Consistency of Local Density Matrices is QMA-complete

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Suppose we have an n-qubit system, and we are given a collection of local density matrices rho₁,...,rhoₘ, where each rhoᵢ describes a subset Cᵢ of the qubits. We say that the rhoᵢ are ``consistent'' if there exists some global state sigma (on all n qubits) that matches each of the rhoᵢ on the subsets Cᵢ. This generalizes the classical notion of the consistency of marginal probability distributions. We show that deciding the consistency of local density matrices is QMA-complete (where QMA is the quantum analogue of NP). This gives an interesting example of a hard problem in QMA. Our proof is somewhat unusual: we give a Turing reduction from Local Hamiltonian, using a convex optimization algorithm by Bertsimas and Vempala, which is based on random sampling. Unlike in the classical case, simple mapping reductions do not seem to work here.