In ZX-calculus, the CNOT gate is represented by this:
Can someone show me why this is true, using just the basic rewriting rules? All books/papers I have seen simply take it without proof, but I can't see why it is true.
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Now, by definition, CNOT controls NOT: $|x_1,x_2\rangle\mapsto|x_1,x_1\oplus x_2\rangle$. In other words, if $|0\rangle$ is fed to the first qubit, the result is $|0\rangle$ on the first qubit, and the identity on the second. If it is fed $|1\rangle$, the result is $|1\rangle$ on the first qubit, and the gate NOT on the second one. Again this can be verified in the graphical calculus:
where the first equality is the "copy" rule, the second is the "spider" rule, and the last one is the fact that a rotation of angle $0$ is the identity.
where the first equality is the "$\pi$-copy" rule, and the second is the "spider" rule.