An inequality for spinor Bose-Einstein condensates

An inequality for spin-$F$ Bose-Einstein condensates (BECs) $F^2(\rho^2-|\Theta|^2)-\boldsymbol{M}^2\ge0$ is reported, where $\rho$, $\Theta$, and $\boldsymbol{M}$ represent the density, singlet pair amplitude, and magnetization vector, respectively. The distribution of high-symmetry spinors in the allowed region by the inequality is elucidated with using the Majorana representation. The result is illustrated by the example of spin-2 BECs.