Black Holes, Qubits and Octonions

The quantized charges x of four-dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U duality and whose U-invariant quartic norm {delta}(x) determines the lowest-order entropy. Here, we introduce a Freudenthal duality x{yields}x-tilde, for which x-tilde-tilde=-x. Although distinct from U duality, it nevertheless leaves {delta}(x) invariant. However, the requirement that x-tilde be an integer restricts us to the subset of black holes for which {delta}(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantized charges A of five-dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest-order entropy. We introduce an analogous Jordan dual A*, with N(A) necessarily a perfect cube, for which A**=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.

Black holes admitting a Freudenthal dual

We give a review of the black-hole/qubit correspondence that incorporates not only the earlier results on black-hole entropy and entanglement measures, seven qubits and the Fano plane, wrapped branes as qubits and the attractor mechanism as a distillation procedure, but also newer material including error-correcting codes, Mermin squares, Freudenthal triples and four-qubit entanglement classification.

https://iopscience.iop.org/article/10.1088/0264-9381/29/22/224008/pdf

The black-hole/qubit correspondence: an up-to-date review

Small representations, string instantons, and Fourier modes of Eisenstein series

We show that the three-qubit entanglement classes, (0) null, (1) separable $A\text{\ensuremath{-}}B\text{\ensuremath{-}}C$, (2a) biseparable $A\text{\ensuremath{-}}BC$, (2b) biseparable $B\text{\ensuremath{-}}CA$, (2c) biseparable $C\text{\ensuremath{-}}AB$, (3) $W$, and (4) Greenberger-Horne-Zeilinger, correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3, and 4 of a Freudenthal triple system defined over the Jordan algebra $\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}$. We also compute the corresponding stochastic local operations and classical communication orbits.

Sergei Krutelevich

Papers

Jordan algebras, exceptional groups, and Bhargava composition