Programmable unknown quantum-state discriminators with multiple copies of program and data: A Jordan-basis approach

The discrimination of any pair of unknown quantum states is performed by devices processing three parts of inputs: copies of the pair of unknown states we want to discriminate are, respectively, stored in two program systems and copies of data, which is guaranteed to be one of the unknown states, in a third system. We study the efficiency of such programmable devices with the inputs prepared with $n$ and $m$ copies of unknown qubits used as programs and data, respectively. By finding a symmetry in the average inputs, we apply the Jordan-basis method to derive their optimal unambiguous discrimination and minimum-error discrimination schemes. The dependence of the optimal solutions on the a priori probabilities of the mean input states is also demonstrated.