Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals

Area of Science:

• Quantum Information Science
• Quantum Communication
• Quantum Cryptography

Background:

• The eigenvalue problem of the Gram matrix is crucial for quantum information science, enabling calculations of key quantities and quantum system simulations.
• Analytic solutions exist for symmetric signals, but asymmetric signals require inefficient universal numerical algorithms.
• Asymmetric signals like amplitude-shift keying coherent-state signals can be simplified by exploiting partial symmetry.

Purpose of the Study:

• To present a method for simplifying the Gram matrix eigenvalue problem for Quadrature Amplitude Modulation (QAM) signals.
• To address the computational inefficiencies associated with asymmetric quantum signal analysis.
• To enhance the applicability of efficient quantum signal processing in quantum communication and cryptography.

Main Methods:

• Exploiting the partial symmetry inherent in QAM signals.
• Developing a simplified approach to the Gram matrix eigenvalue problem for QAM.
• Applying the method to both ordinary and modified QAM signals.

Main Results:

• A clear method for simplifying the Gram matrix eigenvalue problem for QAM signals is established.
• The simplification addresses the computational challenges posed by asymmetric quantum signals.
• The findings are applicable to a wide range of QAM signals used in quantum technologies.

Conclusions:

• The developed method significantly improves the efficiency of analyzing QAM signals in quantum information science.
• This work provides a valuable tool for advancing quantum communication and quantum cipher security.
• The simplification is broadly applicable, supporting the development of secure quantum cryptographic systems.