Black holes, qubits and octonions
File(s)
Accepted version
Author(s)
Borsten, L
Dahanayake, D
Duff, MJ
Ebrahim, H
Rubens, W
Type
Journal Article
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 black hole of supergravity, both of which are given by the invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of supersymmetric 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 -branes whose wrapping around the six compact dimensions provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, , with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan’s invariant. The qubits are naturally described by the seven vertices of the Fano plane, which provides the multiplication table of the seven imaginary octonions, reflecting the fact that has a natural structure of an -graded algebra. This in turn provides a novel imaginary octonionic interpretation of the charges of : the NS–NS charges correspond to the three imaginary quaternions and the R–R to the four complementary imaginary octonions. We contrast this approach with that based on Jordan algebras and the Freudenthal triple system. 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 invariant. Similar analogies exist for magic 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 .
Date Issued
2009-02-01
Date Acceptance
2008-11-13
Citation
Physics Reports, 2009, 471 (3-4), pp.113-219
ISSN
0370-1573
Publisher
Elsevier
Start Page
113
End Page
219
Journal / Book Title
Physics Reports
Volume
471
Issue
3-4
Copyright Statement
© 2009 Elsevier Ltd. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/
Sponsor
Science and Technology Facilities Council (STFC)
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000264967900001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Grant Number
ST/G000743/1
Subjects
Science & Technology
Physical Sciences
Physics, Multidisciplinary
Physics
Black hole
Qubit
Entanglement
Hyperdeterminant
Octonion
MAXWELL-EINSTEIN SUPERGRAVITY
BEKENSTEIN-HAWKING ENTROPY
ELECTRIC MAGNETIC DUALITY
WEAK-COUPLING DUALITY
PURE 3-QUBIT STATES
STRING THEORY
U-DUALITY
JORDAN ALGEBRAS
ENTANGLEMENT MONOTONES
QUANTUM-MECHANICS
Publication Status
Published
Date Publish Online
2009-01-08