# Quantum Teleportation Quantum teleportation transfers an unknown qubit state $\ket\psi=\alpha\ket0+\beta\ket1$ from Alice to Bob. Alice hold the 1st qubit of a Bell state pair $\ket\Phi=\frac{1}{\sqrt 2}(\ket{00}+\ket{11})$ and Bob hold the 2nd qubit. We start with the three-qubit state $\ket\psi\ket\Phi=\frac{1}{\sqrt 2}(\alpha\ket{000}+\alpha\ket{011}+\beta\ket{100}+\beta\ket{111})$ ## Alice apply CNOT The first qubit controls the second: $\frac{1}{\sqrt 2}(\alpha\ket{000}+\alpha\ket{011}+\beta\ket{110}+\beta\ket{101})$ ## Alice applies $H$ to the first qubit $ \begin{align} &(H\otimes I\otimes I)\bigg(\frac{1}{\sqrt 2}(\alpha\ket{000}+\alpha\ket{011}+\beta\ket{110}+\beta\ket{101})\bigg)=\\ &=\frac{1}{\sqrt 2}\big(\alpha H\ket0\ket{00}+\alpha H\ket0\ket{11}+\beta H\ket1\ket{10}+\beta H\ket1\ket{01}\big)\\ &= \frac{1}{2}\big(\alpha (\ket0+\ket1)\ket{00}+\alpha (\ket0+\ket1)\ket{11}+\beta (\ket0-\ket1)\ket{10}+\beta (\ket0-\ket1)\ket{01}\big)\\ &=\frac{1}{2}\big(\alpha\ket{000}+\alpha\ket{100}+\alpha\ket{011}+\alpha\ket{111}+\beta\ket{010}-\beta\ket{110}+\beta\ket{001}-\beta\ket{101}\big)\\ \end{align} $ Group the 2-qubits of Alice: $ \begin{align} & = \frac{1}{2}\bigg(\ket{00}(\alpha\ket0+\beta\ket1) +\ket{01}(\beta\ket0+\alpha\ket1)+\ket{10}(\alpha\ket0-\beta\ket1)+\ket{11}(-\beta\ket0+\alpha\ket1)\bigg) \\ &=\frac{1}{2}\bigg(\ket{00}\ket\psi +\ket{01}X\ket\psi +\ket{10}Z\ket\psi +\ket{11}XZ\ket\psi \bigg) \\ \end{align} $ ## Alice measure her 2 qubits Alice measure classic bits $a,b$ and send then to Bob. ## Bob correct his qubit According to Alice classic bits see table below. note: correction is based that $X^2=Z^2 =I$, $(ZX)(XZ)=I$. | a | b | correction | | --- | --- | ---------- | | 0 | 0 | $I$ | | 0 | 1 | $X$ | | 1 | 0 | $Z$ | | 1 | 1 | $ZX$ | ## The result Bob will have $\ket\psi$ . Alice's original qubit is destroyed by measurement so teleportation does not clone the original qubit. It also does not enable faster-than-light communication because Bob must receive Alice’s two classical bits before recovering $\ket\psi$. ## Created 2026-01-18 18:29