Quantum Inf Process (2016) 15:1309–1345
Swiveled Rényi entropies
· Mark M. Wilde
Received: 27 October 2015 / Accepted: 1 December 2015 / Published online: 15 February 2016
© Springer Science+Business Media New York 2016
Abstract 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 α = 1 and do not converge to the von Neumann entropy-
based measures in the limit as α → 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 α = 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.
Mark M. Wilde
Faculty of Informatics, Masaryk University, Brno, Czech Republic
Department of Physics and Astronomy, Center for Computation and Technology, Hearne Institute
for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, USA