  We study bipartite post measurement entanglement entropy after selective
measurements in quantum chains. We first study the quantity for the critical
systems that can be described by conformal field theories. We find a connection
between post measurement entanglement entropy and the Casimir energy of
floating objects. Then we provide formulas for the post measurement
entanglement entropy for open and finite temperature systems. We also comment
on the Affleck-Ludwig boundary entropy in the context of the post measurement
entanglement entropy. Finally, we also provide some formulas regarding modular
hamiltonians and entanglement spectrum in the after measurement systems. After
through discussion regarding CFT systems we also provide some predictions
regarding massive field theories. We then discuss a generic method to calculate
the post measurement entanglement entropy in the free fermion systems. Using
the method we study the post measurement entanglement entropy in the XY spin
chain. We check numerically the CFT and the massive field theory results in the
transverse field Ising chain and the XX model. In particular, we study the post
meaurement entanglement entropy in the infinite, periodic and open critical
transverse field Ising chain and the critical XX model. The effect of the
temperature and the gap is also discussed in these models.
