Black holes, qubits and octonions

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 (Alice, Bob and Charlie), 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. Moreover the classification of three-qubit entanglements is related to the classification of N = 2 supersymmetric STU black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. 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, vertical bar 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 superset of [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. We contrast this approach with that based on Jordan algebras and the Freudenthal triple system. 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 superset of [SL(3)](3) invariant. Similar analogies exist for magic N = 2 supergravity black holes in both four and five dimensions. Despite the ubiquity of octonions, our analogy between black holes and quantum information theory is based on conventional quantum mechanics but for completeness we also provide a more exotic one based on octonionic quantum mechanics. Finally, we note some intriguing, but still mysterious, assignments of entanglements to cosets, such as the 4-way entanglement of eight qubits to E-8/[SL(2)](8). (C) 2009 Elsevier B.V. All rights reserved.