Bell-pair preparation is built in two visible moves: split the control into two equal branches, then let CNOT copy only the branch where the control is 1. The final ledger has two matching branches, 00 and 11, each with probability one half, rather than four equally likely rows. This is an ideal circuit-state check; real preparation needs calibrated pulses, coherence time, readout efficiency, repeated shots, and uncertainty.
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.