Classical Phase Space Representations of Density Operators with Classical Probability Distributions

This 33-minute talk from the Workshop on "Quantum Harmonic Analysis" at the Erwin Schrödinger International Institute for Mathematics and Physics explores alternative approaches to representing quantum states in phase space. Discover how the classical Gabor transform can be used instead of the Wigner transform to obtain quadratic representations of N-body wave functions that define fermionic random processes with true probability distributions rather than quasiprobabilities. Examine practical examples involving variance and entanglement entropy calculations. The second part introduces a novel approach using quantum Gabor transform with operator window, resulting in a new type of fermionic random point process that accounts for correlations among density operator components while maintaining classical computability. Learn about the hyperuniform nature of this process through variance estimates and examples. This presentation represents joint work with Simon Halvdansson for the second portion and was delivered by Luis Daniel Abreu as part of the ESI workshop held May 5-10, 2025.