Quantum addition imparts less disorder than mixing and commutes with incoherent channels

Abstract

We prove a generalized version of a previously conjectured inequality by Zhang et al (Zhang et al 2018 Phys. Lett. A 382, 1516–23) for the quantum addition operation defined by Datta et al (Audenaert et al 2016 J. Math. Phy. 57, 052202) in the context of proving an entropy power inequality for qubits. We also show that the quantum addition operation commutes with an incoherent channel, which may have possible implications for resource theories of coherence in optical settings.