Discrimination of the change point in a quantum setting

In the change point problem, we determine when the observed distribution has changed to another one. We expand this problem to a quantum case where copies of an unknown pure state are being distributed. That is, we estimate when the distributed quantum pure state is changed. As the most fundamental case, we treat the problem of deciding the true change point t(c) between the two given candidates t(1) and t(2). Our problem is mathematically equal to identifying a given state with one of the two unknown states when multiple copies of the states are provided. The minimum of the averaged error probability is given and the optimal positive operator-valued measure (POVM) is given to obtain it when the initial and final quantum pure states are subject to the invariant prior. We also compute the error probability for deciding the change point under the above POVM when the initial and final quantum pure states are fixed. These analytical results allow us to calculate the value in the asymptotic case.