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Multi-mode NOON states can quantum-enhance multiple-phase estimation in theabsence of photon loss. However, a multi-mode NOON state is known to bevulnerable to photon loss, and its quantum-enhancement can be dissipated bylossy environment. In this work, we demonstrate that a quantum advantage inestimate precision can still be achieved in the presence of photon loss. Thisis accomplished by optimizing the weights of the multi-mode NOON statesaccording to photon loss rates in the multiple modes, including the referencemode which defines the other phases. For practical relevance, we also show thatphoton-number counting via a multi-mode beam-splitter achieves the useful,albeit sub-optimal, quantum advantage. We expect this work to provide valuableguidance for developing quantum-enhanced multiple-phase estimation techniquesin lossy environments.

Optimal multiple-phase estimation with multi-mode NOON states against photon loss