  The quantum central limit theorem derived by Cushen and Hudson provides the
foundations for understanding how subsystems of large bosonic systems evolving
unitarily do reach equilibrium. It finds important applications in the context
of quantum interferometry, for example, with photons. A practical feature of
current photonic experiments, however, is that photons carry their own internal
degrees of freedom pertaining to, e.g., the polarization or spatiotemporal mode
they occupy, which makes them partially distinguishable. The ensuing deviation
from ideal indistinguishability is well known to have observable consequences,
for example in relation with boson bunching, but an understanding of its role
in bosonic equilibration phenomena is still missing. Here, we generalize the
Cushen-Hudson quantum central limit theorem to encompass scenarios with partial
distinguishability, implying an asymptotic convergence of the subsystem's
reduced state towards a multimode Gaussian state defined over the internal
degrees of freedom. While these asymptotic internal states may not be directly
accessible, we show that particle number distributions carry important
signatures of distinguishability, which may be used to diagnose experimental
imperfections in large boson sampling experiments.
