Gibbs-Bogoliubov inequality on Nishimori line

The Gibbs-Bogoliubov inequality states that the free energy of a system is always lower than that calculated by a trial function. In this study, we show that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori line for Ising spin-glass models with Gaussian randomness. Our inequality states that the quenched free energy of a system is always lower than that calculated using a quenched trial function. The key component of the proof is the convexity of the pressure function $\mathbb{E}\left[\log Z_{} \right]$ with respect to the parameters along the Nishimori line, which differs from the conventional convexity with respect to the inverse temperature. When our inequality was applied to mean-field models, such as the Sherrington-Kirkpatrick model and $p$-spin model, the bound coincided with the replica-symmetric solution indicating that the equality holds.