Entanglement criteria via the uncertainty relations in SU(2) and SU(1,1) algebra: detection of SU(2) squeezed states

We derive a class of inequalities, to detect bipartite entanglement for continuous variables, from the uncertainty relations in the SU(1,1) and the SU(2) algebra in conjunction with partial transposition. These inequalities are presented in terms of the SU(2) operators J_x, J_y, and the total photon number N_a+N_b. They include as special cases the inequality derived by Hillery and Zubairy [quant-ph/0507168], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the inequality obtained by Agarwal and Biswas. We show that this optimal inequality can detect entanglement in a broad class of the SU(2) minimum-uncertainty states for any degree of squeezing. Experimental schemes to measure the observables J_x and J_y are also discussed, especially the one using linear optical devices and photo detectors