Stringy black holes and the geometry of entanglement

Recently striking multiple relations have been found between pure state two- and three-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 theories. In particular, we show that finding the frozen values of the moduli in the calculation of the macroscopic black-hole 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 three-qubit pure 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.

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) and by the dyonic charges. The points of intersection of the line with the quadric $\mathcal{Q}$ are the principal null directions $u^{\pm }$ defined by the frozen value of the moduli $S^{(0)}$, 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 $\eta$ and by the dyonic charges. The point of intersection of the line tangent to the quadric $\mathcal{Q}$ 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 $\mathcal{Q}$ 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 $\mathcal{Q}$ 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 $\mathcal{Q}$ is an isotropic line, i.e. it lies entirely inside the quadric and coincides with one from the family of special lines of $\mathcal{Q}$. 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 $\mathcal{Q}$ is an isotropic line, coinciding with one from the other family of special lines of $\mathcal{Q}$. These lines are related to the anti-self-duality of the Plücker matrix and are called $\beta$ lines.

Figure 6

Geometric representation of small black holes corresponding to real states in the totally separable $(A)(B)(C)$ class. Such holes are represented by a single point lying at the intersection of an $\alpha$ line and a $\beta$ line. Here this point is represented by $\xi$. The other point $\eta$ is either projectively equivalent to $\xi$ or can be taken to be zero.