Statistics of Marginal Wave Functions as a Real-Space Diagnostic of Quantum Entanglement

Abstract

We present a statistical framework for extracting spatially resolved entanglement directly from an ensemble of marginal (one-body) wave functions in time-dependent Quantum Monte Carlo (TDQMC). Treating the guide waves as a statistical mixture in Hilbert space, we show that the Gram matrix acts as a covariance operator whose spectrum coincides with the Schmidt spectrum. The associated functional standard deviation closely tracks the von Neumann entanglement entropy both globally and locally via walker partitioning, providing a physically transparent real-space diagnostic of quantum correlations without requiring construction of the full many-body wave function. Applications to one-dimensional two-electron bosonic and Fermionic systems (helium atom and hydrogen-like molecule) demonstrate excellent agreement with strict conditional-wave results for opposite-spin electrons. For same-spin Fermions, TDQMC’s statistical treatment of exchange symmetry avoids the unphysical negative local entropies that arise from naive ln(2) subtraction, yielding spatial profiles that remain positive throughout. The method establishes a direct bridge between classical ensemble statistics and quantum entanglement measures, offering a computationally efficient real-space diagnostic tool for mapping the spatial distribution of correlations in many-body systems.