Constructive counterexamples to the additivity of minimum output Rényi entropy of quantum channels for all $p>1$

We present explicit quantum channels with strictly sub-additive minimum output Rényi entropy for all $p>1$, improving upon prior constructions which handled $p>2$. Our example is provided by explicit constructions of linear subspaces with high geometric measure of entanglement. This construction applies in both the bipartite and multipartite settings. As further applications, we use our construction to find entanglement witnesses with many highly negative eigenvalues, and to construct entangled mixed states that remain entangled after perturbation.