# Matrix definition of the Ising XX gate

On the English Wikipedia, $$XX$$ Ising gates are defined in matrix form as :

$$XX(\phi) = \begin{bmatrix}\cos(\phi)&0&0& -i \sin(\phi)\\ 0&\cos(\phi)&-i \sin(\phi) & 0 \\ 0 & -i \sin(\phi) & \cos(\phi) & 0 \\ -i \sin(\phi) & 0 & 0 & \cos(\phi) \\ \end{bmatrix}$$

However, on the French Wikipedia, the $$XX$$ Ising gates are defined as :

$$XX(\phi) = \frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0& -i e^{i \phi}\\ 0&1&-i & 0 \\ 0 & -i & 1 & 0 \\ -i e^{i \phi} & 0 & 0 & 1 \\ \end{bmatrix}$$

From some sources, I believe the right one is the first one. I would like some confirmations about the second definition being an error, and I would be glad to take any explanation on it. In particular, does this matrix in the second definition corresponds to the definition of another gate?

Your first option is the correct one, being related to $$e^{-i\phi X\otimes X}$$, which is $$XX(\phi) = \begin{bmatrix}\cos(\phi)&0&0& -i \sin(\phi)\\ 0&\cos(\phi)&-i \sin(\phi) & 0 \\ 0 & -i \sin(\phi) & \cos(\phi) & 0 \\ -i \sin(\phi) & 0 & 0 & \cos(\phi) \\ \end{bmatrix}.$$