Counterexamples to Additivity of Minimum Output p-Renyi Entropy for p Close to 0

Toby S. Cubitt, Aram W. Harrow, Debbie Leung +2 more

Gilad Gour, Todd Kemp

Debbie Leung, Benjamin Lovitz, Ion Nechita +1 more

The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order $p>1$, at the von Neumann point $p=1$, and for sufficiently small positive $p$, while much of the interval $0<p<1$ has remained open. In this work, we prove that for every $p>0$ there exist finite-dimensional quantum channels whose minimum output $p$-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for $p>3/4$, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.

Serban T. Belinschi, Benoı̂t Collins, Ion Nechita

Jiunn-Wei Chen, Chang-Tse Hsieh, Han-Hsin Lin

Yun Cheng, Yuan Li