Alice and Bob pre-share one Bell pair, e.g. |Φ+⟩=(|00⟩+|11⟩)/√2. Alice applies one of four local gates to only her half:
- I (00) → stays |Φ+⟩
- X (01) → |Ψ+⟩
- Z (10) → |Φ−⟩
- XZ (11) → |Ψ−⟩
These four Bell states are mutually orthogonal, so a joint measurement on both qubits distinguishes them perfectly — but measuring either qubit alone gives a 50/50 coin flip no matter what Alice did, because the reduced state of one half of a Bell pair is always maximally mixed. The 2 bits only become readable once Bob's CNOT+H circuit merges both qubits together, which is why Alice must physically send her one qubit over.