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 (Formula presented.) and do not converge to the von Neumann entropy-based measures in the limit as (Formula presented.) , 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 (Formula presented.) , 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.
Publication Source (Journal or Book title)
Quantum Information Processing
Dupuis, F., & Wilde, M. (2016). Swiveled Rényi entropies. Quantum Information Processing, 15 (3), 1309-1345. https://doi.org/10.1007/s11128-015-1211-x