Quantum Information Processing

, Volume 15, Issue 3, pp 1309–1345 | Cite as

Swiveled Rényi entropies

  • Frédéric Dupuis
  • Mark M. Wilde


This paper introduces “swiveled Rényi entropies” as an alternative to the Rényi entropic quantities put forward in Berta et al. (Phys Rev A 91(2):022333, 2015). What distinguishes the swiveled Rényi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Rényi entropies are ordered, which is an important property of the Rényi family of entropies. The swiveled Rényi entropies are, however, generally discontinuous at \(\alpha =1\) and do not converge to the von Neumann entropy-based measures in the limit as \(\alpha \rightarrow 1\), instead bounding them from above and below. Particular variants reduce to known Rényi entropies, such as the Rényi relative entropy or the sandwiched Rényi relative entropy, but also lead to ordered Rényi conditional mutual information and ordered Rényi generalizations of a relative entropy difference. Refinements of entropy inequalities such as monotonicity of quantum relative entropy and strong subadditivity follow as a consequence of the aforementioned properties of the swiveled Rényi entropies. Due to the lack of convergence at \(\alpha =1\), it is unclear whether the swiveled Rényi entropies would be useful in one-shot information theory, so that the present contribution represents partial progress toward this goal.


Rényi entropies Conditional mutual information  Monotonicity of quantum relative entropy Strong subadditivity 



We are grateful to Salman Beigi for insightful discussions about the topic of this paper. FD acknowledges the support of the Czech Science Foundation GA CR Project P202/12/1142 and the support of the EU FP7 under Grant Agreement No 323970 (RAQUEL). MMW is grateful to Stephanie Wehner and her group for hospitality during a research visit to TU Delft (May 2015), to Renato Renner and his group for the same during a visit to ETH Zurich (June 2015), and acknowledges support from startup funds from the Department of Physics and Astronomy at LSU, the NSF under Award No. CCF-1350397, and the DARPA Quiness Program through US Army Research Office Award W31P4Q-12-1-0019.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Physics and Astronomy, Center for Computation and Technology, Hearne Institute for Theoretical PhysicsLouisiana State UniversityBaton RougeUSA

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