Simultaneous min-entropy smoothing on multiparty systems

We consider the multiparty typicality conjecture raised by Dutil from a one-shot perspective. Asking for a multipartite state close to the state of the system that is typical on different subsystems simultaneously, this conjecture serves as a placeholder for the general difficulty to transfer the concept of classical joint typicality to the quantum setting. In this work, we reformulate the multiparty typicality conjecture as an optimization problem for min-entropies of different marginals. We find that the resulting one-shot conjecture is satisfied whenever the marginals under consideration commute. In this case we provide an optimal bound on the distance of the optimal state that demonstrates that atypical correlations for different subsystems can form mutually exclusive events on the global system. We furthermore show that our conjecture also holds in the two party quantum case. The techniques are then generalized to a restricted case for more parties given that the marginals to optimize do not overlap. Finally, this leads to a proof of our conjecture for tripartite systems in a pure state.