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

Wojciech Górecki1, Sisi Zhou2,3,4, Liang Jiang2,3,4, and Rafał Demkowicz-Dobrzański1

1Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
2Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA
3Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA
4Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA

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We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a 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 Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.

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