinked physical perspectives, and comment on the operational significance of the ensuing measures for quantum technology tasks such as information encoding, distribution, discrimination and metrology. We then provide a broader outlook of a few applications in which quantumness beyond entanglement looks fit to play a key role.

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“…Different quantum generalizations of the α-Rényi divergence have been introduced [34,30,48] and their monotonicity under quantum operations for certain ranges of the Rényi parameter α has been established [34,19,3]. The Rényi entropy method has since been successfully employed to prove strong converse theorems for classical channel coding with entangled inputs for a large class of quantum channels with additive Holevo capacity [26].…”

Section: Strong Converse Theorems and The Rényi Entropy Methodsmentioning

confidence: 99%

Strong converse theorems using Rényi entropies

Leditzky

Wilde

Datta

2016

Journal of Mathematical Physics

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We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [41], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where e and q denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

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Monotonicity of a relative Rényi entropy

Abstract: We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al.

Cited by 216 publications

References 16 publications

Quantum Information Processing with Finite Resources

Quantum Information Processing with Finite Resources

Measures and applications of quantum correlations

Strong converse theorems using Rényi entropies

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