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We address the problem of how simple a solution can be for a given quantumlocal consistency instance. More specifically, we investigate how small therank of the global density operator can be if the local constraints are knownto be compatible. We prove that any compatible local density operators can besatisfied by a low rank global density operator. Then we study both fermionicand bosonic versions of the N-representability problem as applications. Afterapplying the channel-state duality, we prove that any compatible local channelscan be obtained through a global quantum channel with small Kraus rank.

Rank Reduction for the Local Consistency Problem