Time-energy trade-off in unambiguous-state-discrimination positive operator-valued measures

Unambiguous-nonorthogonal-state discrimination has a fundamental importance in quantum information. Moreover, it can be used for entanglement distillation and secure communication. The discrimination is carried out by a positive operator-valued measure (POVM) generalized measurement, which is typically implemented by coupling the system to an ancilla. We find a trade-off between the needed energy resources and the evolution time needed to implement the POVM and express it in terms of an actionlike cost inequality. We find the realization that minimizes this actionlike cost and show that, in this case, the cost is determined by the maximal population transfer from the system to the ancilla. We demonstrate our findings in an example of a three-level system coupled to a laser.