Distinguishability of Quantum States by Separable Operations

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 2otimes 2 separable operations not being realizable by local operations and classical communication. Before our work only a class of 3otimes 3 nonlocal separable operations was known \[Bennett et al, Phys. Rev. A 59, 1070 (1999)\]. 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 \[Watrous, Phys. Rev. Lett. 95, 080505 (2005)\]. 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.