Applying the CNOT gate to the state |+-⟩ would result in the state |--⟩ as per:
What has occurred is a "phase kickback". The relative negative phase from the target qubit has transferred to the control qubit.
Let's look at the representation using vectors and matrices. The state |+-⟩ can be achieved by starting with the classical state |00⟩, flipping the second qubit using a NOT gate, e.g. (I⊕N)*|00⟩ = |01⟩ and then applying the hadamard gate to each qubit - (H⊕H)= |+-⟩. Using a vector column, this state would look like this:
If I apply the CNOT gate to this state, I would get:
My question is how am I supposed to interpret the resulting vector column? How can I convert from the computational basis after CNOT to the equivalent state in Hadamard basis to actually see that there's a negative phase infront of both qubits? I don't see that in the computational basis.