What surface maximizes entanglement entropy?

TL;DR

This work addresses which entangling surface, among all those of fixed area, maximizes entanglement entropy in a quantum field theory, using both field-theoretic and holographic (Ryu–Takayanagi) perspectives. In $d=4$ the maximization problem is tightly linked to minimizing the Willmore energy, so the round sphere maximizes entropy within each topology, while Lawson and Clifford/Wilmore surfaces govern maximal entropy for higher-genus topologies. The authors generalize to $d>4$ by introducing a normalized Willmore energy to compare surfaces with fixed area, finding that the round sphere remains the global entropy maximizer across topologies and that product topologies $S^m\times S^n$ minimize the normalized Willmore energy within their class. Collectively, these results reveal a deep connection between geometric variational problems (Willmore/Lawson) and entanglement entropy, with implications for holography and higher-dimensional geometry.

Abstract

For a given quantum field theory, provided the area of the entangling surface is fixed, what surface maximizes entanglement entropy? We analyze the answer to this question in four and higher dimensions. Surprisingly, in four dimensions the answer is related to a mathematical problem of finding surfaces which minimize the Willmore (bending) energy and eventually to the Willmore conjecture. We propose a generalization of the Willmore energy in higher dimensions and analyze its minimizers in a general class of topologies $S^m\times S^n$ and make certain observations and conjectures which may have some mathematical significance.

Figures (2)

• Figure 1: Ratio of normalized Willmore energies (ellipsoid to sphere) in dimension $d$.
• Figure 2: $\widehat{W}_r(x)$ for (from left to right) $S^3\times S^1$, $S^2\times S^2$ and $S^1\times S^3$, respectively.