  We present a theory of the entanglement transition tuned by measurement
strength in qudit chains evolved by random unitary circuits and subject to
either weak or random projective measurements. The transition can be understood
as a nonanalytic change in the amount of information extracted by the
measurements about the initial state of the system, quantified by the Fisher
information. To compute the von~Neumann entanglement entropy $S$ and the Fisher
information $\mathcal{F}$, we apply a replica method based on a sequence of
quantities $\tilde{S}^{(n)}$ and $\mathcal{F}^{(n)}$ that depend on the $n$-th
moments of density matrices and reduce to $S$ and $\mathcal{F}$ in the limit
$n\to 1$. These quantities with $n\ge 2$ are mapped to free energies of a
classical spin model with $n!$ internal states in two dimensions with specific
boundary conditions. In particular, $\tilde{S}^{(n)}$ is the excess free energy
of a domain wall terminating on the top boundary, and $\mathcal{F}^{(n)}$ is
related to the magnetization on the bottom boundary. Phase transitions occur as
the spin models undergo ordering transitions in the bulk. Taking the limit of
large local Hilbert space dimension $q$ followed by the replica limit $n\to 1$,
we obtain the critical measurement probability $p_c=1/2$ and identify the
transition as a bond percolation in the 2D square lattice in this limit.
Finally, we show there is no phase transition if the measurements are allowed
in an arbitrary nonlocal basis, thereby highlighting the relation between the
phase transition and information scrambling. We establish an explicit
connection between the entanglement phase transition and the purification
dynamics of a mixed state evolution and discuss implications of our results to
experimental observations of the transition and simulability of quantum
dynamics.
