Quantum Entanglement, Sum Of Squares, And The Log Rank Conjecture

For every constant epsilon > 0, we give an exp((O) over tilde(root n))-time algorithm for the 1 vs 1-epsilon Best Separable State (BSS) problem of distinguishing, given an n(2) x n(2) matrix M corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state rho that M accepts with probability 1, and the case that every separable state is accepted with probability at most 1 - epsilon. Equivalently, our algorithm takes the description of a subspace W subset of F-n2 (where F can be either the real or complex field) and distinguishes between the case that W contains a rank one matrix, and the case that every rank one matrix is at least epsilon far (in l(2) distance) from W.To the best of our knowledge, this is the first improvement over the brute-force exp(n)-time algorithm for this problem. Our algorithm is based on the sum-of-squares hierarchy and its analysis is inspired by Lovett's proof (STOC '14, JACM '16) that the communication complexity of every rank-n Boolean matrix is bounded by (O) over tilde(root n).