Abstract
We generalize the energy-entropy ratio inequality in quantum field theory (QFT) established by one of us from localized states to a larger class of states. The states considered in this paper can be in a charged (non-vacuum) representation of the QFT or may be only approximately localized in the region under consideration. Our inequality is S(Ψ || Ω) ≤ 2π R ( Ψ, H_ρ Ψ ) + log d(ρ) + ε, where S is the relative entropy, where R is a "radius" (width) characterizing the size of the region, d(ρ) is the statistical (quantum) dimension of the given charged sector ρ hosting the quantum state Ψ, Ω is the vacuum state, H_ρ is the Hamiltonian in the charged sector, and ε is a tolerance measuring the deviation of Ψ from the vacuum according to observers in the causal complement of the region.