A solution to two party typicality using representation theory of the symmetric group

We give a proof of the multi-party typicality conjecture for the first nontrivial case when there are only two parties. The conjecture itself is motivated by the study of multi-party state merging protocols on quantum systems. Our approach is based on fundamental group-theoretical properties, thereby providing an opportunity to study the problem from a more systematic perspective. Our proof also covers an extended multiparty typicality conjecture that we state in this work. This extended multiparty typicality conjecture is formulated using arbitrary $k$-norms instead of only the $2$-norm as in the original conjecture.
Comment: 17 pages, no figures. Typos removed, extended introduction. Generalized problem statement