For every ϵ>0, we give an
exp(O~(n/ϵ2))-time algorithm for the 1 vs
1−ϵ \emph{Best Separable State (BSS)} problem of distinguishing, given
an n2×n2 matrix M corresponding to a quantum measurement,
between the case that there is a separable (i.e., non-entangled) state ρ
that M accepts with probability 1, and the case that every
separable state is accepted with probability at most 1−ϵ.
Equivalently, our algorithm takes the description of a subspace W⊆Fn2 (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 ϵ
far (in ℓ2 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 \emph{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~(n).Comment: 23 pages + 1 title-page + 1 table-of-content