What surface maximizes entanglement entropy?

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 that 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 that may have some mathematical significance.

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.