  Information metrics give lower bounds for the estimation of parameters. The
Cencov-Morozova-Petz Theorem classifies the monotone quantum Fisher metrics.
The optimum bound for the quantum estimation problem is offered by the metric
which is obtained from the symmetric logarithmic derivative. To get a better
bound, it means to go outside this family of metrics, and thus inevitably, to
relax some general conditions. In the paper we defined logarithmic derivatives
through a phase-space correspondence. This introduces a function which
quantifies the deviation from the symmetric derivative. Using this function we
have proved that there exist POVMs for which the new metric gives a higher
bound from that of the symmetric derivative. The analysis was performed for the
one qubit case.
