On quantum Rényi entropies: A new generalization and some properties

By Martin Muller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, Marco Tomamichel. 2013

689 citations recorded by the source corpus.

The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/219ead92698df8580076625995ce57f8fc6350a8

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Many-Body Quantum Magic (2020, background): “[53, 54] for more comprehensive discussions on quantum Rényi relative entropies).”
Quantum Conditional Mutual Information and Approximate Markov Chains (2014, neutral): “Let Dα(·‖·) be the α-QuantumRényi Divergence as defined in [40,55].”
Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy (2013, background): “The sandwiched Rényi relative entropy D̃α was independently defined in [22,51,68].”
Description of quantum coherence in thermodynamic processes requires constraints beyond free energy (2014, methodology): “For α → 1, Sα(ρ||σ) is simply the quantum relative entropy [30] and the constraints of Eq.”
The role of quantum information in thermodynamics—a topical review (2015, background): “If we work directly in the single-shot regime, we recover a whole family of monotones [82] based on quantum Rényi relative entropies [91], of which the free energy is a member.”
Entanglement entropy: Holography and renormalization group (2018, methodology): “(44), Sn(ρA||1A/dA) = log dA − Sn(A) , (69) It is also shown to be monotonic with respect the parameter n (Beigi, 2013; Müller-Lennert et al., 2013), ∂nSn(ρ||σ) ≥ 0 , (70) which is considered as a generalization of the inequality Eq.”