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Probe incompatibility in multiparameter noisy quantum metrology

We derive fundamental bounds on the maximal achievable precision in multiparameter noisy quantum metrology, valid under the most general entanglement-assisted adaptive strategy, which are tighter than the bounds obtained by a direct use of single-parameter results. This allows us to study the issue of the optimal probe incompatibility in the simultaneous estimation of multiple parameters in generic noisy channels, while so far the issue has been studied mostly in effectively noiseless scenarios (where the Heisenberg scaling is possible). We apply our results to the estimation of both unitary and noise parameters, and indicate models where the fundamental probe incompatibility is present. In particular, we show that in lossy multiple arm interferometry the probe incompatibility is as strong as in the noiseless scenario. Finally, going beyond the multiple-parameter estimation paradigm, we introduce the concept of \emph{random quantum sensing} and show how the tools developed may be applied to multiple channel discrimination problems. As an illustration, we provide a simple proof of the loss of the quadratic advantage of time-continuous Grover algorithm in presence of dephasing or erasure noise.