# Symmetries of maximally bi-partite pure entangled states

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 :) )

• For any state with full Schmidt rank you can do this, it just won't be $M$ and $M^T$ that are the pairs of operators doi.org/10.48550/arXiv.1906.07731 Commented Sep 11 at 3:07
• @QuantumMechanic Thanks. This ref indeed looks like the kind of things I'm interested in. Commented Sep 11 at 16:34