A fundamental tool to prove inner bounds in classical network information theory is the so-called ‘conditional joint typicality lemma’. In addition to the lemma, one often uses unions and intersections of typical sets in the inner bound arguments without so much as giving them a second thought. These arguments do not work in the quantum setting. This bottleneck shows up in the fact that so-called ‘simultaneous decoders’, as opposed to ‘successive cancellation decoders’, are known for very few channels in quantum network information theory. Another manifestation of this bottleneck is the lack of so-called ‘simultaneous smoothing’ theorems for quantum states. In this paper, we overcome the bottleneck by proving for the first time a one-shot quantum joint typicality lemma with robust union and intersection properties. To do so we develop two novel tools in quantum information theory, which may be of independent interest. The first tool is a simple geometric idea called tilting, which increases the angles between a family of subspaces in orthogonal directions. The second tool, called smoothing and augmentation, is a way of perturbing a multipartite quantum state such that the partial trace over any subset of registers does not increase the operator norm much. Our joint typicality lemma allows us to construct simultaneous quantum decoders for many multiterminal quantum channels. It provides a powerful tool to extend many results in classical network information theory to the one-shot quantum setting.

证明经典网络信息理论内在界限的基本工具是所谓的“条件联合典型性引理”。除了引理之外，人们经常在内界自变量中使用典型集的并集和交集，而无需多加思考。这些论点在量子环境中不起作用。这个瓶颈体现在以下事实：在量子网络信息理论中，很少有人知道所谓的“同时解码器”，而不是“连续抵消解码器”。该瓶颈的另一个体现是缺乏所谓的“同时平滑”量子定理。在本文中，我们通过首次证明具有稳健的结合和交点特性的单次量子接头典型性引理克服了瓶颈。为此，我们开发了量子信息论中的两个新颖的工具，这些工具可能具有独立的意义。第一个工具是一个简单的几何构想，称为倾斜，这会增加子空间族之间在正交方向上的夹角。第二种工具称为平滑和扩充，是一种扰动多部分量子状态的方法，这样，寄存器的任何子集上的部分迹线不会显着增加运算符范数。我们共同的典型性引理使我们能够为许多多端量子通道构造同时量子解码器。它提供了一个强大的工具，可以将经典网络信息理论中的许多结果扩展到一次量子设置。