The CNOT gate output states are clearly defined when the control qubit is in either of the the "pure" state of 0 or 1, as in the following diagram:
However, when the control bit is already in a superposition of states, for example:
As this control bit is already in superposition, I am having trouble solving the algebra in the same fashion as above examples to get the resulting states of the control and target lines after this C operation.
The resulting tensor after applying the C matrix does not seem "factorizable" into two vectors.
Could anyone help by providing the right solution to this problem, please?
Also, if anyone could explain what is (if any) the geometrical meaning (in terms of rotations/flip/etc) of applying CNOT in the case of the control bit in superposition of states, as drawn in the circuit diagram above?
I have read some related answers to questions like this, but many seem to give general "use linear algebra and work it out" answers. I am hoping if someone can give the finished explicit solution to this factoring problem, please.
Thank you for any help/pointers!