I am reading this article on "Fusion-based quantum computation" and they claim there about Fusion Gate type-2 that it collapses the state to 2 types of eigenstates, and returns 2 results - $ZZ$ eigenvalue and $XX$ eigenvalue (Section II, page 3):
I can understand why it is true that it returns $ZZ$ eigenvalue (parity check) from the definition of this gate in this article, where I collapse to a state that is an eigenstate of $ZZ$:
My QUESTION: I can't understand how I am getting $XX$ results (parities of things like $|++\rangle$ (even) or $|+-\rangle$ (odd)) at the same time??
