The Bell Measurement

Quantum teleportation requires Alice to measure her two qubits in the Bell basis — the four maximally-entangled two-qubit states. Rather than building a new kind of detector, a Bell-basis measurement is simply a change of basis followed by ordinary computational-basis measurement. That change of basis is a small two-gate circuit.

The four Bell states

The Bell basis consists of

Each state is labelled by two classical bits once you measure in the Bell basis: the bit pair 00 identifies $|\Phi^+\rangle$, 10 identifies $|\Phi^-\rangle$, 01 identifies $|\Psi^+\rangle$, and 11 identifies $|\Psi^-\rangle$.

The Bell measurement circuit

Creating a Bell pair from $|00\rangle$ takes two gates: $H$ on qubit 0, then $\mathrm{CNOT}(0 \to 1)$. Both gates are their own inverse ($H^2 = I$, $\mathrm{CNOT}^2 = I$), so the inverse circuit — the Bell measurement — is just the same two gates applied in reverse order:

1. $\mathrm{CNOT}(0 \to 1)$, then
2. $H$ on qubit 0,
3. measure both qubits in the computational basis.

We can verify this on $|\Phi^+\rangle$. After the CNOT we get

Applying $H$ to qubit 0 maps $|+\rangle \to |0\rangle$, leaving $|00\rangle$. Measuring now always yields the outcome 00 — exactly the label assigned to $|\Phi^+\rangle$. The same calculation shows that $|\Phi^-\rangle \to |10\rangle$, $|\Psi^+\rangle \to |01\rangle$, and $|\Psi^-\rangle \to |11\rangle$.

Why this matters for teleportation

In the teleportation protocol, Alice holds the qubit she wants to send (qubit 0) and her half of the shared entangled pair (qubit 1). She performs the Bell measurement on these two qubits and obtains one of the four outcomes 00, 01, 10, or 11. Each outcome tells Bob which of four corrections to apply to his qubit — but only once Alice sends the two classical bits over a regular channel. No quantum information travels faster than light.

Try it

The starter code prepares $|\Phi^+\rangle$ by applying $H(0)$ and $\mathrm{CNOT}(0 \to 1)$. Add the Bell measurement — CNOT then H then measure — so that the grader sees the deterministic outcome 00 with probability 1.

The probability bar for 00 should reach 100%, confirming that the Bell measurement correctly identifies $|\Phi^+\rangle$.

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