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

TL;DR

This paper introduces 2-indexed Schatten quasi-norms for operators on tensor product Hilbert spaces, explores their properties, and applies them to quantum information measures, extending key results in quantum channel theory.

Contribution

It defines new Schatten quasi-norms, establishes their properties without operator space theory, and applies them to quantum information measures and channel entropy additivity.

Findings

Super-multiplicativity of the $q o p$ completely bounded co-quasi-norm for quantum channels.

Extension of tensorization of reverse hypercontractivity.

Additivity of maximum output R\'enyi-$\alpha$ entropy for $\alpha \geq 1/2$.

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 $|\frac{1}{q} - \frac{1}{p}| \leq 1$. 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 $q \to p$ 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 R\'enyi conditional entropies for $\alpha \geq \frac{1}{2}$ or the Sandwiched R\'enyi Umlaut information for $\alpha < 1$. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output R\'enyi-$\alpha$-entropy for $\alpha\geq\frac{1}{2}$ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output R\'enyi-$\alpha$ entropy for $\alpha \geq \frac{1}{2}$.