  Random measurements have been shown to induce a phase transition in an
extended quantum system evolving under chaotic unitary dynamics, when the
strength of measurements exceeds a threshold value. Below this threshold, a
steady state with a sub-thermal volume law entanglement emerges, which is
resistant to the disentangling action of measurements, suggesting a connection
to quantum error-correcting codes. Here we quantify these notions by
identifying a universal, subleading logarithmic contribution to the volume law
entanglement entropy: $S^{(2)}(A)=\kappa L_A+\frac{3}{2}\log L_A$ which bounds
the mutual information between a qudit inside region $A$ and the rest of the
system. Specifically, we find the power law decay of the mutual information
$I(\{x\}:\bar{A})\propto x^{-3/2}$ with distance $x$ from the region's
boundary, which implies that measuring a qudit deep inside $A$ will have
negligible effect on the entanglement of $A$. We obtain these results by
mapping the entanglement dynamics to the imaginary time evolution of an Ising
model, to which we can apply field-theoretic and matrix-product-state
techniques. Finally, exploiting the error-correction viewpoint, we assume that
the volume-law state is an encoding of a Page state in a quantum
error-correcting code to obtain a bound on the critical measurement strength
$p_{c}$ as a function of the qudit dimension $d$:
$p_{c}\log[(d^{2}-1)({p_{c}^{-1}-1})]\le \log[(1-p_{c})d]$. The bound is
saturated at $p_c(d\rightarrow\infty)=1/2$ and provides a reasonable estimate
for the qubit transition: $p_c(d=2) \le 0.1893$.
