Exact High-Temperature Quantum Area Law
We prove a $β^{2}$ area law for the quantum mutual information of thermal states of local lattice Hamiltonians: for any bipartitioning $A|B$ and all inverse temperatures $β$ below a critical value $β^{*}$, we show $\mathcal{I}_β(A,B) \leqslant \tilde{f}(β)\, β^{2}\,|\partial_{AB}|$, where $|\partial_{AB}|$ denotes the size of the boundary region and $\tilde{f}(β)$ remains finite as $β\to 0$. This quadratic scaling improves the relationship between high-temperature correlations and macroscopic thermodynamics. While a linear area law implicitly permits a loose, temperature-independent upper bound on extractable work and binding energy, the quadratic scaling ensures that the nonequilibrium free energy available after decoupling regions and the total energy of the interface both are proportional to the area of the interface of $A|B$ and vanish as $β$ at high temperatures. These results show that the high-temperature mutual information and correlation are suppressed, respectively, quadratically and linearly in inverse temperature, which are consistent with the expected thermal decoupling in macroscopic thermodynamic quantities.