Thank you, the answer is simpler then what I thught then. Is this true even in the case of multipartite entanglement? So a entanglement preserving channel/operator can be always written as a convex combination of local unitaries (possibly random).

That's what I thought, but it seems not true. Meaning that given a separable state you cannot entangle it using local operations but given an entangled state you can destroy or decrease its entanglement with local operations.

Sorry I am not a mathematician but the way sam expressed my question is the way I do understand the problem. Is this something that people have studied? If not what are similar proprieties that I can start to look at that people have already studied? A group ‘normalized’ by a set of operators? Thx

Thanks Lspice and Sam. Lspice do you have in mind any pointer to help me getting an idea of H as a random a H⊆End(V)? I have the feeling that even for simple groups it a difficult object to characterize.

Yes Sam! thanks and some pointer in the literature.

Let me try again: H is a set the of operators that are mapped into themselves (modulo permutation) by G-conjugation (G given). Why is the definition of H circular?

I mean the set of operators $H$ with the property that for all $h\in H$ $ghg^{-1}=h',\;\;\exists h'\in H\;\; \forall g\in G$.

Thanks LSpice, I corrected it, typo

Thanks paul, can you point me out a easy paper on the topic?

thanks carlo, why is that? what about other groups like SU(n) for example?

Thanks! I also edited the question to make it more clear.

Thank you, you are right those conditions are needed. Do you think there is a way around them? Does your intuition suggest that the solution without those conditions need to much more complicated?