Entropy constraints on effective field theory

In effective field theory, the positivity bounds of higher derivative operators are derived from analyticity, causality, and unitarity. We show that the positivity bounds on some operators of the effective field theory, e.g., dimension-eight term of a single massless scalar field, the Standard Model Effective Field Theory dimension-eight $SU(N)$ gauge bosonic operators, and higher-derivative operators in the Einstein-Maxwell theory, generated by interactions between heavy and light degrees of freedom can be derived by the non-negativity of relative entropy. For such effective field theories, we prove that the interactions increase thermodynamic entropy at a fixed charge and an extremal point of energy, which is intimately connected with the extremality relations of black holes exhibiting Weak-Gravity-Conjecture. These arguments are applicable when corrections from the interactions involving higher-derivative operators of light fields are not dominant in the effective field theories. The entropy constraint is a consequence of the Hermiticity of Hamiltonian, and any theory violating the non-negativity of entropy would not respect the second law of thermodynamics.