  Despite its fully unitary dynamics, the bosonic Kitaev chain (BKC) displays
key hallmarks of non-Hermitian physics including non-reciprocal transport and
the non-Hermitian skin effect. Here we demonstrate another remarkable
phenomena: the existence of an entanglement phase transition (EPT) in a variant
of the BKC that occurs as a function of a Hamiltonian parameter g, and which
coincides with a transition from a reciprocal to a non-reciprocal phase. As g
is reduced below a critical value, the post-quench entanglement entropy of a
subsystem of size l goes from a volume-law phase where it scales as l to a
super-volume law phase where it scales like lN with N the total system size.
This EPT occurs for a system undergoing purely unitary evolution and does not
involve measurements, post-selection, disorder or dissipation. We derive
analytically the entanglement entropy out of and at the critical point for the
$l=1$ and $l/N \ll 1$ case.
