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International series of numerical mathematics

The uncertainty inequality in quantum holography, Bohr’s indeterminacy principle, and synchronized neural networks

Hanlin Chen, Walter Schempp

Published Jan 1, 1992• 1 citation
Publisher

Abstract

A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation is insensitive to fine structures of the interference pattern generated by a Mach-Zehnder interferometer. Actually Niels Bohr’s indeterminacy principle cannot be based on any of the known uncertainty principles. It is shown how the holographic transform allows to circumvent the difficulties with the standard deviation by using a group theoretical implementation of the canonical commutation relation of quantum electrodynamics. The geometric quantization approach makes the heuristic arguments concerning quantum parallelism rigorous by considering wave packets as symplectic spinors on the hologram plane. It includes the standard uncertainty inequality as a special case. As an application to the area of quantum computers, we study synchronized neural networks and their photonic implementations as suggested by recent experiments in brain research. As another result of the geometric quantization approach to holography, the uncertainty minimizing Gabor wavelets arise which provide useful wavelet expansions for image analysis, segmentation, and compression. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-7565-3_32

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A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation is insensitive to fine structures of the interference pattern generated by a Mach-Zehnder interferometer. Actually Niels Bohr’s indeterminacy principle cannot be based on any of the known uncertainty principles. It is shown how the holographic transform allows to circumvent the difficulties with the standard deviation by using a group theoretical implementation of the canonical commutation relation of quantum electrodynamics. The geometric quantization approach makes the heuristic arguments concerning quantum parallelism rigorous by considering wave packets as symplectic spinors on the hologram plane. It includes the standard uncertainty inequality as a special case. As an application to the area of quantum computers, we study synchronized neural networks and their photonic implementations as suggested by recent experiments in brain research. As another result of the geometric quantization approach to holography, the uncertainty minimizing Gabor wavelets arise which provide useful wavelet expansions for image analysis, segmentation, and compression. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A rigorous proof of quantum parallelism cannot be based on the Heisenberg inequality because the standard deviation of self-adjoint operators in complex Hilbert space is insensitive to fine structures of the collective stationary interference distribution generated by a Mach-Zehnder interferometer from a coherent primary beam. Actually Niels Bohr's indeterminacy principle of spatio-temporal quantum electrodynamics cannot be based on any of the known uncertainty principles. It is shown how the holographic transform allows to circumvent the difficulties with the standard deviation by using a group theoretical implementation of the canonical commutation relations of quantum electrodynamics. The geometric quantization approach combined with the emitter-absorber transaction model of quantum dynamics on the whole real line R allows one to describe by a Liouville density the flow and counter-flow of single optical photons in split fan-in/fan-out coherent photonic channels. It makes the heuristic arguments concerning quantum parallelism rigorous by considering the collective stationary interference distributions of coherent wavepacket densities as symplectic spinors over the linear symplectic manifold modelled on the hologram plane. Consequently the symplectic spinorial organizational form governs photonic holograms. It implies the existence of single-photon holograms and includes the standard uncertainty inequality as a special case.
Quantum holography is a well established theory of mathematical physics based on harmonic analysis on the Heisenberg Lie group G. The geometric quantization is performed by projectivization of the complexified coadjoint orbit picture of the unitary dual Ĝ of G in order to achieve a geometric adjustment of the quantum scenario to special relativity theory. It admits applications to various imaging modalities such as synthetic aperture radar (SAR) in the microwave range, and, most importantly for the field of non-invasive medical diagnosis, to the clinical imaging modality of magnetic resonance tomography (MRI) in the radio frequency range. Quantum holography explains the quantum teleportation phenomemon through Einstein-Podolsky-Rosen (EPR) channels which is a consequence of the non-locality of phase coherent quantum field theory, the concept of absolute simultaneity of special relativity theory which provides the Einstein equivalence of energy and Fitzgerald-Lorentz dilated mass, and the perfect quantum holographic replication process of molecular genetic information processing. It specifically reveals what was before unobservable in quantum optics, namely the interference phenomena of matter wavelets of Bose-Einstein condensates, and what was before unobservable in special relativity, namely the light in flight (LIF) recording processing by ultrafast laser pulse trains. Finally, it provides a Lie group theoretical approach to the Kruskal coordinatized Schwarzschild manifold of relativistic cosmology with large scale applications to general relativity theory such as gravitational instanton symmetries and the theory of black holes. The direct spinorial detection of gravitational wavelets emitted by the binary radio pulsar PSR 1913+16 and known only by anticipatory system computation so far will also be based on the principles of quantum holography applied to very large array (VLA) radio interferometers.
We propose a geometric formulation of quantum indeterminacy based on polar duality between convex bodies representing position and momentum data. A pair (X,P) of centrally symmetric convex bodies is said to satisfy the indeterminacy condition when the h-polar of X is included in P. We show that this condition naturally arises from the geometry of quantum blobs and is closely related to Hardy's uncertainty principle and the Donoho--Stark inequalities. Using symplectic capacities and John ellipsoids, we associate a canonical covariance ellipsoid with every quantum polar pair and prove that it satisfies the quantum condition. The Robertson--Schrödinger inequalities follow as a consequence. This provides a non-statistical formulation of quantum indeterminacy from which the usual uncertainty relations emerge. which the usual uncertainty relations emerge.
Quantum holography is a well established theory of mathematical physics based on harmonic analysis on the Heisenberg Lie group G. The geometric quantization is performed by projectivization of the complexified coadjoint orbit picture of the unitary dual G of G in order to achieve a geometric adjustment to special relativity theory. It admits applications to various imaging modalities such as synthetic aperture radar (SAR), and most importantly for the field of non-invasive medical diagnosis, to the clinical imaging modality of magnetic resonance tomography (MRI). Quantum holography explains the quantum teleportation phenomenon through Einstein-Podolsky-Rosen (EPR) channels which is a consequence of the non-locality of quantum physics. It specifically reveals what was before unobservable in special relativity, namely the light in flight (LIF) recording processing by ultra fast laser pulse trains. Finally, it provides a Lie group theoretical approach to the Kruskal coordinatized Schwarzschild manifold of quantum cosmology with large scale applications to general relativity theory such as gravitational instanton symmetries and the theory of black holes. The direct spinorial detection of gravitational wavelets emitted by the binary radio pulsar PSR 1913+16 will also be based on the principles of quantum holography.
In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. Quantum mechanics stands here for the quantum-mechanical description, at a given instant of time, of a non-relativistic particle. These uncertainty principles suggest that there should exist a common mathematical structure behind both quantum mechanics and the theory of signals. It is the aim of the present article to establish that the concept of real Heisenberg nilpotent group Ã(ℝ) lies at the foundations of both of these fields. The crucial point is to endow the time-frequency plane ℝ ⊕ ℝ with the structure of a two dimensional real symplectic vector space so that it gets symplectomorphic to the tangent plane to the “Schrodinger coadjoint orbit” at the point 1 in the Kirillov orbit picture for the unitary dual of Ã(ℝ). Various applications of this geometric relationship between signal theory and nil potent harmonic analysis are pointed out. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
It is computationally demonstrated how quantum associative networks, implemented using quantum holography, could be harnessed for object recognition. These simulated quantum nets alone execute efficient image recognition, i.e., reconstruction of an image selected from associative memory (hologram). However, optically implementable neural-net preprocessing of object-images is needed for appearance-based viewpoint-invariant recognition of objects. We present computer simulation results of two methods: Moore-Penrose orthogonalization and encoding of object-images with Gabor wavelets. A computer-supported quantum Gabor-wavelet holography is proposed. © 2005 Optical Society of America