= Solution
<Quantum teleportation> transfers an unknown <qubit> state using a previously shared <Bell pair>, local operations and two classical bits. It requires no description of the unknown complex amplitudes and produces no extra copy.
Let the input be $|\chi\rangle_S=\alpha|0\rangle+\beta|1\rangle$, and let Alice and Bob share $|\Phi^+\rangle_{AB}$. Alice applies a <controlled-NOT gate> from $S$ to $A$, then a <Hadamard gate> to $S$. Expanding the circuit gives the <Bell-basis teleportation identity> in computational-output form:
$$
|\chi\rangle_S|\Phi^+\rangle_{AB}\longmapsto\frac12\sum_{a,b\in\{0,1\}}|a\rangle_S|b\rangle_A\,X^bZ^a|\chi\rangle_B.
$$
Her <projective measurement> yields each bit pair $(a,b)$ with probability $1/4$. She sends the two classical bits, and Bob applies \b[$Z^aX^b$] to recover $|\chi\rangle$. The order is the inverse of the displayed conditional operator; interchanging these Pauli corrections changes only an irrelevant global phase when both bits are one.
Before receiving the bits, Bob has
$$
\rho_B=\frac14\sum_{a,b}X^bZ^a|\chi\rangle\langle\chi|Z^aX^b=\frac I2.
$$
Thus teleportation respects the <no-communication theorem> and cannot transmit the usable unknown state faster than the classical message. Alice's input has been measured, and the shared entanglement resource has been consumed, in agreement with the <no-cloning theorem>. The protocol transfers the state, not the material particle initially carrying it.
A physical implementation must distribute and preserve an entangled pair, implement a Bell-basis measurement and transmit its result to a controllable output system. Imperfect entanglement, lost outcomes and noisy corrections reduce <quantum fidelity>; a heralded probabilistic experiment must report its success probability separately from conditional fidelity. Ideal deterministic teleportation assumes every Bell outcome can be identified and corrected.
Back to article page
