We study the numerical accuracy ( ) and time cost ( ) of estimating the quantum Van Trees information, for a qubit encoding parameter on its phase, see Section 2.3.1. ( ) A numerical estimation was done using the RSM method with (green triangles) and (purple triangles) samples and using the REMS method with (cyan circles) and (red diamonds) samples. The ansatz ( ), resulting in Eq. ( ), is shown as a blue line. We consider different families of states parametrized by , see Eq. ( ). The number of outcomes of the POVM is fixed to 4, as this is the maximum number of outcomes for an extremal POVM in a 2-dimensional Hilbert space . Note that the RESM method gives much better results with one sample than the RSM method with 1000. ( ) Error in the numerical estimation of , see Eq. ( ), sampling directly the whole space of POVMs (RMS) or only its extremal points (RESM) for , with respect to the computational time invested. The slope for RSM case is and for RESM is . The error with the method proposed decreases much faster using RESM rather than RMS.
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