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