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We study the maximum entanglement that can be produced by a global unitarytransformation for systems of two and three qubits constrained to the fullysymmetric states. This restriction to the symmetric subspace appears naturallyin the context of bosonic or collective spin systems. We also study thesymmetric states that remain separable after any global unitary transformation,called symmetric absolutely separable states (SAS), or absolutely classical forspin states. The results for the two-qubit system are deduced analytically. Inparticular, we determine the maximal radius of a ball of SAS states around themaximally mixed state in the symmetric sector, and the minimal radius of a ballthat contains the set of SAS states. As an application of our results, we alsoanalyse the temperature dependence of the maximum entanglement that can beobtained from the thermal state of a spin-1 system with a spin-squeezingHamiltonian. For the symmetric three-qubit case, our results are mostlynumerical, and we conjecture a 3-parameter family of states that achieves themaximum negativity in the unitary orbit of any mixed state. In addition, wederive upper bounds, apparently tight, on the radii of balls containingonly/all SAS states.

Maximum entanglement of mixed symmetric states under unitary transformations