Rank Reduction for the Local Consistency Problem

We address the problem of how simple a solution can be for a given quantum
local consistency instance. More specifically, we investigate how small the
rank of the global density operator can be if the local constraints are known
to be compatible. We prove that any compatible local density operators can be
satisfied by a low rank global density operator. Then we study both fermionic
and bosonic versions of the N-representability problem as applications. After
applying the channel-state duality, we prove that any compatible local channels
can be obtained through a global quantum channel with small Kraus rank.