✦Quantum teleportation

Transfer an unknown qubit using entanglement and classical bits.

1. Core idea

Alice and Bob share a Bell pair. Alice entangles her unknown input with her half, measures two qubits, and sends two classical bits. Bob uses those bits to choose Pauli corrections. The original input is consumed; no copy is made, and no information travels faster than light.

2. Mathematical foundation

The notation describes a precise relationship among states, transformations or probabilities. Identify each symbol, check the dimensions of the objects, and distinguish amplitudes from the probabilities produced by a measurement.

3. Worked example

4. Reasoning flow

1. Prepare the input qubit and an entangled Bell pair.
2. Alice applies CX and H, then measures her two qubits.
3. Alice sends the two classical outcomes to Bob.
4. Bob conditionally applies X and Z.
5. Bob holds the original quantum state, without a duplicate.

Qiskit implementation

Run this standalone Python script with Qiskit installed. It uses exact statevector simulation rather than noisy hardware shots.

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
qc = QuantumCircuit(3)
qc.ry(0.8, 0)        # unknown-to-Bob test state on input qubit
qc.h(1); qc.cx(1, 2) # create shared Bell pair
qc.cx(0, 1); qc.h(0) # Alice's Bell-basis transform
# Coherent corrections implement deferred measurement principle
qc.cx(1, 2)         # X correction controlled by Alice bit 1
qc.cz(0, 2)         # Z correction controlled by Alice bit 0
print(qc.draw())
print(Statevector.from_instruction(qc).probabilities_dict())
# This models teleportation coherently, not classical communication latency.

5. Check your understanding

6. What to watch out for

Ideal-state calculations assume perfect gates and measurements. When assessing an algorithm, separate its mathematical guarantee from the cost of building oracles, preparing inputs, repeating experiments and correcting physical noise.