There is apparently a Hungarian day in physics today. Péter Lévay from Budapest

points out mathematical similarities between non-trivial problems in quantum information theory on one side and mathematical aspects of BPS black holes on the other side. A quick analysis indicates that the author knows what he is talking about on both sides of the conjectured "duality".

First of all, I should mention a related paper by Michael Duff from January 2006:

This "older" paper found the square root of Cayley's hyperdeterminant introduced in 1845 at two seemingly unrelated places:

- in three-qubit quantum entanglement - think about the GHZ state /000>+/111> whose importance and bizarre properties were pointed out to me by Herman Verlinde
- in the entropy formula for extremal black holes in d=4, N=2, STU theories
- well, OK, meant three places: Michael Duff has also pointed out that the hyperdeterminant appears in the Nambu-Goto action for a 2+2-dimensional target space: see here

If you don't know what STU backgrounds are, they are heterotic strings on six-tori and the three number S,T,U are three complex scalar fields in d=4, namely the dilaton/axion, the complexified Kähler structure, and complex structure, each of which gives you a single SL(2,Z) duality group. Alternatively, you may describe this background using a dual description with type II on K3.

The letters S-T-U are interesting because there is a triality symmetry permuting all these three fields - and this triality is a part of the full U-duality group (entertainingly enough, the general U-duality group is a reconciliation of S-dualities and T-dualities, but these STU letters are unrelated to the S-T-U letters describing the particular background). Such a discrete symmetry is very constraining. These symmetries consequently tell you that the entropy, as a function of four electric and four magnetic charges, must have a very special form.

If you organize the eight charges into a "2 x 2 x 2 hypermatrix", the squared entropy is given by Cayley's hyperdeterminant. This hyperdeterminant is actually a polynomial of fourth (not second) order in the hypermatrix elements (the entropy of four-dimensional string theory black holes is always a square root of¨duality-invariant, *quartic* polynomials in charges), with six copies of a two-dimensional epsilon symbol (to get rid of 12 indices):

- HyperDet M = -0.5 e(ab)e(cd)e(ef)e(gh)e(ij)e(kl) M(ace)M(bdg)M(ikf)M(jlh)

In the formula above, it is assumed that Cayley had known Einstein's sum rule back in 1845, i.e. 34 years prior to Einstein's birth, and all indices are either 0 or 1.

While the fact that the black hole entropy depends on the hyperdeterminant may be easy for a string theorist, the fact that it is useful to study three-qubit quantum entanglement is simple for a quantum information theorist. At least at the level of this interesting algebraic structure by Cayley, the two fields of physics seem to converge.

They also converge sociologically at the Perimeter Institute where some of our friends also want them to converge towards loop quantum gravity which seems slightly less realistic. ;-)

It is up to you whether you find these similarities to be just mathematical coincidences or signals of a profound connection. During Preskill's lectures, I was also thinking how to represent various operators in the multi-qubit systems by gamma matrices acting on the spinors. But I, for one, find it likely that the similarity is just a coincidence.

Péter Lévay continues in this line of reasoning. He is not satisfied just with the cute entropy formula written using the object defined above. He also writes the entropy formula in terms of the squared norm of a three-qubit state associated with the horizon. See equation 99, for example, whose general structure cannot hide a similarity with the Ooguri-Strominger-Vafa formula. However, his equation 99 has a very different interpretation of the wavefunction than OSV have. The author does not cite OSV, as far as I see. Moreover, there is one more important difference: in the OSV picture, the squared wavefunction (squared topological partition sum) is equal to the exponentiated entropy, not the entropy itself.

The paper also finds an interpretation of the attractor mechanism based on quantum computation - as some canonical optimization problem.

At any rate, quantum black holes do carry quantum information, the relevant entropy is, at least in some cases, attributable to the entanglement entropy, and the research of quantum entanglement in this context could turn out to be very fruitful which is why many of us should look at these "coincidences" more carefully. They may very well clarify the origin and mathematical details of the black hole complementarity.

But notice that this particular similarity does not really translate the full quantum information of the black hole to the qubits of the corresponding quantum computer. Instead, the quantum computer only "remembers" the macroscopic features of the black hole, and therefore the motivation described in the previous paragraph is a bit of fraud.

## snail feedback (1) :

Dear Lubos,

thank you very much for interesting information.

There is a triality symmetry permuting 3 complex scalar fields (dilaton, the complexified Kahler structure and complex structure). Naturally this is subgroup of the U-duality group permuting moduli spaces of this fields.

These indications of the entanglement entropy refer to the equivalence between moduli space of the generalized CY-fold (U-dual modular group) and quantum coherent state of the K-theoretic condensates of the octonionic black hole throats (infinite-order orbifold of the generalized conifold).

So the octonionic black hole throat is an irreducible representation of the U-dual root lattice,

thus:

Off schell duality cascade = octonionic matrix-valued Hagedorn tachyon = multi-flux warped octonionic throat = Z_top = 0, right ?

Salute, Planckeon

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