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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):

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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 $ZZ$:

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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??

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  • $\begingroup$ I can see why the state is eigen value of XX and ZZ, but the success state is collapsing to a state which the eigenvalue of XX is predetermined. How can I have all the 4 options to measure? $\endgroup$
    – Ron Cohen
    Commented May 14, 2022 at 17:42

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