  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.
