Quantum Nonlocality without Entanglement

We exhibit an orthogonal set of product states of two separated three-state particles that nevertheless cannot be reliably distinguished by any sequence of local operations and classical communication between the separate observers. It is proved that there is a finite gap between the mutual information obtainable by a joint measurement on these states and a measurement in which only local actions are permitted. This result implies the existence of separable superoperators that cannot be implemented locally. A set of states are found involving three two-state particles which also appear to be non-measurable locally. These and other multipartite states are classified according to the entropy and entanglement costs of preparing and measuring them by local operations