A finite entanglement entropy and the c-theorem

TL;DR

The paper defines a finite entanglement quantity F(A,B) built from standard entropies, F(A,B)=S(A)+S(B)-S(A∩B)-S(A∪B), which remains finite in QFT and equals the relative entropy between joint and product states. In 1+1D CFTs, F for one-component intersecting sets is fixed to $F(A,B)=k\log\left(\frac{r_A r_B}{r_{A\cap B} r_{A\cup B}}\right)$ with $k=(c+\bar c)/6$, while S(A) diverges as $S(A)=(c+\bar c)/6\log(r_A/\epsilon)$. An entropic c-theorem follows from SSA: the function $G(r)$ is increasing (in log variables concave), and $C(r)=rG'(r)$ is nonnegative, decreasing along RG flow, approaching $k$ at fixed points, reproducing Zamolodchikov’s result in an entropy framework. The two-component (non-intersecting) case introduces a cross-ratio function $U(\eta)$ with $U(\eta)=U(1-\eta)$ and boundedness, and numerical tests show the ability to distinguish theories with the same central charge. The work also discusses potential physical interpretations, including dimensional reduction and entropy flux in Unruh/Hawking contexts, suggesting deep links between entanglement structure and QFT classification.

Abstract

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with non zero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincare symmetry and the mathematical properties of the entropy. In particular, for one component sets in two dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for 1+1 dimensional QFT. We propose this well defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.

Figures (3)

• Figure 1: (a) The spatial surfaces $A$ and $A^\prime$ have the same causal domain of dependence given by the diamond shaped set in this picture. The same entropy should correspond to $A$ and $A^\prime$. Thus, we do not make distinctions between a diamond set and any of its Cauchy surfaces. (b) Two spatial surfaces $A$ and $B$ included in the same global Cauchy surface. Writting $\text{area}(X)$ for the $d-1$ dimensional volume of the boundary of a $d$ dimensional spatial set $X$ we have $\text{area}(A)+\text{area}(B)=\text{area}(A\cap B)+\text{area}(A\cup B)$.
• Figure 2: (a) Two intersecting one component sets $A$ and $B$ whose straight Cauchy surfaces have sizes $r_A$ and $r_B$ respectively. The diamonds formed by the intersection and union (followed by causal completion) of $A$ and $B$ have sizes $r_{A\cap B}$ and $r_{A\cup B}$. This configuration of sets is uniquely determined by the position of the points $x_1, x_2, x_3$ and $x_4$ forming the spatial corners of the diamonds. (b) Configuration of two intersecting sets of size $\sqrt{r_1 r_2}$ from which eq. (\ref{['relativista']}) can be obtained by means of the SSA inequality.
• Figure 3: The Rindler wedge $A$ and an exterior diamond set $B$. The curves of constant $\nu$ are lines through the origin.