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 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.