The n-Category Café

Yes, well, have fun at the Comlab conference, and say hi to Louis and Bob and Jamie and them from me. (I’ll just be sitting in my icy cold room, as usual, eating my beans and noodles and wondering about how I am going to get my next waitressing job.)

Regarding your Oxford slides, your first two slides are identical, and you have a bad line break at slide 12.

On slide 16, should you say something about normalizing rows to produce probability? I know we’ve discussed this before. Along with the idea of relative probabilities, you had the idea of matrix mechanics over Durov’s generalized rings.

Hmm, perhaps I should return to all that Progic material.

Oxford slide no. 40: There is a classical no-cloning theorem?

(Or is it a QM no-cloning theorem that’s become a classic? :-)

Thanks for this very classical TWF episode! It had starring almost all my favorite characters (octonions, E8, higher gauge theory, etc.) and had as always an excellent storytelling. All that makes it very hard to accept that TWF will be off the air very soon!

There will be a movie, eh a book one day, right?

I noticed that at Woit’s some wrestling has been going on concerning how classification of tripartite entangled states (under SLOCC) may be connected to Duff et all string theory stuff. Maybe of interest here at the cafe is that Aleks Kissinger and I found that this SLOCC-classification is actually directly connected to the classification of commutative Frobenius algebras on 2d Hilbert space in Hilb; see col 1 in arXiv:1002.2540 which states that there are only two kinds of commutative Frobenius algebras, special ones and “anti-special ones” (yes, we have been told many times that this is a horrible name), and this follows directly from the entanglement classification result.

I’m pleased that hyperdeterminants are slowly gethering more interest. They are hard to get into but rewarding when you do.

The work on connections between M-theory and quantum information entanglement is intriguing. I hope it leads somewhere. It is curious that this work links the 2x2x2 hyperdeterminant and the octonions which were both worked on by Cayley. I wonder how long Cayley thought about possible connections just because they are both algebraic structures defined on 8 component vectors.

There is another curious historical coincidence for the 2x2x2x2 hyperdeterminant that was constructed by Schläfli. It is a degree 24 polynomial. Schläfli also discovered the 24-cell and in both cases the appearance of the number 24 can be linked to the importance of the number 24 in bosonic string theory.

Not remotely connected with anything in the TWF, but this link gives an attempt to decode a crop circle recently found in England as a statement of Euler’s trigonometric identity. (This story appears other places, so I’m pretty sure that the physical crop circle does exist – rather than being photoshopped – but beyond that…). On the assumption that it was done by human beings, there’s still niggling questions (surely the “corrected-version” brackets would, by the principle of only putting things in when they’re needed, imply the pi isn’t in the exponent?). Anyway, it’s mildly amusing to see how people will spend their time.

Mike Duff’s students William Rubens and Leron Borsten are here at the school on Foundational Structures in Quantum Computation and Information. Over coffee William just told me about this new paper:

• L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, W. Rubens, Four-qubit entanglement from string theory

It gives an actual map relating a certain class supersymmetric black holes to patterns of entanglement of 4 qubits. I don’t understand it, but one key piece seems to be the Kostant-Sekiguchi correspondence between nilpotent orbits of $SO_0(4,4)$ in $so(4,4)$ and nilpotent orbits of $SL(2,\mathbb{C})^4$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$.

Here I am just parroting the paper. But what does the second appearance of ‘nilpotent’ in the above sentence mean? I know what a nilpotent element of a Lie algebra is, but not what a nilpotent element of a vector space is. My best guess is that we should use the vector space isomorphism

$$so(4,4) \cong (sl(2,\mathbb{C})^4 \oplus \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$$

to interpret elements of $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ as elements of $so(4,4)$. Then we can say whether or not they are nilpotent.

The entanglement classes of 4 qubits are all the nonzero orbits of $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ under the action of $(sl(2,\mathbb{C})^4$. I don’t know what we leave out if only consider nilpotent orbits.

Leron Borsten told me something like this. The allowed charges for a certain class of STU black holes correspond to points in a certain 8-dimensional lattice… perhaps the integral split octonions. We can think of this lattice as sitting inside $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$. Then the entropy of the black hole is the square root of the absolute value of the $2 \times 2 \times 2 \times 2$ hyperdeterminant.

He said this was nicely explained here:

• Péter Lévay and Szilárd Szalay, The attractor mechanism as a distillation procedure.

I’ve added an addendum to week298 — a letter from Péter Lévay explaining more about quantum entanglement, black holes and exceptional algebraic structures.