Black Holes, Qubits and Octonions

TL;DR

This work uncovers deep mathematical correspondences between black hole entropy in string/M-theory and quantum entanglement in multi-qudit systems. Central to the bridge are invariants: Cayley’s hyperdeterminant for three qubits and Cartan’s quartic $E_{7(7)}$ invariant for seven-qubit entanglement, with the Fano plane and octonions providing a natural algebraic language; Freudenthal triple systems unify four- and five-dimensional descriptions. The authors develop a comprehensive dictionary linking black hole charge configurations, dualities, and horizon data to entanglement classes (GHZ, W, Bell), SLOCC/LOCC orbits, and multi-qudit entanglement measures, extending from STU models to N=8 supergravity and 5D theories. They also explore higher-order corrections, attractor behavior, and error-robustness analogies in the quantum information setting, revealing a rich, structurally unified picture that invites further cross-pollination between gravity and quantum information. The significance lies in a rigorous algebraic framework that translates gravitational invariants into entanglement language, enabling new insights and potential experimental probes of octonionic and Freudenthal structures through quantum information experiments.

Abstract

We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits, known as the 3-tangle, and the entropy of the 8-charge STU black hole of N=2 supergravity, both of which are given by the [SL(2)]^3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. There are further relationships between the attractor mechanism and local distillation protocols. At the microscopic level, the black holes are described by intersecting D3-branes whose wrapping around the six compact dimensions T^6 provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, |ABC> (A,B,C=0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N=8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan's E_7 supset [SL(2)]^7 invariant. The qubits are naturally described by the seven vertices ABCDEFG of the Fano plane, which provides the multiplication table of the seven imaginary octonions, reflecting the fact that E_7 has a natural structure of an O-graded algebra. This in turn provides a novel imaginary octonionic interpretation of the 56=7 x 8 charges of N=8: the 24=3 x 8 NS-NS charges correspond to the three imaginary quaternions and the 32=4 x 8 R-R to the four complementary imaginary octonions. N=8 black holes (or black strings) in five dimensions are also related to the bipartite entanglement of three qutrits (3-state systems), where the analogous measure is Cartan's E_6 supset [SL(3)]^3 invariant.

Figures (9)

• Figure 1: The 3-index quantity $a_{ABC}$ is an example of a hypermatrix, here depicted as a cube. In 1845 Cayley generalised the determinant of a $2\times 2$ matrix to the hyperdeterminant of a $2 \times 2 \times 2$ hypermatrix. The hyperdeterminant may also be assembled from the $2\times 2$ matrices formed by partitioning the hypermatrix cube across its planes of symmetry as shown in \ref{['sec:Freudenthal-Fano']}.
• Figure 2: A diagrammatic comparison of the hyperdeterminant against the ordinary determinant. The encircled letters denote tensors and the lines are index contractions. The similarities are manifest as the building blocks are virtually identical, the principle difference being the extra indices on the $a$'s. The hyperdeterminant is simply the determinant of the determinant-like object $\gamma$. (The third indices on the $a$'s are the free indices of the $\gamma$'s).
• Figure 3: A schematic of four of the entanglement measures of a three-qubit system: the 2-tangles, $\tau_{AB}, \tau_{AC}$, and $\tau_{BC}$ and the 3-tangle, $\tau_{ABC}$. The 2-tangles give the bipartite entanglements between pairs, while the 3-tangle is a measure of the genuine three-way entanglement.
• Figure 4: The vertices of the hypermatrix cube from \ref{['fig:BinaryFanoBasisCube']} are transformed under the dictionary \ref{['eq:charges7']} to electric and magnetic charges of the $STU$ black hole. The black hole entropy is related to the hyperdeterminant of this hypermatrix, and can also be computed from the three pairs of slicings of the cube along its symmetry planes (see \ref{['sec:Freudenthal-Fano']}).
• Figure 11: The $E_7$ entanglement diagram corresponding to the decomposition \ref{['eq:56Decomp']} and the state \ref{['eq:7QubitState']}. Each of the seven vertices $A, B, C, D, E, F, G$ represents a qubit and each of the seven triangles $ABD, BCE, CDF, DEG, EFA, FGB, GAC$ describes a tripartite entanglement. As discussed in \ref{['sec:octandfan']} the oriented triangles correspond to quaternionic cycles in the multiplication table of imaginary octonions.