Swiveled Rényi entropies

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, 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 informations 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.

Recoverability in quantum information theory

“…This theorem has already been put to good use in characterizing local state transformations [53] and in obtaining chain rules for Rényi entropies [28], for example, and it should be interesting to find further applications of it in the context of quantum information theory. A recent application, related to the developments here, is in the study of swiveled Rényi entropies [54]. …”

Multivariate Trace Inequalities

“…4 for more details). Extensions and further applications of this approach are discussed by Dupuis and Wilde [18]. Hirschmann's refinement was first studied in this context by Junge et al [35], where the following theorem essentially appeared: Theorem 3.1 (Stein-Hirschman).…”

Superadditivity of Quantum Relative Entropy for General States

“…However, in the same paper, he provides a correct generalization of this inequality for three operators. This result has recently been extended by Sutter et al in [21] via de so-called multivariate trace inequalities (see also the subsequent paper by Wilde [25], where similar inequalities are derived following the statements of [6]). …”