Explicit formula for the Holevo bound for two-parameter qubit estimation problem

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The main contribution of this paper is to derive an explicit expression for the fundamental precision bound, the Holevo bound, for estimating any two-parameter family of qubit mixed-states in terms of quantum versions of Fisher information. The obtained formula depends solely on the symmetric logarithmic derivative (SLD), the right logarithmic derivative (RLD) Fisher information, and a given weight matrix. This result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models; the Holevo bound coincides with the SLD Cramer-Rao bound and it does with the RLD Cramer-Rao bound. One of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property: The structure of the Holevo bound changes smoothly when the weight matrix varies. In particular, it always coincides with the RLD Cramer-Rao bound for a certain choice of the weight matrix. Several examples illustrate these findings.