Stringy Black Holes and the Geometry of Entanglement

TL;DR

This work builds a bridge between string-theoretic black holes in the STU model and three-qubit entanglement by showing that moduli stabilization in the attractor mechanism corresponds to obtaining a canonical GHZ form under complex SLOCC. The entropy, expressible as S_{BH} = pi sqrt{-D} = (pi/2) sqrt{tau_{ABC}}, is tied to the Cayley hyperdeterminant and to real-entanglement measures through a twistor-based geometric classification in CP^3, where lines intersect a fixed quadric to distinguish GHZ, W, and separable regimes. The analysis provides a unified dictionary linking black hole charge configurations to three-qubit amplitudes, and reveals a deep, geometry-driven embedding of STU black holes into the broader landscape of complex entanglement, with implications for holography via entanglement measures. Overall, the paper shows that attractor dynamics, black hole entropy, and entanglement theory share a common mathematical backbone rooted in hyperdeterminants, twistor geometry, and complex SLOCC transformations.

Abstract

Recently striking multiple relations have been found between pure state 2 and 3-qubit entanglement and extremal black holes in string theory. Here we add further mathematical similarities which can be both useful in string and quantum information theory. In particular we show that finding the frozen values of the moduli in the calculation of the macroscopic entropy in the STU model, is related to finding the canonical form for a pure three-qubit entangled state defined by the dyonic charges. In this picture the extremization of the BPS mass with respect to moduli is connected to the problem of finding the optimal local distillation protocol of a GHZ state from an arbitrary pure three-qubit state. These results and a geometric classification of STU black holes BPS and non-BPS can be described in the elegant language of twistors. Finally an interesting connection between the black hole entropy and the average real entanglement of formation is established.

Figures (6)

• Figure 1: Geometric representation of large black holes corresponding to real states in the GHZ class. The line is defined by the vectors $\xi$ and $\eta$ of Eq. (86) defined by the dyonic charges. The points of intersection of the line with the quadric ${\cal Q}$ are the principal null directions $u^{\pm}$ defined by the frozen value of the moduli $S$ Eq. (76) on the horizon.
• Figure 2: Geometric representation of small black holes corresponding to real states in the W class. The line is defined by the vectors $\xi$ and $\eta$ defined by the dyonic charges. The point of intersection of the line tangent to the quadric ${\cal Q }$ corresponds to the two coincident principal null directions $u^{\pm}$.
• Figure 3: Geometric representation of small black holes corresponding to real states in the A(BC) class. The point off the quadric ${\cal Q}$ is defined by the vector $\xi$ of dyonic charges. The other vector $\eta$ is either projectively equivalent to $\xi$ or vanishing. The dashed lines intersecting at a point refer to the existence of two families of lines on ${\cal Q}$ ruling it.
• Figure 4: Geometric representation of small black holes corresponding to real states in the $C(AB)$ class. The line through the points $\xi$ and $\eta$ lying now on ${\cal Q}$ is an isotropic line, i.e. it lies entirely inside the quadric and coincides with one from the family of special lines of ${\cal Q}$. These lines are related to the self-duality of the Plücker matrix and are called $\alpha$-lines.
• Figure 5: Geometric representation of small black holes corresponding to real states in the $B(AC)$ class. The line through the points $\xi$ and $\eta$ lying now on ${\cal Q}$ is an isotropic line, coinciding with one from the other family of special lines of ${\cal Q}$. These lines are related to the anti-self-duality of the Plücker matrix and are called $\beta$-lines.