# Why does changing $X\to iX$ in the definition of the CNOT make an important difference?

How to find a ket $$|\psi \rangle$$ that illustrates how changing X to iX in the definition of the CNOT gate makes an important difference because of what happens when CNOT is applied to $$|\psi \rangle$$?

The definition of the CNOT gate is: $$CNOT = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes X$$

What is the physical meaning of this?

Assuming your question is "What $$| \psi_i \rangle$$ would have different resulting $$| \psi_f \rangle$$ having undergone a $$C-X$$ gate vs. a $$C-iX$$ gate?"

Consider $$| \psi_i \rangle = \frac{1}{\sqrt2} \Big( |00 \rangle + |11 \rangle \Big)$$

$$C-X | \psi_i \rangle = \frac{1}{\sqrt2} \Big( |00 \rangle + |10 \rangle \Big)$$ $$C-iX | \psi_i \rangle = \frac{1}{\sqrt2} \Big( |00 \rangle + i|10 \rangle \Big)$$

Obviously these two resulting states are different since there is a relative phase. To make the difference more clear you can try to consider measurements in the Y-basis. Just as a note a $$C-U$$ gate is a "Controlled $$U$$ gate", and often times you will find $$CNOT$$ written as $$C-X$$ to be consistent with $$C-H$$ (Controlled Hadamard), $$C-Z$$ (Controlled Z gate), etc.

The Physical meaning of this is that adding a global phase to a single Qubit Gate on a multi Qubit system results in a change in the multi Qubit system. Notice how if we only had a single Qubit, there would be no way of knowing the difference between $$X$$ and $$iX$$, because it would be a global phase; however in a 2 Qubit system, that phase shows itself as a relative phase which indeed is measurable.

These two gates only differ by a local unitary, namely $$\begin{pmatrix}1&0\\ 0&i\end{pmatrix}$$ on the control qubit. Thus, there is no important difference between them, at least in terms of their entangling power and and non-local properties - they are exactly equivalent in that respect. But of course, it depends what you deem an important difference, which you would have to clarify in your question.

Gates $$X$$ and $$iX$$ differs in global phase which is $$\pi/2$$ (since $$i = e^{i\frac{\pi}{2}}$$). As two states (gates) which differ in global phase only cannot be distinguished, there is no difference between $$X$$ and $$iX$$.

However, the difference appears in controlled versions of the gates. Controlled $$iX$$ is composed of controlled $$X$$, i.e. CNOT, and controlled global phase which is in this case described by gate $$U1(\pi/2) \otimes I$$ where $$U1= \begin{pmatrix} 1 & 0 \\ 0 & e^{i\frac{\pi}{2}} \end{pmatrix}$$ So $$C-iX = \text{CNOT} (U1(\pi/2) \otimes I)$$.