Optimal probes and error-correction schemes in multi-parameter quantum metrology.

We derive a necessary and sufficient condition for the possibility of preserving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling can be preserved, we provide a quadratic semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cram\'er-Rao (HCR) bound for pure states (Matsumoto theorem). We provide examples of significant advantages offered by joint-parameter QEC protocols, that sense all the parameters utilizing a single error-protected subspace, over separate-parameter QEC protocols where each parameter is effectively sensed in a separate subspace.

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