  We show that the topological phase transition for a Kitaev chain embedded in
a cavity can be identified by measuring experimentally accessible photon
observables such as the Fano factor and the cavity quadrature amplitudes.
Moreover, based on density matrix renormalization group numerical calculations,
endorsed by an analytical Gaussian approximation for the cavity state, we
propose a direct link between those observables and quantum entropy
singularities. We study two bipartite entanglement measures, the von Neumann
and R\'enyi entanglement entropies, between light and matter subsystems. Even
though both display singularities at the topological phase transition points,
remarkably only the R\'enyi entropy can be analytically connected to the
measurable Fano factor. Consequently, we show a method to recover the bipartite
entanglement of the system from a cavity observable. Thus, we put forward a
path to experimentally access the control and detection of a topological
quantum phase transition via the R\'enyi entropy, which can be measured by
standard low noise linear amplification techniques in superconducting circuits.
In this way, the main quantum information features of Majorana polaritons in
photon-fermion systems can be addressed in feasible experimental setups.
