# Black Holes, Qubits and Octonions

**Authors:** L. Borsten, D. Dahanayake, M. J. Duff, H. Ebrahim, W. Rubens  
**arXiv:** [0809.4685](https://arxiv.org/abs/0809.4685)  
**Date:** 2008-09-26  
**Summary page:** https://www.summarizepaper.com/en/paper/0809.4685v4/

## Key points

- Study explores connections between black hole entropy in string theory and quantum entanglement of qubits and qutrits
- Tripartite entanglement of qubits linked to entropy of 8-charge STU black holes in N=2 supergravity
- Measure of tripartite entanglement (3-tangle) intricately linked to black hole entropy through Cayley's [SL(2)]^3 invariant hyperdeterminant
- Black holes described as intersecting D3-branes wrapping around six compact dimensions T^6, establishing correspondence with qubits
- Analysis extends to 56-charge N=8 black holes and seven-qubit tripartite entanglement using Cartan's E_7 supset [SL(2)]^7 invariant
- Qubits represented by vertices on Fano plane based on E_7's O-graded algebra structure, linking to imaginary octonions multiplication table
- Interpretation links N=8 black hole charges to imaginary quaternions and octonions, revealing underlying mathematical structures
- Relationships explored between N=8 black holes/strings in five dimensions and bipartite entanglement of three qutrits using Cartan's E_6 supset [SL(3)]^3 invariant

## Summary

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In their paper "Black Holes, Qubits and Octonions," authors L. Borsten, D. Dahanayake, M. J. Duff, H. Ebrahim, and W. Rubens explore the fascinating connections between black hole entropy in string theory and quantum entanglement of qubits and qutrits in quantum information theory. The study delves into the intricate relationship between tripartite entanglement of qubits and the entropy of 8-charge STU black holes in N=2 supergravity. The researchers highlight how the measure of tripartite entanglement, known as the 3-tangle, is intricately linked to black hole entropy through Cayley's [SL(2)]^3 invariant hyperdeterminant introduced in 1845. They further investigate connections between the attractor mechanism and local distillation protocols within this framework. At a microscopic level, they describe black holes as intersecting D3-branes wrapping around six compact dimensions T^6 to interpret charges in string theory. By associating three-qubit basis vectors with corresponding wrapping cycles, they establish a correspondence between black holes and qubits. The study extends its analysis to 56-charge N=8 black holes and seven-qubit tripartite entanglement using Cartan's E_7 supset [SL(2)]^7 invariant. The researchers illustrate how qubits can be naturally represented by vertices on the Fano plane—reflecting a multiplication table for imaginary octonions based on E_7's O-graded algebra structure. Moreover, they introduce an innovative interpretation linking N=8 black hole charges to imaginary quaternions and octonions—providing insights into the underlying mathematical structures at play in these complex systems. Finally, the paper explores relationships between N=8 black holes (or strings) in five dimensions and bipartite entanglement of three qutrits using Cartan's E_6 supset [SL(3)]^3 invariant—a concept that sheds light on additional layers of connection between quantum information theory and string theory. Overall, this comprehensive study offers a deep dive into the interplay between fundamental concepts in physics such as black hole entropy, quantum entanglement, and algebraic structures—opening up new avenues for exploration at the intersection of theoretical physics and information science.