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Quantum Classical Correspondence Using Coherent State Measurements and Husimi Q Probability Distributions

Youheng Zheng

TL;DR

The paper proposes a lean quantum-classical correspondence protocol that interleaves brief Schrödinger evolutions with POVMs in the coherent-state basis to generate Husimi Q distributions and collapse to coherent states, enabling trajectories that closely track classical Newtonian motion for times longer than pure Schrödinger dynamics allow. By deriving and testing two regime-separating inequalities relating $Δt$ and $ħ$, the authors identify a semiclassical window where quantum and classical trajectories strongly coincide; simulations across multiple 1D potentials demonstrate that smaller $ħ$ extends convergence times and improves phase-space overlap when $Δt$ is suitably tuned. The work combines a split-operator quantum propagator, RK4 classical integration, and a rigorously framed POVM construction to produce a practical framework for observing quantum-to-classical transition in a controlled, measurement-assisted setting. The findings highlight a viable pathway for recovering classical motion from quantum postulates and offer insight into parameter regimes where quantum effects yield to classical determinism over extended periods, with potential implications for mesoscopic and nano-scale systems.

Abstract

We propose and simulate a protocol to evolve a quantum particle forward in time such that its trajectory closely matches that of the particle's Newtonian counterpart. Using short bursts of Schrödinger time-evolution interleaved with positive operator-valued measurements (POVMs) in the coherent basis, we demonstrate quantum-classical convergence for durations far beyond Schrödinger time-evolution alone. We examine the impact of the time between measurements $Δt$ and the reduced Planck's constant $\hbar$ on divergence time. Results indicate that for appropriate values of $Δt$, smaller values of $\hbar$ lead to longer divergence times. This method suggests a elegant, intuitive bridge to recover classical motion from quantum postulates.

Quantum Classical Correspondence Using Coherent State Measurements and Husimi Q Probability Distributions

TL;DR

The paper proposes a lean quantum-classical correspondence protocol that interleaves brief Schrödinger evolutions with POVMs in the coherent-state basis to generate Husimi Q distributions and collapse to coherent states, enabling trajectories that closely track classical Newtonian motion for times longer than pure Schrödinger dynamics allow. By deriving and testing two regime-separating inequalities relating and , the authors identify a semiclassical window where quantum and classical trajectories strongly coincide; simulations across multiple 1D potentials demonstrate that smaller extends convergence times and improves phase-space overlap when is suitably tuned. The work combines a split-operator quantum propagator, RK4 classical integration, and a rigorously framed POVM construction to produce a practical framework for observing quantum-to-classical transition in a controlled, measurement-assisted setting. The findings highlight a viable pathway for recovering classical motion from quantum postulates and offer insight into parameter regimes where quantum effects yield to classical determinism over extended periods, with potential implications for mesoscopic and nano-scale systems.

Abstract

We propose and simulate a protocol to evolve a quantum particle forward in time such that its trajectory closely matches that of the particle's Newtonian counterpart. Using short bursts of Schrödinger time-evolution interleaved with positive operator-valued measurements (POVMs) in the coherent basis, we demonstrate quantum-classical convergence for durations far beyond Schrödinger time-evolution alone. We examine the impact of the time between measurements and the reduced Planck's constant on divergence time. Results indicate that for appropriate values of , smaller values of lead to longer divergence times. This method suggests a elegant, intuitive bridge to recover classical motion from quantum postulates.
Paper Structure (13 sections, 40 equations, 10 figures, 1 table)

This paper contains 13 sections, 40 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Simulation
  4. Physics Setup
  5. Numerical Techniques
  6. Data Handling and Divergence Metric
  7. Computational Implementation
  8. Parameters
  9. Delineating Regimes
  10. POVM Framework
  11. Results and Discussion
  12. Acknowledgments
  13. Appendix

Figures (10)

  • Figure 1: Husimi Q Wikipedia
  • Figure 2: Coherent State
  • Figure 3: $\hbar=0.1$, $\Delta t=0.1$
  • Figure 6: Root-mean-square (RMS) deviation over time
  • Figure 7: Divergence time heatmap
  • ...and 5 more figures