My question focuses on bi-partite pure systems, but I am also interested in answers focusing on 2-qubit systems (this is actually the case I'm mostly interested in).
While being written in a different manner, many entangled state are actually equivalent.
For instance, one can show that:
$$|00\rangle+|11\rangle=|++\rangle+|--\rangle$$
More generally, calling $\{|i\rangle\}$ an orthonormal basis, for any operator $M$, we have the property that: $$ \sum_{i} (\mathbb{I} \otimes M) |ii\rangle = \sum_i (M^T \otimes \mathbb{I}) |ii\rangle,$$
where the transposition of the operator is taken in the basis $\{|i\rangle\}$ (I recall that transposition is a basis-dependent property).
Q1: Do all symmetries about bi-partite pure entangle state consequences of this properties (or there are symmetries that are not a consequence of the property I mention)?
Q2: Are there other interesting/important symetries that exists regarding bi-partite pure entangled states?
(For Q2: if you have examples that are consequences of the symetry I mention but are not directly apparent, I could also be interested. I know that "not directly apparent" might be subjective :) )
