Iterative methods for finding optimal quantum measurements under minimum-error and minimax criteria

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 Ježek et al. [Phys. Rev. A 65, 060301 (2002)] for maximizing the correct probability. We prove that, for a linearly independent pure state set, Ježek 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.

Figure 1

Example of the result of our method for finding a minimum-error measurement in the case of a binary pure state set: (a) the transition of $w_m^{\left(r\right)}$; (b) the transition of ${\xi }_m{Y}_m^{\left(r\right)}$; (c) the lower and upper bounds for $P_{\mathrm{C}}^{\mathrm{opt}}$ vs number of iterations.

Figure 2

Example of numerical calculations for minimum-error measurements for an $M$-ary optical ASK coherent state set. Number of iterations to achieve difference between upper and lower bounds for maximum correct probability, ${\overline{P_{\mathrm{C}}}}^{\left(w^{\left(r\right)}\right)}-{\underline{P_{\mathrm{C}}}}^{\left(w^{\left(r\right)}\right)}$, less than ${10}^{-9}$ is shown.

Figure 3

Example of numerical calculations for minimax solutions for an $M$-ary optical ASK coherent state set. Number of iterations to achieve difference between upper and lower bounds for minimax value, ${\overline{P_{\mathrm{C}}^{☆}}}^{\left(w^{\left(r\right)}\right)}-{\underline{P_{\mathrm{C}}^{☆}}}^{\left(w^{\left(r\right)}\right)}$, less than ${10}^{-9}$ is shown.

Figure 4

Accuracy of minimax value vs number of iterations for a ternary ASK coherent state set.

Figure 5

Example of numerical calculations for minimum-error measurements for four pure ($R=1$) and mixed ($R>1$) states. Number of iterations to achieve difference between upper and lower bounds for maximum correct probability less than ${10}^{-9}$ is shown.

Figure 6

Example of numerical calculations for minimax solutions for four pure ($R=1$) and mixed ($R>1$) states. Number of iterations to achieve difference between upper and lower bounds for minimax value less than ${10}^{-9}$ is shown.