  We extend the results of arXiv:2209.12903 by studying local projective
measurements performed on subregions of two copies of a CFT${}_2$ in the
thermofield double state and investigating their consequences on the bulk
double-sided black hole holographic dual. We focus on CFTs defined on an
infinite line and consider measurements of both finite and semi-infinite
subregions. In the former case, the connectivity of the bulk spacetime is
preserved after the measurement. In the latter case, the measurement of two
semi-infinite intervals in one CFT or of one semi-infinite interval in each CFT
can destroy the Einstein-Rosen bridge and disconnect the bulk dual spacetime.
In particular, we find that a transition between a connected and disconnected
phase occurs depending on the relative size of the measured and unmeasured
subregions and on the specific Cardy state the measured subregions are
projected on. We identify this phase transition as an entangled/disentangled
phase transition of the dual CFT system by computing the post-measurement
holographic entanglement entropy between the two CFTs. We also find that bulk
information encoded in one CFT in the absence of measurement can sometimes be
reconstructed from the other CFT when a measurement is performed, or can be
erased by the measurement. Finally, we show that a purely CFT calculation of
the Renyi entropy using the replica trick yields results compatible with those
obtained in our bulk analysis.
