  We study constraints on the effective field theory (EFT) from the relative
entropy between two theories: we refer to these as target and reference
theories. The consequence of the non-negativity of the relative entropy is
investigated by choosing some reference theories for a given target theory
involving field theories, quantum mechanical models, etc. It is found that the
constraints on EFTs, e.g., the single massless scalar field with the
dimension-eight operator, and SMEFT dimension-eight $SU(N)$ gauge bosonic
operators, are consistent with the positivity bounds from the unitarity and
causality when the higher-derivative operators are generated by the interaction
between heavy and light fields. The constraints on Einstein-Maxwell theory with
higher-derivative operators from the non-negativity of relative entropy are
also investigated. The constraints on such EFTs from the relative entropy hold
under an assumption that perturbative corrections from the interaction
involving higher-derivative operators of light fields are not dominant in the
EFTs. The consequence of this study on the weak gravity conjecture and the
second law of thermodynamics is also discussed.
