Revisiting Nishimori multicriticality through the lens of information measures

The quantum error correction threshold is closely related to the Nishimori physics of random statistical models. We extend quantum information measures such as coherent information beyond the Nishimori line and establish them as sharp indicators of phase transitions. We derive exact inequalities for several generalized measures, demonstrating that each attains its extremum along the Nishimori line. Using a fermionic transfer matrix method, we compute these quantities in the 2d pm 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. We further analyze the domain-wall free energy distribution and confirm its scale invariance at the multicritical point.