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Abstract

Suppose we have an n-qubit system, and we are given a collection of local density matrices ρ 1,...,ρ m , where each ρ i describes a subset C i of the qubits. We say that the ρ i are “consistent” if there exists some global state σ (on all n qubits) that matches each of the ρ i on the subsets C i . 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yi-Kai Liu
    • 1
  1. 1.Computer Science and EngineeringUniversity of CaliforniaSan Diego

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