[Paper Review] Quantum Tomography Approach in Signal Analysis

Margarita A. Man’ko|ArXiv.org|Jun 2, 1999

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper introduces a quantum tomography framework for signal analysis, linking the fractional Fourier transform (FrFT) to the tomographic representation of optical signals via the quantum harmonic oscillator's Green function. It establishes a mathematical bridge between quantum mechanics and signal processing, demonstrating that the FrFT emerges naturally from the tomographic transform, offering a novel perspective for information processing applications.

ABSTRACT

Some properties of the fractional Fourier transform, which is used in information processing, are presented in connection with the tomography transform of optical signals. Relation of the Green function of the quantum harmonic oscillator to the fractional Fourier transform is elucidated.

Motivation & Objective

• To establish a theoretical connection between quantum tomography and signal processing techniques.
• To explore how the fractional Fourier transform (FrFT) arises from the tomographic representation of optical signals.
• To clarify the role of the Green function of the quantum harmonic oscillator in relation to the FrFT.
• To provide a quantum mechanical foundation for signal analysis using tomographic methods.
• To enable new insights into information processing through the lens of quantum tomography.

Proposed method

• Utilizes the tomographic transform of optical signals as a framework for signal representation.
• Applies the quantum harmonic oscillator's Green function to derive properties of the fractional Fourier transform.
• Establishes a mathematical correspondence between the tomographic transform and the FrFT through the Green function's kernel.
• Employs the Wigner function and quasiprobability distributions to represent signals in phase space.
• Uses the time evolution operator of the harmonic oscillator to generate the FrFT as a rotation in phase space.
• Analyzes the structure of the FrFT via the tomographic reconstruction formula, linking it to quantum measurement theory.

Experimental results

Research questions

• RQ1 How can quantum tomography be applied to the analysis of optical signals?
• RQ2 What is the relationship between the fractional Fourier transform and the tomographic representation of signals?
• RQ3 How does the Green function of the quantum harmonic oscillator relate to the fractional Fourier transform?
• RQ4 Can the fractional Fourier transform be derived from a quantum mechanical tomographic framework?
• RQ5 What are the implications of this connection for signal processing and information theory?

Key findings

• The fractional Fourier transform is shown to emerge naturally from the tomographic representation of optical signals via the quantum harmonic oscillator's Green function.
• The Green function of the harmonic oscillator provides the kernel that generates the FrFT as a phase-space rotation.
• The tomographic transform offers a quantum mechanical interpretation of the FrFT, linking it to quasiprobability distributions.
• The method establishes a direct correspondence between signal processing operations and quantum measurement processes.
• The framework enables a unified description of signal transformations using tools from quantum mechanics.
• The results suggest that quantum tomography provides a deeper mathematical foundation for understanding the fractional Fourier transform in signal analysis.