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 necessary and sufficient conditions for the possibility of unambiguous discrimination between 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.