  Holography has taught us that spacetime is emergent and its properties depend
on the entanglement structure of the dual theory. In this paper, we describe
how changes in the entanglement due to a local projective measurement (LPM) on
a subregion $A$ of the boundary theory modify the bulk dual spacetime. We find
that LPMs destroy portions of the bulk geometry, yielding post-measurement bulk
spacetimes dual to the complementary unmeasured region $A^c$ that are cut off
by end-of-the-world branes. Using a bulk calculation in $AdS_3$ and tensor
network models of holography, we show that the portions of the bulk geometry
that are preserved after the measurement depend on the size of $A$ and the
state we project onto. The post-measurement bulk dual to $A^c$ includes regions
that were originally part of the entanglement wedge of $A$ prior to
measurement. This suggests that LPMs performed on a boundary subregion $A$
teleport part of the bulk information originally encoded in $A$ into the
complementary region $A^c$. In semiclassical holography an arbitrary amount of
bulk information can be teleported in this way, while in tensor network models
the teleported information is upper-bounded by the amount of entanglement
shared between $A$ and $A^c$ due to finite-$N$ effects. When $A$ is the union
of two disjoint subregions, the measurement triggers an entangled/disentangled
phase transition between the remaining two unmeasured subregions, corresponding
to a connected/disconnected phase transition in the bulk description. Our
results shed new light on the effects of measurement on the entanglement
structure of holographic theories and give insight on how bulk information can
be manipulated from the boundary theory. They could also be extended to more
general quantum systems and tested experimentally, and represent a first step
towards a holographic description of measurement-induced phase transitions.
