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.

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• Classical statistical mechanics
• Open quantum systems & decoherence
• Quantum error correction
• Quantum measurements
• Quantum memories