Marius Junge

Abstract Trace inequalities are powerful techniques in studying quantum entropy. The physics of quantum field theory and holography nonetheless motivate entropy inequalities in von Neumann algebras that lack a useful notion of a trace. Haagerup and Kosaki L_p spaces enable re-expressing trace inequalities in scenarios that start with a non-tracial von Neumann algebra, and we show how the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta & Tomamichel 2017) port to general von Neumann algebras. Using an approximation method of Haagerup, we prove the tightened recovery map correction to data processing for relative entropy in (potentially non-tracial) von Neumann algebras. We also prove a p-fidelity version. Furthermore, we prove that non-decrease of relative entropy is equivalent to the existence of an L_1-isometry implementing the channel on both input states.

The random unitary group channels are in general non-degradable. We give upper and lower bounds on their quantum capacity, which differ only up to a factor 2. Our method relies heavily on operator space tools, in particular complex interpolation.