Quantum nonlocality without entanglement

We exhibit an orthogonal set of product states of two three-state particles that nevertheless cannot be reliably distinguished by a pair of separated observers ignorant of which of the states has been presented to them, even if the observers are allowed 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 measuremen t on these stales and a measurement in which only local actions are permitt ed. This result implies the existence of separable superoperators that cann ot be implemented locally. A set of states are found involving three two-st ate particles that also appear to be nonmeasurable locally. These and other multipartite states are classified according to the entropy and entangleme nt costs of preparing and measuring them by local operations. [S1050-2947(9 9)00302-9].