Black holes, qubits and octonions

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 (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge ST U 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 ST U 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, |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 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G 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 × 8 charges of N = 8: the 24 = 3 × 8 NS-NS charges correspond to the three imaginary quaternions and the 32 = 4 × 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 ⊃ [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 .

D = 3 unification of curious supergravities

“…[17], and refs. therein) for maximal supergravity in D = 4 → D = 3, and then proceeds with a chain of maximal, symmetric and rank-preserving embeddings which has already been considered in [7,14,18]:…”

“…The chain of embeddings (4.1)-(4.4) has been used in [18] (also cfr. [14]) to study the tripartite entanglement of seven qubits inside E 7 . Moreover, it was recently exploited in [7] in order to obtain the N = 1, D = 4 theory with 7 WZ multiplet given in the fourth line of (4.5).…”

Black holes admitting a Freudenthal dual

“…where the index i refers to the three triality related versions of P, Q transforming as a (2, 4) of [10,54]. The idea was then further generalised in [28] giving the appropriate definition for N = 8, D = 4 black holes which we adopted here (65).…”

“…It is well known that the four dimensional supergravities that arise from string and M-theory, such as the N = 2 ST U , N = 2 "magic", N = 4 heterotic and N = 8 M/Type II, may all be described by a Freudenthal triple system (FTS) M(J) [1,2,3,4,5,6,7,8,9,10], where J is a cubic Jordan algebra underlying the corresponding 5D supergravity [3,11,12]. The corresponding continuous U-duality is given by the automorphism group Aut(M(J)), e.g.…”

“…"Large" BPS, "small" BPS and large non-BPS correspond to ∆(x) > 0, ∆(x) = 0 and ∆(x) < 0, respectively. In this continuous case, the black hole entropy, U-duality orbits and generating solutions are well understood [5,7,10,14,15,16,17,18,19,20,21,22,23,24]. In the fully quantised string theory, however, black hole charges x must be integer valued and hence assigned to * leron.borsten@imperial.ac.uk † duminda.dahanayake@imperial.ac.uk ‡ m.duff@imperial.ac.uk § william.rubens06@imperial.ac.uk elements of an integral FTS M(J) where J is an integral cubic Jordan algebra [25,26,27,28].…”