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Quantum interference is known to become extinct with distinguishinginformation, as illustrated by the ubiquitous double-slit experiment or thetwo-photon HOM effect. In the former case single particle interference isdestroyed with which-path information while in the latter bunching interferencetails-off as photons become distinguishable. It has been observed that whenmore than two particles are involved, these interference patterns are ingeneral a non monotonic function of the distinguishability. Here we perform acomprehensive characterization, both theoretically and experimentally, offour-photon interference by analyzing the corresponding correlation functions,contemplating several degrees of distinguishability across differentparameters. This study provides all the necessary tools to quantify the impactof multi-photon interference on precision measurements of parameters such asphase, frequency, and time difference. We apply these insights to quantify theprecision in the estimation of an interferometric phase in a two-portinterferometer using a four-photon state. Our results reveal that, for certainphase values, partially distinguishable multi-photon states can achieve higherFisher information values compared to the two-photon experiment. These findingshighlight the potential of distinguishable multi-photon states for enhancedprecision in quantum metrology and related applications.

Unraveling quantum phase estimation: exploring the impact of multi-photon interference on the quantum Fisher information