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

• URL: http://arxiv.org/abs/2604.14055v1
• Date: Wed, 15 Apr 2026 16:35:46 GMT
• Title: Two-Indexed Schatten Quasi-Norms with Applications to Quantum Information Theory
• Abstract summary: We define 2-indexed $(q,p)$-Schatten quasi-norms for any $q,p > 0$ on operators on a tensor product of Hilbert spaces.<n>For linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms.<n>Our results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-$$-entropy for $geqfrac12$ or the Sandwiched Rényi Umlaut information
• Score: 9.048611509540079
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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 $α\geq \frac{1}{2}$.

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