Two-Indexed Schatten Quasi-Norms with Applications to Quantum Information Theory

We define 2-indexed (q,p)-Schatten quasi-norms for any q,p > 0 on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that |frac{1}{q} - frac{1}{p}| leq 1. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the q → p completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for q geq p>0, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as Rényi conditional entropies for αgeq frac{1}{2} or the Sandwiched Rényi Umlaut information for α< 1. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-α-entropy for αgeqfrac{1}{2} extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-α entropy for αgeq frac{1}{2}.