Split operator propagator

This is a frameshot from the propagation of a 2D wavefunction

Conservation of Shannon information in classical mechanics

There are many places where to find the proof of the conservation of the quantum Shannon information under unitary evolution. However, I just could not find the equivalent proof that everybody talks in classical mechanics, so here it goes.





The Lioville equation can be expressed in terms of a self-adjoint operator as observed by Koopman and von Neumann, such that



Given that the Liouvillian follows the Leibniz rule, which is true for conservative systems



it is now easy to prove that



This fact is in complete contrast with respect to the thermodynamical entropy, which can be fundamentally irreversible.


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DUST

DUST stands for Differential Unitary Space Time coding. This coding method is used in the transmission of information from multiple antennas to multiple antennas.

The transmission of information in a block can be modelled with the following equation



where S is the input matrix, H is the channel matrix, \rho is a scalar coefficient that modulates the channel and W is the noise.

DUST encodes the information in a unitary matrix U_n, which is indirectly encoded through a sequence of two S unitary blocks as



If H is full-rank and the channel does not change significantly in the time the two blocks are transmitted. The unitary matrix U can be recovered from the sequence of two received unitary blocks X without! the need to know the channel H as



where the new noise is defined as


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Random Unitary Matrices

The generation of random unitary matrices has many applications in many fields.
The simplest method is based on the QR decomposition of a random complex matrix with independent elements following a Normal distribution.

In Mathematica one can generate a random complex number with Normal distribution for both the real and imaginary parts with the following function

NormalRandomComplex[] := (#[[1]] + #[[2]]*I) &@
  RandomReal[MultinormalDistribution[{0, 0}, {{1, 0}, {0, 1}}]]

where {0, 0} specifies the origin and {{1, 0}, {0, 1}} the covariance matrix (standard deviation 1)

The function that generates the random n x n unitary matrix is

RandomUnitaryMatrix[n_] := Module[{z, q, r},
  z = Table[NormalRandomComplex[], {n}, {n}];
  {q, r} = QRDecomposition[z];
  q.DiagonalMatrix[Sign[Tr[r, List]]]

]

In this code, the QR decomposition of the complex random matrix with normal distribution outputs a unitary matrix "q" and a triangular matrix "r". The sought random unitary matrix is a correction of "q", which must be multiplied by a diagonal matrix filled with 1 and -1 according to the sign of the corresponding diagonal element of "r".

New numerical algorithms

Two recent developments on numerical analysis have called my attention

A new very efficient sparse Fourier transform
http://web.mit.edu/newsoffice/2012/faster-fourier-transforms-0118.html

A significantly faster algorithm for matrix multiplication
http://www.newscientist.com/article/mg21228422.500-mathematical-matrix-multiplier-sees-first-advance-in-24-years.html

De Broglie waves and relativity

Another extremely interesting connection between relativity and quantum mechanics is that the quantum De Broglie waves can be understood as an effect of the relativistic desynchronization of clocks given that the clocks rotate with a frequency proportional to the total energy of the particle.   

Imagine a train with a long chain of synchronized clocks from one extreme to the other with the synchronization established according to an observer moving along with the train. Another person standing still on the train station will conclude that the clocks are actually not synchronized (what he would see is a different thing). If the train is moving to the right, the clocks on the right will be ahead respect to the clocks on the left. The De Broglie wave length will be the length that stretches 12 hours of desynchronization.   

[1] W. Baylis, Canadian Journal of Physics, 2007, 85:(12) 1317-1323, 10.1139/p07-121

The ball is round!

The first thing we learn in a course of relativity is the relativistic length contraction effect for moving objects relative to an observer. However, what one would actually see is a different thing ! because another important effect must be considered; the time required for the light to arrive from the different points of the observed object to our eyes. The combined effect of the Lorentz contraction along with the time delay was investigated reaching to curious results. For example, a relativistic ball remains with a round appearance!, as discussed by Penrose in the following paper

http://adsabs.harvard.edu/abs/1959PCPS...55..137P

A nice visulaization can be found at

http://www.spacetimetravel.org/ueberblick/ueberblick1.html