State transformations within entanglement classes containing permutation-symmetric states

The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit-states cannot be obtained from nor transformed to any state, which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that this situation persists for generic n-qubit symmetric states, we consider particular families for which we can determine that on the contrary they are endorsed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. The only classes found for which every state can be converted to a more weakly entangled state are the n-qubit GHZ and W classes. In the course of the study of LOCC transformations we also characterize the local symmetries of symmetric states.