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

Abstract

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 $|\frac{1}{q}-\frac{1}{p}|\le 1$. 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 $q\to p$ completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for $q\ge p>0$, 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 $\alpha \ge \frac{1}{2}$ or the Sandwiched Rényi Umlaut information for $\alpha <1$. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-$\alpha$-entropy for $\alpha \ge \frac{1}{2}$ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-$\alpha$ entropy for $\alpha \ge \frac{1}{2}$.

63 pages Added Theorem 4.3 and Corollary 4.3 answering one of the initial open questions Expanded the statements of Theorem 3.3 and Lemma 3.3 and corrected some minor mathematical inconsistencies Simplified Lemma C.1