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