Criteria for particle entanglement in many-body systems of bosons

Basing on the analogy between the coherent states of light and separable states of $N$ bosons, we demonstrate that the violation Cauchy-Schwarz inequality for any-order correlation function signals the entanglement among the constituent particles. Rather than restricting to the correlations between the positions of particles, we consider the broadest set of measurements allowed by quantum mechanics. Our result is general -- it applies to any quantum system of bosons, even when the number of particles is not fixed, provided that there is no coherence between different number states. We also demonstrate that the compact expression for the separable state of bosons can be used to relate some known metrological quantities to the particle entanglement in a very simple way.