In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order $α\in[2,\infty)$ . The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order $α\in[2,\infty)$. The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.

quant-pharXiv:2607.27015Jul 30, 20260 citations

The α → 1 limit of the sharp quantum Rényi divergence

Bjarne Bergh, Robert Salzmann, Nilanjana Datta

Fawzi and Fawzi [Quantum 5, 387 (2021)] recently defined the sharp Rényi divergence, Dα#, for α ∈ (1, ∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit α → 1 of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin–Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences for which we prove various properties, including the data-processing inequality.Fawzi and Fawzi [Quantum 5, 387 (2021)] recently defined the sharp Rényi divergence, Dα#, for α ∈ (1, ∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit α → 1 of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin–Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences for which we prove various properties, including the data-processing inequality.

arXiv:2102.06576Journal of Mathematical PhysicsSep 1, 20213 citations

Quantum Rényi and f-Divergences from Integral Representations

Christoph Hirche, Marco Tomamichel

arXiv:2306.12343Communications in Mathematical PhysicsAug 20, 202425 citations

On quantum Renyi entropies: a new definition and some properties.

Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr +2 more

Jun 13, 201326 citations

The

\alpha \to 1

Limit of the Sharp Quantum Rényi Divergence

α\in[2,\infty)

. The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.