SuperDense quantum teleportation: same amplitude with 2D Donut relative Void noncommutative phase shift as per my former QM professor Bernstein

So this is a discussion by my former Quantum Mechanics Professor Herbert J. Bernstein.

 Bernstein, H.J. SuperDense Quantum Teleportation. Quantum Inf Process 5, 451–461 (2006). https://doi.org/10.1007/s11128-006-0030-5

https://sci-hub.se/10.1007/s11128-006-0030-5

 

OK.... so the noncommutative phase exists within the ONE entangled amplitude or "collapsed" entangled measured state.

So this is called "remote state preparation" as it is only one entangled bit with the relative phase changed within the three particles (photons) that are entangled. 

In kicks the noncommutative matrix math.

fascinating. 

http://people.na.infn.it/~lizzi/chapter6.pdf

So Bernstein is using the Weyl map to describe the noncommutative phase - hence the "Donut" or the Void as the Emptiness that is yin-yang.

The "donut" cares more information due to the noncommutative phase - in contrast to assuming a spherical geometry of space.

So here is the Donut or noncommutative torus - notice that the "classical" limit is the logarithmic value (just as in music theory).

And yet under quantum mechanics this "conversion" process is interactive:

OK now back to Bernstein:

So the qubits as relative phase are a 2D noncommutative torus.

fascinating.

a 2d torus should then be a line segment rotated around a point. ... but this shape have a hole in it,

Exactly. https://www.reddit.com/r/learnmath/comments/cufbwf/what_is_a_2d_torus/

Lets say I had a square, and I wanted to glue the left edge of the square to the right edge. I could wrap it around to do this, and the shape I would get would be a cylinder, right? Now lets say I also wanted to glue the top of the square to the bottom. Then I could take my cylinder, pull the top around and attach it to the bottom, to get a torus. This is usually how mathematicians think of a torus: a torus is a square (or rectangle) with opposite sides identified.

So the deal is that with a noncommutative 2D torus this is not the same symmetry. Instead you have a 5D time-frequency as relative phase within each "point" of the torus.

Note that since a circle is a one-dimensional manifold, the n-torus is a n-dimensional manifold.

When many people say or write "torus" with no qualification they usually mean a specific embedding of T² in R³ (3-dimensional space), which looks like a hollow bagel or doughnut.

Tori are exactly the same, except instead of lines, each coordinate is a circle. More precisely, each coordinate is an angle, and a full revolution in a coordinate gets you back to the same point.

So a noncommutative tori means that each "coordinate" is a 720 degree circle.

If a 3d torus is a circle rotated around a line, then a sphere is in fact a 3d torus.

Generally when the torus is described in this way, it's explicitly mentioned that the line (the axis of revolution) lies outside (or at worst tangent to) the circle.

OK that helps.

 https://ncatlab.org/nlab/files/TorusBranchedCoverOverSphere.jpg

https://arxiv.org/pdf/1703.02470.pdf
 

a [Downconverted] photon spontaneously splits into two other photons of lower energies.

so....

amplitude is related to the number of photons.

Planck constant stands in for the "amplitude" when the photon is conceived as a particle (i.e. all photons have a constant amplitude, and the only variable about them is their frequency). Obviously, the aggregate amplitude can be increased by increasing the number of photons involved, but what this does say is that the amplitude of light must always be an integer multiple of the Planck length (i.e. it "steps"), whereas the frequency of light is infinitely variable.

 https://physics.stackexchange.com/questions/383426/photon-energy-depends-on-frequency-and-or-amplitude

 Back to Bernstein:

And so then with equal amplitude you get a guaranteed entangled relative phase called Mutually Unbiased Bases or MUB