Density Operators and Time by Carl A Brannen

Tomography means measuring something by taking slices out of it, more or less. In this case, quantum tomography means determining a quantum wave function by taking measurements of it.

Here's the calculations for quantum tomography. (Uh, I don't have a reference for this, but I think it's obvious enough that I'm not likely to have made too many errors typing this in on the fly.)

For the case of spin-1/2, one obtains a bunch of particles all in the same identical (but unknown) state. One uses 1/3 of them to measure the spin in the x direction. Another 1/3 is used to measure spin in the y direction. And the last 1/3 are used to measure spin in the z direction.

With all three spin measurements, you get a probability between 0 and 1 for spin in the + direction (say +x). This number is only approximated, but you can get as accurate as you wish by sampling enough particles.

So you've got p_x, p_y, and p_z as your three probabilities. Convert these to three numbers x, y, and z by

:

x = 2p_x - 1,

y = 2p_y - 1,

z = 2p_z - 1.

By the laws of quantum mechanics, you should be able to prove that xx + yy + zz

Ooops it ate the rest of the post.

So you should be able to prove that

xx yy zz is less than or equal to 1.

If it were = to 1, then you have a pure density matrix state and the vector is on the Bloch sphere. Either way, your estimate for the density matrix for the quantum states is:

(1 x sigma_x y sigma_y z sigma_z)/2

where sigma_n are the Pauli spin matrices.

So quantum tomography is the process of figuring out a wave function from measurements. A complete set of mutually unbiased bases defines a measurement system that optimizes the process of quantum tomography.

I was probably not clear in what I wrote. Quantum tomography is a way of entangling various spin states with a spin and then using ancillary measurements to estimate the state of the one spin, My statement was meant to indicate that I think this could be used to estimate internal state amplitude of a quantum black hole. AS Bekenstein found a black hole is a one dimensional channel, and the state through a black hole could in principle be teleported through a black hole.

Lawrence B. Crowell

I had to read your most recent post several times over several days before it made sense to me. I looked it up on arXiv, i.e. gr-qc/0603046, and they do use the method.

By the way, getting back to the E8 feature, you wrote "The miracle of E_8 is that the symmetry of the root space is that of the group." I like the way wikipedia puts it: It's unique among simple groups in that its nontrivial representation of smallest dimension is the adjoint representation acting on the Lie algebra of E8.

This gets back to how E8 can arise naturally at low temperatures from bound states of less complicated things.

To connect with density matrices, the Leech lattice is in Jacobi functions ~ @^3(E_8), and so there are three E_8's there. The Leech lattice decomposes to S^3xSL(2,7), or a Fano plane at each point of a 3-sphere. So this appears to define a Bloch sphere type of construction. The projective Fano plane defines a set of three E_8s on a three ball (an 8^3). The breaking of this system then freezes one of these E_8's into its lattice of roots and the other two persists as E_8xE_8. The lattice of roots then defines a tessellation of AdS. So there are connections with some established physics here.

This might be a different direction from what you might be thinking for the Leech lattice has 196,560 elements and things appear to be vastly complicated. The automorphisms over these of course lead to the Fischer-Greiss (monster) group. I might imagine things could go into those domains as well. Yet I think that to make physics work there must be a master quantum error correction code, such as a Hamming distance for E_8 [8, 4, 4] or the Steiner group for the Mathieu group.

We might in a coarse grained sense say that the E_8's emerge from a sort of chaos or "simplicity," for obviously we are not going to cast about finding irreps for the Leech lattice, or find all of them. So these massive groups exist as a sort of ensemble space for various low energy (eg < 10^2 E_{planck}) configurations. And of course I don't propose getting into monster group considerations in any considerable way.

Tony Smith seems to want to push things that far, in fact with systems of monster groups or moonshines and so forth. I have a hard time making sense of some of what he writes, it seems at times almost autistic in a strange way.

My paper for this essay contest, which garnered a vote or two from the fqxi and not many public votes, is a part of the physical arguments I am laying down for this.

Lawrence B. Crowell

"So there are connections with some established physics here."

The things you are talking about here, Fano planes and Jacobi functions, are things that some of my correspondents also talk about. In particular, Michael Rios see math-ph/0503015. Marni Sheppeard in her blog:

http://kea-monad.blogspot.com/2007/08/m-theory-lesson-80.html

talks about Fano planes and references a blog post of mine which link is out of date. To see what she is talking about, with respect to Fano planes, see the diagrams at this post:

http://carlbrannen.wordpress.com/2007/10/04/fict

Uh, in the comments, "Kea" is Marni Sheppeard, and "Kneemo" is Michael Rios. The drawings show how to calculate topological phase for the Pauli MUBs. I don't know if they have much to do with Fano planes but the higher math types seemed interested.

By the way, regarding your essay, an interesting paper has come by on arXiv that sort of agrees with both of our papers. See:

http://arxiv.org/abs/0901.4917

and other papers by Walter Smilga.

The paper gives a derivation of a rather accurate formula for the fine structure constant originally found by Wyler. It's based on the assumption that the correct symmetry group is SO(3,2) rather than Poincare.

This fits in with what I'm doing because I've got two copies of the the time coordinate, that is, the usual time and the absolute age of the universe. So it's quite natural that one would need a larger symmetry group for this. My original work on the fermions was based on extending the Dirac algebra by adding one hidden dimension to it, which amounts to the same thing (since the Dirac algebra is complex, the sign of the additional dimension doesn't effect the algebra any).

And it fits in with what you're doing because this symmetry is related to AdS in some manner. I'm not a gravity guy and can't explain this further.