Programmable discrimination with an error margin

The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two program ports, each fed with a number of identically prepared qubits---the data and the programs. The device aims at correctly identifying the data state with one of the two program states. This scheme has the unambiguous and the minimum-error schemes as extremal cases, when the error margin is set to zero or it is sufficiently large, respectively. Analytical results are given in the two situations where the margin is imposed on the average error probability---weak condition---or it is imposed separately on the two probabilities of assigning the state of the data to the wrong program---strong condition. It is a general feature of our scheme that the success probability rises sharply as soon as a small error margin is allowed, thus providing a significant gain over the unambiguous scheme while still having high confidence results.