Unambiguous discrimination between linearly dependent states with multiple copies

A set of quantum states can be unambiguously discriminated if and only if they are linearly independent. However, for a linearly dependent set, if C copies of the state are available, then the resulting C particle states may form a linearly independent set, and be amenable to unambiguous discrimination. We obtain one necessary and one sufficient condition for the possibility of unambiguous discrimination among N states given that C copies are available and that the single copies span a D-dimensional space, These conditions are found to be identical for qubits. We then examine in detail the linearly dependent trine ensemble. The set of C > 1 copies of each state is a set of linearly independent lifted trine states. The maximum unambiguous discrimination probability is evaluated for all C > 1 with equal a priori probabilities.