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

2026-04-15

AI Summary

This work addresses the absence of a suitable two-parameter norm framework in quantum information theory by systematically constructing $(q,p)$-Schatten quasi-norms on tensor products of Hilbert spaces and introducing completely bounded quasi-norms and co-quasi-norms. Without invoking operator space theory, the authors employ tools from matrix analysis and operator convexity to prove that the completely bounded co-quasi-norm is supermultiplicative whenever $q \geq p > 0$. This framework not only yields concise expressions for the Rényi conditional entropy (for $\alpha \geq 1/2$) and the sandwiched Rényi mutual information (for $\alpha < 1$), but also uncovers their intrinsic connection to reverse hypercontractivity. Moreover, it establishes for the first time the additivity of both minimal and maximal output Rényi-$\alpha$ entropies for all $\alpha \geq 1/2$.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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 Rényi conditional entropies for $α\geq \frac{1}{2}$ 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 $α\geq\frac{1}{2}$ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-$α$ entropy for $α\geq \frac{1}{2}$.

Problem

• Schatten quasi-norms
• quantum information theory
• Rényi entropy
• tensor products
• completely bounded norms

Innovation

• Schatten quasi-norms
• completely bounded norms
• quantum channels
• Rényi entropy
• super-multiplicativity