  We study the distinguishability of multipartite quantum states by separable
operations. We first present a necessary and sufficient condition for a finite
set of orthogonal quantum states to be distinguishable by separable operations.
An analytical version of this condition is derived for the case of $(D-1)$ pure
states, where $D$ is the total dimension of the state space under
consideration. A number of interesting consequences of this result are then
carefully investigated. Remarkably, we show there exists a large class of
$2\otimes 2$ separable operations not being realizable by local operations and
classical communication. Before our work only a class of $3\otimes 3$ nonlocal
separable operations was known [Bennett et al, Phys. Rev. A \textbf{59}, 1070
(1999)]. We also show that any basis of the orthogonal complement of a
multipartite pure state is indistinguishable by separable operations if and
only if this state cannot be a superposition of 1 or 2 orthogonal product
states, i.e., has an orthogonal Schmidt number not less than 3, thus generalize
the recent work about indistinguishable bipartite subspaces [Watrous, Phys.
Rev. Lett. \textbf{95}, 080505 (2005)]. Notably, we obtain an explicit
construction of indistinguishable subspaces of dimension 7 (or 6) by
considering a composite quantum system consisting of two qutrits (resp. three
qubits), which is slightly better than the previously known indistinguishable
bipartite subspace with dimension 8.
