  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 maximal 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.

Quantum Physics