Fig. 6: Benchmark of non-asymptotic QI phase sensing.
Here we numerically evaluate the square root of the weighted mean-square error (rWMSE) in the optical scenario with signaling photon number NS = 200, background photon number NB = 0.01, m = 1600 effective transmitters, \({m}_{{{{\rm{re}}}}}=2000\) transmitter modes for saturation, and transmissivity ratios ηj ~ 0.7. The classical benchmarks ϵθ,c is evaluated by assuming that the whole energy mNS is concentrated in a single probe in each experiment trail. Given a fixed experiments round number ν, we optimize the phase number \(J,{J}^{{\prime} }=\nu /{\nu }^{{\prime} }\) to be estimated while leaving the other phases at random guess. The upper bound of benchmark is evaluated by \({\nu }^{{\prime} }\)-shot homodyne measurement with MSE \({E}_{{{{\rm{c}}}},{{{\rm{homo}}}},{{{\rm{k}}}}}\approx 6.31/{\nu }^{{\prime} }\). The lower bound of benchmark is evaluated by the Bayesian bound with \({E}_{{{{\rm{Baye}}}}}=3.53/{\nu }^{{\prime} }\). The explicit calculation of the cases where two classical probes are used, i.e. \(J=2\nu /{\nu }^{{\prime} }\) is an open problem as it induces negative Beyasian bound in this case.
