[Paper Review] Is the EMI model a QFT? An inquiry on the space of allowed entropy functions

César A. Agón, Pablo Bueno+1 more|arXiv (Cornell University)|May 24, 2021

Black Holes and Theoretical Physics85 references23 citations

TL;DR

This paper investigates whether the Extensive Mutual Information (EMI) model—a geometric formula for mutual information (MI) that enforces tripartite information vanishing—can describe a conformal field theory (CFT). Using long-distance limits and conformal block matching, it shows EMI exactly reproduces the free fermion current conformal block in d dimensions for boosted spheres, but subleading corrections rule out any CFT or limit of CFTs. The work exposes gaps in current constraints for allowed entropy functions in QFT.

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 Poincar\'e 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, $I_3(A,B,C)\equiv 0$), a closed geometric formula involving integrals over $\partial A$ and $\partial B$ 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 $I_{ m \scriptscriptstyle EMI}(A,B)$ 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 $I_{ m \scriptscriptstyle EMI}(A,B)$ for two arbitrarily boosted spheres in general $d$ exactly matches the result for the free fermion current conformal block $G^d_{\Delta=(d-1),J=1}$. 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.

Motivation & Objective

• To determine whether the EMI model, which enforces vanishing tripartite information, can describe a genuine CFT in general dimensions.
• To analyze the long-distance behavior of EMI mutual information for spatially separated regions, especially spheres under Lorentz boosts.
• To test whether EMI's MI matches known conformal block structures of CFTs, particularly those of conserved currents.
• To assess whether EMI could represent a limit of CFTs by examining subleading corrections in the large separation regime.
• To identify missing constraints in the set of axioms governing allowed entropy functions in QFT, based on the failure of EMI to correspond to any CFT.

Proposed method

• Derive the EMI mutual information formula as a double integral over the boundaries ∂A and ∂B, involving normal vectors and a Green's function kernel with |xA−xB|−2(d−2) dependence.
• Compute the EMI MI for two boosted spherical regions in d dimensions using conformal coordinates and cross-ratio parametrization.
• Match the EMI result to the conformal block of a conserved current with dimension ∆=d−1 and spin J=1 via analytic continuation and integral representations.
• Perform a large-distance expansion of the EMI MI and compare the subleading term to the corresponding free fermion result using modular flow techniques.
• Use the modular flow of spherical regions to compute the coefficient of the first subleading term in the free fermion MI, enabling quantitative comparison with EMI.
• Apply the Markov property and Poincaré invariance to constrain the space of allowed entropy functions and assess the physical consistency of EMI.

Experimental results

Research questions

• RQ1 Does the EMI model, which enforces I3(A,B,C)=0, correspond to a valid CFT in general d-dimensional spacetime?
• RQ2 Does the EMI mutual information for two boosted spheres exactly match the conformal block of a conserved current in d dimensions?
• RQ3 Are the subleading corrections in the long-distance expansion of EMI MI consistent with those of any known CFT, such as the free fermion?
• RQ4 Can the EMI formula be realized as a limit of CFTs, given its structural similarities to free fermion behavior?
• RQ5 What constraints are missing in the current set of axioms (positivity, monotonicity, clustering, etc.) to fully characterize allowed entropy functions in QFT?

Key findings

• The EMI mutual information for two arbitrarily boosted spheres in general d dimensions exactly matches the conformal block of a conserved current with ∆=d−1 and J=1.
• The long-distance expansion of EMI MI contains a subleading term that does not match the corresponding term in the free fermion mutual information, ruling out EMI as a description of any CFT.
• Despite matching the leading-order conformal block, EMI cannot represent a CFT or any limit of CFTs due to mismatched subleading corrections.
• The EMI model necessarily includes a free fermion in its spectrum if it were to describe a CFT, but additional operators would also be required.
• The failure of EMI to match subleading terms in the free fermion MI highlights the incompleteness of the current set of axioms for allowed entropy functions in QFT.
• The analysis reveals that the known axioms—positivity, monotonicity, clustering, Poincaré invariance, and the Markov property—are insufficient to uniquely characterize the space of allowed entropy functions.