Maximum entanglement of mixed symmetric states under unitary transformations

TL;DR

This paper investigates the maximum entanglement achievable through unitary transformations in symmetric two- and three-qubit states, analyzing separability, thermal states, and providing bounds on classical states within the symmetric subspace.

Contribution

It provides analytical and numerical results on maximum entanglement in symmetric states, including bounds on absolutely separable states and a conjecture for three-qubit maximum negativity.

Findings

Analytical results for two-qubit symmetric states.

Numerical analysis and conjecture for three-qubit symmetric states.

Bounds on the radii of balls of absolutely separable states.

Abstract

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.