I Understanding Entanglement Measures and the Bloch Sphere Representation

[edguy99]

"To do this, the scientists turned the difficult analytical problem into an easy geometrical one. They showed that, in many cases, the amount of entanglement between states corresponds to the distance between two points on a Bloch sphere, which is basically a normal 3D sphere that physicists use to model quantum states."

Not sure I understand "we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range". What is "degree 2"?

https://phys.org/news/2016-02-physicists-easy-entanglementon-sphere.html

 

[Misha]

My understanding is that there are two "two"'s here. One is the rank of the density matrix and another is the degree of a polynomial measure of entanglement. Degree 2 corresponds to the linear entropy.

To he honest, I have troubles understanding what's new in this paper. For instance, the fact that there are measures of entanglement based on some representations of the density matrix in terms of Bloch vectors is known for some time. In a concise form, the geometrical reasoning is presented here starting from Eq. (23) to the end of the section. In particular, it's proven there that for SU(2) all entanglement measures can be expressed in terms of the linear entropy. For SU(N) this is no longer true but limiting cases (completely disentangled and fully entangled) still turn out choice-independent. At length, the multipartite SU(2) case is considered here.