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Phys Rev Lett. 2017 Sep 29;119(13):130504. doi: 10.1103/PhysRevLett.119.130504. Epub 2017 Sep 29.

Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases.

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QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy.
Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy.
Department of Physics, Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom.
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom.
Dipartimento di Scienze, Università degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy.


A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this Letter, we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the ultimate theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.

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