Sharing a Bell Pair

Quantum teleportation requires two parties — conventionally called Alice and Bob — to share a pre-entangled pair of qubits before any quantum information can be transferred. This shared pair is the entanglement resource, and it must be in a specific maximally-entangled state known as a Bell state.

The Bell state $|\Phi^+\rangle$

The four Bell states are the most-entangled states of two qubits. The one used as the teleportation resource is

$|\Phi^+\rangle = \frac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr).$

Each term carries amplitude $\tfrac{1}{\sqrt{2}}$, so measuring both qubits yields 00 or 11 with equal probability — $\left|\tfrac{1}{\sqrt{2}}\right|^2 = \tfrac{1}{2}$ each. Crucially, the two qubits are correlated: you never see 01 or 10. Neither qubit alone has a definite state; they form one inseparable two-qubit system.

Building it with two gates

Starting from $|00\rangle$ (both qubits in $|0\rangle$), a two-step circuit creates $|\Phi^+\rangle$:

1. Hadamard on qubit 0. This turns $|0\rangle$ into the equal superposition $\tfrac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr)$, so the two-qubit state becomes

   $\frac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr)\otimes|0\rangle = \frac{1}{\sqrt{2}}\bigl(|00\rangle + |10\rangle\bigr).$

2. CNOT (controlled-X) with control qubit 0 and target qubit 1. The CNOT flips qubit 1 whenever qubit 0 is $|1\rangle$. Applying it to the state above maps $|00\rangle \to |00\rangle$ and $|10\rangle \to |11\rangle$, giving

   $\frac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr) = |\Phi^+\rangle.$

The resulting state cannot be written as a product $|\psi_A\rangle \otimes |\psi_B\rangle$ for any single-qubit states $|\psi_A\rangle$ and $|\psi_B\rangle$; that is the mathematical signature of entanglement.

Try it

Prepare $|\Phi^+\rangle$ by applying a Hadamard to qubit 0 followed by a CNOT from qubit 0 to qubit 1. The grader checks the full state vector; the amplitudes on $|00\rangle$ and $|11\rangle$ must each equal $\tfrac{1}{\sqrt{2}} \approx 0.707$ and the amplitudes on $|01\rangle$ and $|10\rangle$ must be zero.

After running, inspect the Amplitudes panel. You should see exactly two non-zero entries — at index $0$ (bitstring 00) and index $3$ (bitstring 11) — each with magnitude $\approx 0.707$. That is the Bell pair both parties will share in teleportation.

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