Abstract-In the context of network information theory, one often needs a multiparty probability distribution to be typical in several ways simultaneously. When considering quantum states instead of classical ones, it is in general difficult to prove the existence of a state that is jointly typical. Such a difficulty was recently emphasized and conjectures on the existence of such states were formulated.In this paper, we consider a one-shot multiparty typicality conjecture. The question can then be stated easily: is it possible to smooth the largest eigenvalues of all the marginals of a multipartite state ρ simultaneously while staying close to ρ? We prove the answer is yes whenever the marginals of the state commute. In the general quantum case, we prove that simultaneous smoothing is possible if the number of parties is two or more generally if the marginals to optimize satisfy some non-overlap property.

Abstract. We introduce a novel method to compute approximations of integrals over implicitly defined hypersurfaces. The new method is based on a weak formulation in L 2 (0, 1), that uses the coarea formula to circumvent an explicit integration over the hypersurfaces. As such it is possible to use standard quadrature rules in the spirit of hp/spectral finite element methods, and the expensive computation of explicit hypersurface parametrizations is avoided. We derive error estimates showing that high order convergence can be achieved provided the integrand and the hypersurface defining function are sufficiently smooth. The theoretical results are supplemented by numerical experiments including an application for plasma modeling in nuclear fusion.