A Bell state is a two-qubit nonproduct state: measurement outcomes are linked, so the qubits cannot be described independently. The first state permits only 00 and 11, each with probability 0.5, while two other states use 01 and 10; sign changes distinguish the four states. section 4.2.1
Represent the four Bell states with amplitude vectors ordered by the two-qubit basis states:
- Φ⁺ = [√0.5, 0, 0, √0.5], supporting 00 and 11. section 4.2.1
- Φ⁻ = [√0.5, 0, 0, −√0.5], retaining the same possible outcomes with a relative sign change. section 4.2.1
- Ψ⁺ = [0, √0.5, √0.5, 0], supporting 01 and 10. section 4.2.1
- Ψ⁻ = [0, √0.5, −√0.5, 0], retaining 01 and 10 with a relative sign change. section 4.2.1
Preparing Bell states with a circuit
A simulator prepares Φ⁺ by placing the first qubit in superposition and entangling the pair: q = QuantumRegister(2); qc = QuantumCircuit(q); qc.h(q[0]); qc.cx(q[0], q[1]); state = qc.run(). Adding an X gate to the second qubit before the controlled-X prepares Ψ⁺. The remaining states are an exercise in circuit construction. section 4.5.5
q = QuantumRegister(2) qc = QuantumCircuit(q) qc.h(q[0]) qc.cx(q[0], q[1]) state = qc.run()
Inspect the executable circuit that prepares the first Bell state and returns its simulated state. section 4.5.5
