# Controlled Z gate entanglement

In order to prove that the Controlled Z gate can create entanglement, I'm trying to show that using these two arbitrary $$2\times 2$$ matrices and their tensor product: $$\begin{bmatrix}a_1&b_1\\c_1&d_1\end{bmatrix}\quad \begin{bmatrix}a_2&b_2\\c_2&d_2\end{bmatrix}$$ that the controlled Z gate creates a contradiction and cannot be represented as the tensor product of two matrices. How would I go about doing this?

## Criterion for entanglement in $$2$$-qubit states
A two-qubit state with amplitudes $$[a,b,c,d]^T$$ is a product state if and only if it can be written as the Krokecker product $$[u,v]^T\otimes[x,y]^T=[ux,uy,vx,vy]^T$$. This is the case if and only if $$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=0.\tag1$$ Moreover, the criterion is unaffected by global phase and normalization of the state.
Now, the result of applying the CZ gate to the product state $$[1,1]^T\otimes[1,1]^T=[1,1,1,1]^T$$ is $$[1,1,1,-1]^T$$. But $$\det\begin{bmatrix}1&1\\1&-1\end{bmatrix}=-2\ne 0\tag2$$ so the CZ gate can create entanglement.