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Quantum teleportation is a protocol for transmitting an unknown qubit state from one party (Alice) to another (Bob) using a pre-shared Bell pair and two classical bits of communication. The quantum state itself is not physically transmitted — instead, the entanglement channel and classical corrections reconstruct the state on Bob’s qubit.
Quantum teleportation does not enable faster-than-light communication. Bob cannot recover the teleported state until he receives the two classical bits from Alice, which travel at or below the speed of light.

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Protocol walkthrough

1

Prepare the state to teleport

Create a qubit phi in the state |φ⟩ = a|0⟩ + b|1⟩ where |a|² = 0.2 and |b|² = 0.8. The qsim.New(1, 2) constructor accepts unnormalized amplitudes and normalizes them automatically.
qsim := q.New()

// generate qubits of |phi>|0>|0>
phi := qsim.New(1, 2)
q0 := qsim.Zero()
q1 := qsim.Zero()

// |phi> is normalized. |phi> = a|0> + b|1>, |a|^2 = 0.2, |b|^2 = 0.8
for _, s := range qsim.State(phi) {
  fmt.Println(s)
}

// [0][  0]( 0.4472 0.0000i): 0.2000
// [1][  1]( 0.8944 0.0000i): 0.8000
The amplitudes 0.4472 ≈ 1/√5 and 0.8944 ≈ 2/√5 correspond to |a|² = 0.2 and |b|² = 0.8. This is the state Alice wants to send to Bob.
2

Create a Bell pair between q0 and q1

Alice and Bob each hold one qubit of a shared entangled pair. Apply Hadamard to q0 then CNOT to entangle q0 (Alice’s qubit) and q1 (Bob’s qubit).
qsim.H(q0)
qsim.CNOT(q0, q1)
This creates the Bell state (|00⟩ + |11⟩)/√2 between q0 and q1.
3

Alice entangles phi with her half of the Bell pair

Alice applies a CNOT gate with phi as control and q0 as target, then applies a Hadamard to phi.
qsim.CNOT(phi, q0)
qsim.H(phi)
These two operations entangle Alice’s qubit-to-teleport (phi) with her half of the Bell pair (q0).
4

Alice measures and sends classical bits to Bob

Alice measures both phi and q0. The results are two classical bits — mz from measuring phi and mx from measuring q0.
// Alice sends mz, mx to Bob
mz := qsim.Measure(phi)
mx := qsim.Measure(q0)
Alice transmits these two bits to Bob through a classical channel.
5

Bob applies conditional corrections

Bob uses Alice’s classical bits to apply corrective gates to his qubit q1:
  • If mx is 1, apply an X gate (bit flip)
  • If mz is 1, apply a Z gate (phase flip)
// Bob Applies X and Z
qsim.CondX(mx.IsOne(), q1)
qsim.CondZ(mz.IsOne(), q1)
CondX and CondZ are no-ops when the condition is false, so at most two corrections are applied.
6

Verify Bob's qubit matches the original state

After the corrections, q1 holds exactly the state phi originally had.
// Bob got |phi> state with q1
for _, s := range qsim.State(q1) {
  fmt.Println(s)
}

// [0][  0]( 0.4472 0.0000i): 0.2000
// [1][  1]( 0.8944 0.0000i): 0.8000
Bob’s qubit now has amplitudes 0.4472 and 0.8944 — identical to the original phi state. The teleportation is complete.

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Why it works

The teleportation protocol exploits the structure of the Bell basis. When Alice performs her CNOT and Hadamard, the three-qubit system can be rewritten as a sum over the four Bell states on Alice’s qubits, each paired with a corresponding single-qubit state on Bob’s qubit. Alice’s measurement projects onto one of these four terms, and Bob’s conditional corrections undo the corresponding transformation — leaving Bob with exactly |φ⟩. The original qubit phi is destroyed in the process (Alice’s measurement collapses it), consistent with the no-cloning theorem.

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Complete example

qsim := q.New()

// generate qubits of |phi>|0>|0>
phi := qsim.New(1, 2)
q0 := qsim.Zero()
q1 := qsim.Zero()

// |phi> is normalized. |phi> = a|0> + b|1>, |a|^2 = 0.2, |b|^2 = 0.8
for _, s := range qsim.State(phi) {
  fmt.Println(s)
}

// [0][  0]( 0.4472 0.0000i): 0.2000
// [1][  1]( 0.8944 0.0000i): 0.8000

qsim.H(q0)
qsim.CNOT(q0, q1)
qsim.CNOT(phi, q0)
qsim.H(phi)

// Alice sends mz, mx to Bob
mz := qsim.Measure(phi)
mx := qsim.Measure(q0)

// Bob Applies X and Z
qsim.CondX(mx.IsOne(), q1)
qsim.CondZ(mz.IsOne(), q1)

// Bob got |phi> state with q1
for _, s := range qsim.State(q1) {
  fmt.Println(s)
}

// [0][  0]( 0.4472 0.0000i): 0.2000
// [1][  1]( 0.8944 0.0000i): 0.8000

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