Iterative methods for finding optimal quantum measurements under minimum-error and minimax criteria
Kenji Nakahira, Kentaro Kato, Tsuyoshi Sasaki Usuda · Physical Review A · 2015
We investigate the problem of computing optimal quantum measurements in both minimal measuring and minimax strategies. A Belavkin weighted square-root measurement (BWSRM) with appropriate weights can represent the measurement that maximizes the correct probability for any given prior probabilities of quantum states. Using this fact, we propose methods for computing optimal solutions by optimizing the weights of the BWSRM. First, we explain the conditions for the BWSRM to be optimal. In particular, we argue that if a BWSRM with certain weights is a minimax measurement, then the minimax probabilities can be immediately obtained. Next, we propose an extension of the iterative algorithm developed by Je\ifmmode \check{z}\else \v{z}\fi{}ek et al. [Phys. Rev. A 65, 060301 (2002)] for maximizing the correct probability. We prove that, for a linearly independent pure state set, Je\ifmmode \check{z}\else \v{z}\fi{}ek et al.'s algorithm converges to an optimal measurement. We also propose an iterative algorithm for a minimax solution and prove that, for a pure state set, our algorithm monotonically decreases the difference between estimated and true minimax values. Finally the performance of our algorithms is evaluated through numerical experiments.