Is the EMI model a QFT? An inquiry on the space of allowed entropy functions

Abstract

The mutual information I(A, B) of pairs of spatially separated regions satisfies, for any d-dimensional CFT, a set of structural physical properties such as positivity, monotonicity, clustering, or Poincaré invariance, among others. If one imposes the extra requirement that I(A, B) is extensive as a function of its arguments (so that the tripartite information vanishes for any set of regions, I3(A, B, C ) ≡ 0), a closed geometric formula involving integrals over ∂A and ∂B can be obtained. We explore whether this “Extensive Mutual Information” model (EMI), which in fact describes a free fermion in d = 2, may similarly correspond to an actual CFT in general dimensions. Using the long-distance behavior of IEMI(A, B) we show that, if it did, it would necessarily include a free fermion, but also that additional operators would have to be present in the model. Remarkably, we find that IEMI(A, B) for two arbitrarily boosted spheres in general d exactly matches the result for the free fermion current conformal block $$ {G}_{\Delta =\left(d-1\right),J=1}^d $$ G ∆ = d − 1 , J = 1 d . On the other hand, a detailed analysis of the subleading contribution in the long-distance regime rules out the possibility that the EMI formula represents the mutual information of any actual CFT or even any limit of CFTs. These results make manifest the incompleteness of the set of known constraints required to describe the space of allowed entropy functions in QFT.

Local measures of entanglement in black holes and CFTs

“…The additivity property is something that distinguishes partial entanglement entropy from most other quantum information quantities. Mutual information is not a good partial entanglement entropy candidate because it is not an extensive quantity, except for special cases such as the 2d free massless fermion CFT [27,28]. Compared to (I.)…”

Is the EMI model a QFT? An inquiry on the space of allowed entropy functions

“…Five dimensions. Moving to d = 4 + 1, we can consider for instance the strip coefficient k (5) and the one appearing in the entanglement entropy across a S 3 , which we denote F (5) . As explained in appendix A, both can be obtained from the mutual information of a pair of concentric spheres of radii…”

“…Before diving into a detailed analysis of this result, let us study certain limiting cases. 5 To arrive at this expression we also use the identity…”

“…calling Λ to a Lorentz transformation, and taking X, Y , as regions with spatial boundary in a common null plane, we have (Markov property) (8) Positivity and monotonicity are properties of MI as a measure of correlations, monotonicity encoding strong subadditivity of the entropy [6]. The symmetry properties (5) tell that MI is in fact a combination of (undefined) single region functions as in (1). Poincaré invariance and clustering follow the same properties of ordinary correlators.…”

“…For example, it is known that for a CFT the MI for faraway regions has a series expansion in the distance from where the conformal dimensions of primary fields may be extracted [3,4]. The rest of the CFT data, the coefficient of the three point functions, may be obtained from the MI involving three or more distant regions [5].…”