A simple two-gate sequence prepares a two-qubit state with exactly two correlated branches. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Put the control qubit in two branches
Start with the control qubit in zero. An H gate makes two equal control branches with amplitude one over square root of 2.
H∣0⟩=21∣0⟩+21∣1⟩
Attach a target initialized to zero
Before CNOT, the two-qubit state has the same two control branches and a target zero in each branch.
21∣0,0⟩+21∣1,0⟩
CNOT flips only the target branch with control one
CNOT leaves zero-zero unchanged and maps one-zero to one-one. The two surviving branches now carry matching bit values.
input∣0,0⟩∣1,0⟩output∣0,0⟩∣1,1⟩
The output is a correlated two-qubit state
The state is not four equally likely rows. It has exactly two branches, each with probability one half.