M Theory Lesson 288

I suspect that there are still a few people out there who wonder why some bloggers occasionally ramble on about twistors, although it is mind boggling to think that the advances in twistor theory made by stringy people over the last few years have entirely escaped their attention.

One of the important results about entanglement is this twistor geometry paper by Peter Levay. Recall the exchange relation from lesson 285:

$P_{13}P_{24} = P_{12}P_{34} + P_{23}P_{14}$

for Plucker coordinates. In terms of twistor geometry, we can write these variables as

$P_{\mu \nu} = Z_{\mu} W_{\nu} - Z_{\nu} W_{\mu}$

where I am not going to worry about whether the indices should be up or down. The twistor geometry is extremely helpful, because now we can write the (norm of the) hyperdeterminant for three qubit entanglement as

$\frac{1}{2} | P^{\mu \nu} P_{\mu \nu} | = | (Z \cdot Z) (W \cdot W) - (Z \cdot W)^{2} |$

which could hardly be simpler. Moreover, the other three qubit measure is

$\frac{1}{4} \tau_{A(BC)} = \| Z \|^{2} \| W \|^{2} - | \langle Z | W \rangle |^{2}$

The $W$ state condition corresponds to null twistors. So associahedra type polytopes really are beginning to look nice in twistor spaces.

See also Levay's more recent papers on black hole entropy and finite geometries.