Epsilon-net method for optimizations over separable states

We give algorithms for the optimization problem: $\max_\rho \left\langle Q , \rho\right\rangle $, where Q is a Hermitian matrix, and the variable ρ is a bipartite separable quantum state. This problem lies at the heart of several problems in quantum computation and information, such as the complexity of QMA(2). While the problem is NP-hard, our algorithms are better than brute force for several instances of interest. In particular, they give PSPACE upper bounds on promise problems admitting a QMA(2) protocol in which the verifier performs only logarithmic number of elementary gate on both proofs, as well as the promise problem of deciding if a bipartite local Hamiltonian has large or small ground energy. For Q≥0, our algorithm runs in time exponential in ||Q||F. While the existence of such an algorithm was first proved recently by Brandao, Christandl and Yard [Proceedings of the 43rd annual ACM Symposium on Theory of Computation , 343---352, 2011], our algorithm is conceptually simpler.