Proof of the entropic $c$-theorem

Abstract

We prove an entropic $c$-theorem in two dimensions by constructing a $c$-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular $p$-nuclearity), and a finite ultraviolet bound on the resulting $c$-functions. We also note that, formally, the $c$-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.

The paper asks whether the two-dimensional entropic $c$-theorem can be proved directly in continuum quantum field theory, without introducing ultraviolet-regulated entanglement entropies. It replaces entropy strong subadditivity with an increasing-differences inequality for mutual information defined through relative entropy. For spacelike-separated wedges $L_a$ and $R_b$, potential infrared divergences are handled by defining $\mathcal D_R(a,A;b)=\sup_{Y\subset R_b}[I(L_a:Y)-I(L_A:Y)]$, with $Y$ a bounded causal diamond, and an analogous left-moving quantity. The resulting functions $C_R=6a\,\partial_A\mathcal D_R|_{A=a}$ and $C_L$ depend only on $r=\sqrt{ab}$ and are nonnegative and nonincreasing. Assumptions include the Haag–Kastler axioms, split inclusions with finite mutual information across bounded-region gaps, and finite ultraviolet bounds $\mathcal D_R\le(c_R^{\rm UV}/6)\log(A/a)$ and its left-moving counterpart. The proof establishes increasing differences using type-I approximations, the relative-entropy chain rule and monotonicity; additivity and boost invariance then reduce monotonicity to concavity of a one-variable function. At dilation-invariant fixed points, both functions are constant. For chiral-product conformal theories and finite-index rational extensions, $C_{R,L}=6\kappa_{R,L}$, where $\kappa$ is the leading adjacent-interval mutual-information logarithmic coefficient. Equality with stress-tensor central charges remains conditional in general, with rigorous identification available for a subclass. The ultraviolet and infrared limits are not proved to equal the corresponding limiting fixed-point values. A connection to the minimum averaged null energy required to reproduce wedge correlations is stated formally.