Maximum entanglement of mixed symmetric states under unitary transformations

We study the maximum entanglement that can be produced by a global unitary
transformation for systems of two and three qubits constrained to the fully
symmetric states. This restriction to the symmetric subspace appears naturally
in the context of bosonic or collective spin systems. We also study the
symmetric states that remain separable after any global unitary transformation,
called symmetric absolutely separable states (SAS), or absolutely classical for
spin states. The results for the two-qubit system are deduced analytically. In
particular, we determine the maximal radius of a ball of SAS states around the
maximally mixed state in the symmetric sector, and the minimal radius of a ball
that contains the set of SAS states. As an application of our results, we also
analyse the temperature dependence of the maximum entanglement that can be
obtained from the thermal state of a spin-1 system with a spin-squeezing
Hamiltonian. For the symmetric three-qubit case, our results are mostly
numerical, and we conjecture a 3-parameter family of states that achieves the
maximum negativity in the unitary orbit of any mixed state. In addition, we
derive upper bounds, apparently tight, on the radii of balls containing
only/all SAS states.