  Measurements can detect errors in a decohered quantum memory allowing active
error correction to increase the memory time. Previous understanding of this
mechanism has focused on evaluating the performance of error correction
algorithms based on measurement results. In this work, we instead intrinsically
characterize the information dynamics in a quantum memory under repeated
measurements, using coherent information and relative entropy. We consider the
dynamics of a $d$-dimensional stabilizer code subject to Pauli errors and noisy
stabilizer measurements and develop a $(d+1)$-dimensional statistical mechanics
model for the information-theoretic diagnostics. Our model is dual to the model
previously obtained for the optimal decoding algorithm, and the potential
decoding transition in the quantum memory again manifests as a thermal phase
transition in the statistical mechanics model. We explicitly derive the model
and study the phase transition in information encoding in three examples:
surface codes, repetition codes, and the XZZX code.
