  What is the relation between spin squeezing and entanglement? To clarify
this, we derive the full set of generalized spin squeezing inequalities for the
detection of entanglement. These are inequalities for the mean values and
variances of the collective angular momentum components J_k. They can be used
for the experimental detection of entanglement in a system of spin-1/2
particles in which the spins cannot be individually addressed. We present
various sets of inequalities that can detect all entangled states that can be
detected based on the knowledge of: (i) the mean values and variances of J_k in
three orthogonal directions, or (ii) the variances of J_k in three orthogonal
directions, or (iii) the mean values of J_k^2 in three orthogonal directions or
(iv) the mean values and variances of J_k in arbitrary directions. We compare
our inequalities to known spin squeezing entanglement criteria and discuss to
which extent spin squeezing is related to entanglement in the reduced two-qubit
states. Finally, we apply our criteria for the detection of entanglement in
spin models, showing that they can be used to detect bound entanglement in
these systems.
