Entanglement and spin-squeezing properties for three bosons in two modes

We discuss the canonical form for a pure state of three identical bosons in two modes, and classify its entanglement correlation into two types, the analogous Green-Horne-Zeilinger and the $W$ types as well known in systems of three distinguishable qubits. We have performed a detailed study of two important entanglement measures for such a system, the concurrence $\mathcal{C}$ and the triple entanglement measure $\ensuremath{\tau}$. We have also calculated explicitly the spin squeezing parameter $\ensuremath{\xi}$ and the result shows that the $W$ state is the most ``antisqueezing'' state, for which the spin-squeezing parameter cannot be regarded as an entanglement measure.