Research on quantum carriers of information is an active area. From a perspective of quantum information and its applications, as actual as one opened by Albert Einstein, about a century ago:

I spent my life to find out what a photon is, and I still don't know it.

(Albert Einstein)

### "Spaghetti problem" and 3 data clusters bond

Richard Feynman taught us that nearly everything is interesting if one goes deep down into it. His famously quoted "spaghetti cut" problem stretches imagination on asymptotic freedom of movements, singularities, multidimensional scaling… In mathematics, data ordering and entanglement.

In our theory of multidimensional scaling, two operators, divergence and rotor, mediate computation in the scale-space wave information propagation.

The operators are applied along with two principal components of data scatter matrices. A coupled oscillator of 3 data clusters is brought to an equilibrium dynamically, along the 5th manifold, the scale parameter β. It binds data clusters in closed-loop structure information, within a quadrupole1.

### Computational dynamics of quantum information carriers

I got my first Rubik's cube while a high school student at the Mathematical Gymnasium in Belgrade. It brought me many sleepless nights until I solved it. My strategy was to put it in order in two steps - first the middle cubes, then the corners. I came up with a solution to assemble it in one minute on average, one move per second.

Making just random moves, with no feedback, it is estimated that would take more than the age of the known Universe to put it back together again. In our work, we refer to this kind of feedback as a coupled system information exchange.

A partition function of the configuration space is decomposed by utilizing two physical principles – the maximum entropy, loosely associated with time, and the least action principle, holding the configuration space in the minimal loop structure.

### Scale-space tunnelling

Data clusters exchange information via scale-space tunnelling2. The dynamics of the information exchange are formulated by two, up and downscale waves coupling. It has been shown also that this synergy exchange is not instantaneous. Evolving patterns project multi-dimensional information of the coupled data clusters, at a scale equilibrium β.

The expression of information bonding is given with the field equation3,4. From a practical point, the numerical implementation simplifies in 2D, as proven by Olga Ladyzhenskaya. It also generalizes a multidimensional matrix formulation of the uncertainty of information, across the scales. The information of pattern formation is evaluated at the equilibrium points of the free energy F, by computing its first and second-order approximations.

### Asymptotic freedom and the scale-space field

The scale dimension limits asymptotic freedom of movement, at β = 0. Decomposition of the partition function is singular in scale at both ends of the conjoined space - at the infinitely large and at the smallest.

At the smallest scale, a polynomial distribution limits the asymptotic freedom of information exchange that bonds the clusters together2.

An extension to Einstein's equation, E=mc3, is derived. In our work, we refer to mass and energy interchangeably, conveniently using the term information. The stability of the decomposition structure is assessed with the generalized uncertainty relation, across the scales.

### Scale-space network model

The mathematical formulation of the theory of stochastic resonance synergies predicts the connection between the largest and the smallest structures in a coupled network. In addition, it lays down a quantum computing approach to the analysis and application of the networked data structure. The expression of the exploration of the configuration space is given by the path integral.

In mathematics, the scale-space wave function preserves the information of the distribution of the prime numbers. It follows from the decomposition of 3, 6·k integer scales in a closed-loop structure. The bonding structure is reducing to 9 possible cases.

This structure model applies to physical systems, as well. Properties of a physical system in its relativistic space-time domain are derived from the network of synergistically coupled oscillators, in our view. We propose a genotype information processing in mirroring a holographic representation of multidimensional information in the networked data structure.

#### References

1 Jovovic, M., Stochastic Resonance Synergetics. Quantum Information Theory for Multidimensional Scaling, Journal of Quantum Information Science, 5/2:47-57, 2015. 2 Jovovic, M., Hierarchical scale quantization and coding of motion information in image sequences, Informacione Tehnologije VI, Zabljak, 2002.
3 Jovovic, M., H. Yahia, and I. Herlin, Hierarchical scale decomposition of images: singular features analysis, INRIA, 2003.
4 Jovovic, M., and G. Fox, Multi-dimensional data scaling: dynamical cascade approach, Indiana University, 2007.