发表机构
Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎电信学院-信号与图像实验室，巴黎理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入多指标 Schatten 拟范数与反范数，扩展了 Pisier 范数及双指标情形，并证明完全超压缩性张量化及条件 Rényi 熵的可加性与链式法则。
AI 中文摘要
我们引入并研究了多指标 Schatten 空间，这些空间将交换的 $\ell_{q_1}\\![\ell_{q_2}\\![\ell_{q_3}\\![\dots]]]$ 空间提升到量子工具箱中。它们扩展了 Pisier 的算子值 Schatten 范数以及最近引入的双指标 Schatten 拟范数。对于每个有序元组 $\mathcal{Q}=(q_1,\dots,q_n)$，其中指标非零，我们在 $n$ 重 Hilbert 空间张量积上的算子定义了拟范数（对于正指标）或反范数（对于负指标）。我们针对满足互容条件 $|\frac{1}{q_i}-\frac{1}{q_j}|\leq1$ 的同号指标进行此操作。除了通过结合算子空间和矩阵分析技术的分解公式建立这些拟范数和反范数的理想性质外，我们还将开创性工作 [Devetak, Junge, King, Ruskai, CMP 2006] 的几个核心工具和结果提升到多指标和拟范数设定中。这些包括一个涉及交换系统的通用非交换 Minkowski 不等式和两个恒等式移除等式。作为推论，完全超压缩性和完全逆超压缩性对所有相容指标（包括负指标）张量化。关于量子信息理论，我们证明了对于所有 $\alpha\geq\frac12$，完全有界最小和最大输出条件熵在量子信道张量积下是可加的。这扩展了 [Fawzi, Kochanowski, Rouzé, Van Himbeeck, CMP 2026] 中 $\alpha>1$ 的情况以及 [Devetak, Junge, King, Ruskai, CMP 2006] 的无条件结果。除张量积外，我们为 $\tilde H^\uparrow_\alpha$（$\alpha\in[\frac12,1)$）建立了链式法则，适用于可能发信号的通道的复合，将用于设备无关纠缠认证的平滑最大熵链式法则扩展到整个区间 $\alpha\in[1/2,1)$。
英文摘要
We introduce and study multi-indexed Schatten spaces which lift the commutative $\ell_{q_1}\![\ell_{q_2}\![\ell_{q_3}\![\dots]]]$ -spaces to the quantum toolbox. They extend Pisier's operator-valued Schatten norms and the recently introduced two-indexed Schatten quasi-norms. For every ordered tuple $\mathcal{Q}=(q_1,\dots,q_n)$ of non-zero indices we define a quasi-norm (for positive indices) or an anti-norm (for negative indices) on operators on an $n$-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition $|\frac{1}{q_i}-\frac{1}{q_j}|\leq1$. Beyond establishing desirable properties of these quasi- and anti-norms via factorization formulas that combine operator space and matrix-analytic techniques, we lift several central tools and results of the seminal [Devetak, Junge, King, Ruskai, CMP 2006] to the multi-indexed and quasi-normed setting. These include a general non-commutative Minkowski inequality relating swapped systems and two identity-removal equalities. As a consequence, complete hypercontractivity and complete reverse hypercontractivity tensorize for all compatible indices, including negative ones. Concerning quantum information theory we prove that the completely bounded minimal and the maximal output conditional entropies are additive under tensor products of quantum channels for all $α\geq\frac12$. This extends the case $α>1$ of [Fawzi, Kochanowski, Rouzé, Van Himbeeck, CMP 2026] and the unconditional results of [Devetak, Junge, King, Ruskai, CMP 2006]. Beyond tensor products, we establish chain rules for $\tilde H^\uparrow_α$, $α\in[\frac12,1)$, for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval $α\in[1/2,1)$.
Comments84 pages, comments welcome
