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Published in:
2006 | OriginalPaper | Chapter
Consistency of Local Density Matrices Is QMA-Complete
Author : Yi-Kai Liu
Published in: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Publisher: Springer Berlin Heidelberg
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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.