Revisiting Nishimori multicriticality through the lens of information measures

The coherent information and quantum error correction threshold are closely related to the Nishimori physics of random statistical models. We extend quantum information measures—such as coherent information and error-correction success probability—beyond the Nishimori line and establish them as sharp indicators of phase transitions. We further derive exact inequalities for generalized measures like coherent information, demonstrating that they attain their extrema along the Nishimori line. Using a refined fermionic transfer matrix method, we compute these quantities in the two-dimensional ±J random-bond Ising model—corresponding to a surface code under bit-flip noise—on system sizes up to 512 and over 10⁷ disorder realizations. All critical points extracted from statistical and information-theoretic indicators coincide with high precision at p_c = 0.1092212(4), with the coherent information exhibiting the smallest finite-size effects. At this point we confirm the scale invariance of the domain-wall free energy distribution, establishing that statistical and information-theoretic thresholds are identical.