How could I prove in ZX-calculus that these two diagrams are equal (up to a global phase), using axioms from this paper (Fig. 1) for example? Any intuition is welcome!
I'm starting to learn about ZX-Calculus, and I wanted to try to see the different ways to perform a CZ gate (rotation of $\pi$ angle). And it appears that there are (at least) three different ways (sorry for the bad quality of the drawings):
Version 2: (note the minus sign on the bottom qubit)
Now, version 2 is very easy to derive from version 1 if we take the axiom (EU) of this paper (we will call this axiom (EU1)), and version 3 is also easy to derive from version 1 if we take the axiom (EU) of this paper (we will call this axiom (EU2)).
But I can't find how to go from version 2 to version 3 by using only the axioms in the first paper (including EU1), or the other way around. Basically, it should be enought to prove that these two diagrams are equal (and I made the computation: they are equal, up to a global phase):
Any idea how to prove the equality of these two diagrams with axioms only from the first paper for example? It seems super easy (just 1/2 nodes!)... but I can't make it... Any intuition is welcome!