Partition a Quantum Pure-state Set into Unambiguously Discriminable Subsets

The problem of partitioning a quantum pure-state set into unambiguously discriminable subsets is investigated in this paper. We prove that the problem is decomposable into sub-problems of two-subset partitioning. A recursive algebro-geometric method based on Generalized Principal Component Analysis(GPCA) is proposed. When multiple feasible partitions exist, our method is able to return all of them by introducing Brill’s equations. Through this method, the efficiency of a kind of collective photon-number-splitting(PNS) attack is discussed.