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Quantum Electronics

Simulation of quantum logic by linear recording of superimposed Fourier holograms: Linda phenomenon

Alexander V. Pavlov

Published Jul 31, 2019• 2 citations
Publisher

Abstract

Abstract The classical mechanism of the quantum-like cognitive phenomenon of ‘Linda’ is demonstrated, which operates by linear recording of Fourier superimposed holograms in a phase-conjugate 4f-scheme. The results of numerical experiments are presented.
DOI
10.1070/qel16939

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A Fourier-holography scheme with superimposed holograms and phase conjugation is considered. It is shown that the sub-holograms, arising from the recording of superimposed holograms due to quadratic nonlinearity of the exposure characteristics of holographic recording media, form the connections necessary for the realisation of the quantum-logic mechanism of sequential projection of the input vector onto the subspaces of the recorded images. A classical mechanism of the 'Linda' cognitive phenomenon belonging to the category of quantum-like ones is described. An experimental illustration of the self-sustained formation of sub-holograms due to the nonlinearity of the exposure characteristics of holographic recording media is given.
Subject of study. A 4 f Fourier-holography scheme with phase conjugation in the correlation plane and the corresponding Grossberg “Instar” neural-network model were examined to assess the feasibility of implementing quantum-like models of information processing. Aim of study. The aim was to develop a Fourier-holography scheme model consistent with the mathematical formalism of quantum mechanics used in quantum-like cognitive models. Method. Analytical modeling and numerical simulations were performed by representing the processed information with a homogeneous isotropic random field model. Main results. The results showed that the Fourier-holography setup incorporating nonlinear phase conjugation in the correlation plane could serve as a limited analog of a quantum system in terms of parallel processing and choice from a superposition. In this case, the choice was not random; rather, it was determined by the nonlinear transfer function of the phase-conjugating mirror. The shift-invariance, ensured by a thin Fourier hologram, imparted an additional capability—making decisions in the form of a superposition described by a linear prediction model; some researchers consider this approach as a possible mechanism underlying the phenomenon of intuition. Practical significance. An analytical description, consistent with the information-processing model, is required for developing holographic simulators of quantum information-processing systems and computing devices.
Knill, Laflamme, and Milburn [1] have shown that probabilistic quantum logic operations can be implemented using linear optical elements, such as beam splitters and phase shifters, additional photons (ancilla) and post-selection based on the results of measurements on the ancilla. This may seem somewhat surprising, since logic operations are inherently nonlinear. Here the nonlinearity is produced by the quantum measurement process used in the post-selection. Nonlinearities of this kind are inherently quantum-mechanical and have no classical analogy.
The article is dedicated to the search for a biologically motivated mechanism of the cognitive phenomenon of violation of the classical formula of total probability for the disjunction of incompatible events, which is considered by a number of researchers as a quantum-like phenomenon. A classical mechanism implemented by the 6f Fourier holography scheme of the resonant architecture that does not require reference to quantum mechanics either in its physical nature or at the level of formalism is demonstrated. In the analysis, the decision-making is interpreted as a choice of alternatives by using the non-cooperative game "Prisoner's Dilemma". The approach to the task is based on the search for a mechanism for forming a conditional estimate under a condition that contradicts the rule of monotonous decision logic. It is demonstrated that this estimate, in contrast to the unconditional and conditional one with a non-contradictory condition, is formed by logic with exception. The ring architecture of the holographic setup corresponds to the biologically inspired neural network concept of the excitation ring and implements cognitive dissonance on logic with exception. Conditions and ranges of violation of the classical formula of total probability in relation to the correlation radius of the reference image recorded in a hologram storing the monotone logic inference rule are analytically determined. The analytical model is confirmed by a quantitative coincidence of the results of numerical modeling with the published results of natural experiments.
Ever since Knill, Laflamme and Milburn [Nature (London) 409, 46 (2001)] showed that nondeterministic quantum logic operations could be performed with linear optical elements, additional photons (ancilla) and projective measurements, the idea of linear-optics quantum computation has attracted considerable interest. Our group has recently demonstrated several devices of this kind. We give an overview of recent experimental results, including the quantum parity check, the destructive controlled-NOT, and a cyclical quantum memory. The need for high-efficiency detection of single photons, and for detectors capable of distinguishing photon number will be discussed. Some experimental improvements towards meeting that need will be presented.
A hybrid encryption and decryption technique for optical information security is proposed. In this method, the iterative Fourier transform algorithm is employed to optimize the encrypted hologram and the decryption key as binary phase-only diffractive optical elements, which were fabricated by electron-beam lithography. In a simple optical setup, the optical decryption is implemented by superimposing the encrypted hologram and the decryption key. Numerical simulation and optical experiment confirm the proposed technique as a simple and easy implementation for optical decryption.