Commutative version of the local Hamiltonian problem and common eigenspace problem

Abstract

We study the complexity of a problem Common Eigenspace --- verifying consistency of eigenvalue equations for composite quantum systems. The input of the problem is a family of pairwise commuting Hermitian operators H_1,\ldots,H_r on a Hilbert space (\CC^d)^{\otimes n} and a string of real numbers \lambda=(\lambda_1,\ldots,\lambda_r). The problem is to determine whether the common eigenspace specified by equalities H_a|\psi\ra=\lambda_a|\psi\ra, a=1,\ldots,r has a positive dimension. We consider two cases: (i) all operators H_a are k-local; (ii) all operators H_a are factorized. It can be easily shown that both problems belong to the class \QMA --- quantum analogue of \NP, and that some \NP-complete problems can be reduced to either (i) or (ii). A non-trivial question is whether the problems (i) or (ii) belong to \NP? We show that the answer is positive for some special values of k and d. Also we prove that the problem (ii) can be reduced to its special case, such that all operators H_a are factorized projectors and all \lambda_a=0.