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 $α\geq \frac{1}{2}$.

翻译：我们定义了对任意$q,p > 0$在希尔伯特空间张量积算子上的$(q,p)$-Schatten双索引拟范数，自然推广了Pisier算子值Schatten空间理论中的范数。在假设$|\frac{1}{q} - \frac{1}{p}| \leq 1$的条件下，我们建立了这些拟范数的若干理想性质，如关系一致性和在块对角算子上的行为。实际上，我们证明了该条件对于保持自然性质而言是本质上必要的。此外，针对此类拟范数空间之间的线性映射，我们引入了完全有界拟范数和余拟范数。我们证明了对于量子通道的张量积，当$q \geq p>0$时，$q \to p$完全有界余拟范数具有超乘性，这推广了[Devetak, Junge, King, Ruskai, 2006]的重要结果。我们的证明依赖于初等矩阵分析和算子凸性工具，无需算子空间理论。在应用方面，我们展示了这些拟范数可用于表达相关量子信息度量，如$α\geq \frac{1}{2}$时的Rényi条件熵或$α<1$时的夹心Rényi Umlaut信息。我们的乘性结果蕴含了逆超收缩性的张量化概念，推广了[Devetak, Junge, King, Ruskai, 2006]另一重要结果：当$α\geq\frac{1}{2}$时完全有界最小输出Rényi-$α$熵的可加性，以及当$α\geq \frac{1}{2}$时最大输出Rényi-$α$熵的可加性。