Consistency of Local Density Matrices Is QMA-Complete

Suppose we have an n-qubit system, and we are given a collection of local density matrices ρ1,...,ρm, where each ρi describes a subset Ci of the qubits. We say that the ρi are “consistent” if there exists some global state σ (on all n qubits) that matches each of the ρi 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.