  Entanglement of spatial bipartitions, used to explore lattice models in
condensed matter physics, may be insufficient to fully describe itinerant
quantum many-body systems in the continuum. We introduce a procedure to measure
the R\'enyi entanglement entropies on a particle bipartition, with general
applicability to continuum Hamiltonians via path integral Monte Carlo methods.
Via direct simulations of interacting bosons in one spatial dimension, we
confirm a logarithmic scaling of the single-particle entanglement entropy with
the number of particles in the system. The coefficient of this logarithmic
scaling increases with interaction strength, saturating to unity in the
strongly interacting limit. Additionally, we show that the single-particle
entanglement entropy is bounded by the condensate fraction, suggesting a
practical route towards its measurement in future experiments.
