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. 
Jianxin Chen, Zhengfeng Ji, Alexander Klyachko, David W. Kribs and Bei   Zeng
quant-ph