Thursday, April 12, 2007
E6 and Entanglement of Three Qutrits
Michael Duff and Sergio Ferrara posted a follow-up to their papers Black hole entropy and quantum information and E_7 and the tripartite entanglement of seven qubits entitled E_6 and the bipartite entanglement of three qutrits. In order to relate quantum information to 5D black holes, they invoked the use of qutrits, the ternary logic generalizations of qubits. This is a very natural choice and has its roots in the structure of Jordan algebras of degree three. Jordan algebras of degree three underlie the N=2 5D magical supergravities where the charge space of N=2 5D BPS black holes is equated with the Jordan algebra as a vector space over R. In the octonionic case, the Jordan algebra of degree three is called the exceptional Jordan algebra, and the black hole charge space has real dimension 27.
At the end of the paper, Duff and Ferrara state that the "analogy between black holes and quantum information remains, for the moment, just that" and that they "know of no physics connecting them." They mention a comment by Murat Gunaydin (via private communication) suggesting that "the appearance of octonions and split-octonions implies a connection to quaternionic and/or octonionic quantum mechanics." Murat's comment seem quite accurate, but the details of the quaternionic/octonionic quantum mechanics are quite subtle. In 1934, Pascual Jordan, John von Neumann and Eugene Wigner investigated quaternionic/octonionic quantum mechanics and found problems in their Hilbert space formulations. Bischoff in 1993, eventually gave a Hilbert space formulation using the regular representation of the Jordan algebras of degree three. Support for this approach was given by Dray and Manogue's Exceptional Jordan eigenvalue problem. My most recent paper combines the work of Bischoff, Dray and Manogue and elucidates the role of eigenmatrices in the description of N=2 extremal black holes with symmetric moduli space E6(-26)/F4.
The interplay between extremal black holes, entangled qutrits and octonionic quantum mechanics is quite elegant. I tend to take the view that quantum information is the most fundamental, however. Jordan algebras, Freudenthal triple systems and octonions may ultimately be just convenient representations for the description of the quantum logic of nature. As the year progresses, we'll see how this story unfolds, and if we're lucky it will reveal secrets about the nature of spacetime itself.