An optimum quantum measurement that minimizes the average probability of error is considered for symmetric quantum states. The positive operator-valued measure (POM) which satisfies the necessary and sufficient condition for the minimization of the average probability of error is derived by using the quantum Bayes strategy. It is also shown that the mutual information obtained in the optimum quantum measurement becomes extremum. Furthermore, an optimum quantum measurement for parameter estimation is found for symmetric quantum states by applying the maximum-likelihood estimation. The optimum POM for the parameter estimation has the same structure as that for the quantum state discrimination.
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