# Is it necessary to apply CNOT for the sole purpose of "establishing a correlation" between two qubits

Disclaimer: I recently started learning quantum information.

I've been exploring creating the $$|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)$$ Bell state (starting with the state $$|00\rangle$$).

I know we can apply the $$H$$ gate to the leftmost qubit, then use the same qubit as a control for the $$CNOT$$ gate applied on both qubits to entangle them.

Before the $$CNOT$$ gate is applied, the two qubits have no correlation. They are separable qubits, where one is in a superposition.

So I wonder: Is it necessary, in this case, to apply a first $$CNOT$$ gate to the two qubits before applying the $$H$$ gate and the final $$CNOT$$ that creates the entanglement?

(I know that using both methods will still create the $$|\Phi^+\rangle$$ Bell state. I just want to find out if it's necessary to create any correlation between the two qubits before performing any operations on them)

No, it is not necessary to apply a $$CNOT$$ gate before the typical construction of $$|\Phi^+\rangle$$, that is applying $$H$$ on the control qubit followed by a $$CNOT$$.
Indeed, non-local quantum gates like $$CNOT$$ (but there are many others like $$CZ$$ for example) are necessary to create a correlation between qubits. However, such correlation (also called entanglement) is created when the control qubit is in superposition, otherwise the result of applying the non-local gate is deterministic and no entanglement is created.
Regarding the specific example of the $$|\Phi^+\rangle$$ Bell state, it is easy to demonstrate (let the leftmost qubit be "qubit $$0$$" and the rightmost qubit be "qubit $$1$$"):
1. Let $$|\Phi_0\rangle = |00\rangle$$ be the initial state of the system, as you mentioned.
2. Applying $$CNOT(0,1)$$ now would do nothing.
3. Applying $$H$$ on qubit 0 leads us to the state $$|\Phi_1\rangle = \frac{|00\rangle + |10\rangle}{\sqrt{2}}$$.
4. Applying $$CNOT(0,1)$$ now leads us to the final state $$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$$.
When the control qubit is in state $$|1\rangle$$ before applying a $$CNOT$$ an action will take place, but still the result would be deterministic due to lack of superposition.