Unions, intersections and a one-shot quantum joint typicality lemma

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