## Duff, String Theory, Entanglement and Hyperdeterminants

September 2, 2010

Mike Duff has been back in the science news with the publication of one of his papers and a suitably hyped press release from Imperial College.  The research does not actually propose a test of string theory, it merely uses some mathematical ideas in a way inspired by string theory to analyse the entanglement of qubits. Even so, the work is still pretty exciting because of this connection.

Duff’s work in this area began when he noticed that hyperdeterminants come up in the theory of entanglement and also as U-duality invariants defining the entropy of black holes in string theory. At the time, not many applications of hyperdeterminants were well-known, so their appearance in two parts of physics at the same time was taken as a sign that there may be some connection, Duff and his collaborators have been exploring this idea ever since.

The hyperdeterminant is a generalization of determinants to multi-dimensional matrices, or hypermatrices. For a 2x2x2 hypermatrix the hyperdeterminant is a homogeneous degree four polynomial in the 8 components of the hypermatrix, and is known as Cayley’s hyperdeterminant. It can be an invariant characterising the entanglement of three qubits, or an invariant of U-duality for a black hole.

If you look at one of his early papers on this you may notice that he actually cites one of my number theory papers, so you can see that I have some personal interest in this subject. He only cited me because he liked my formula for the hyperdeterminant in terms of Levi-Civita symbol which is

$|A|$  = $-\frac{1}{2}$ $\epsilon^{ab} \epsilon^{cd} \epsilon^{ij} \epsilon^{kl} \epsilon^{rs} \epsilon^{tu}$  $a_{air} a_{bjt} a_{cks} a_{dlu}$

In fact the connection is much more interesting than that because my paper makes a link between hyperdeterminants and elliptic curves. Further work on this has shown that the next hyperdeterminant up for a 2x2x2x2 hypermatrix is related to the j-invariant and of course the j-invariant has been connected to the entropy of black holes too. This larger hyperdeterminant is a polynomial of degree 24, and here the number 24 is connected to the well-known importance of this number in string theory.  I’d be happy to explain this to Duff over a pint if he wants to get in touch 🙂

Duff has taken his knowledge of these invariants in black hole entropy and applied it to count the number of possible states of entanglement for 4 qubits. That is what the latest paper is about. I think the real excitement is the idea that there may be some connection that is more than just a similarity of the mathematics. The question is, can the work be extended to much larger numbers of qubits in a way that makes string theory look like a theory of entanglement with the qubits playing the role of quantized information at a fundamental level? I don’t know if that is what Duff is thinking of, or if he has some deeper reason to expect something like that to be true, but it is much more interesting than the non-idea that this work provides a test for string theory.

Update: Here are some other articles worth linking to: Kea is covering hyperdeterminants as M Theory lessons 345, 346, 347, 348, 349, 350 and of course she has written earlier stuff that you can search for. Lubos has of course mentioned this topic before and even used my formula for the hyperdeterminant here. Another reasonable report on the latest findings can be found at Universe Today.