# Quantum teleportation of unknown qubit when the entangled state is not a Bell state

Assume Bob and Alice have two particles with a prior entanglement: $$A$$ and $$B$$. The entangled state $$|Ψ⟩$$ is maximally entangled, and $$|Ψ⟩ = \frac{1}{\sqrt{2}}(|00⟩ + j|11⟩)\,,$$ where $$j$$ is a complex number.

Alice has a qubit $$C$$ in the state $$|ψ⟩ = a|0⟩ + b|1⟩\,.$$

After the teleportation is complete, can Bob apply a unitary transformation different than identity and Pauli $$X, Y, Z$$ to eliminate the effect of $$j$$ from the result?

• Welcome to QCSE! What are your thoughts on this question? What have you attempted till now to answer this? Commented Jan 24 at 21:12
• Just note that because of general complex number $j$ the state $|\Psi\rangle$ is not normalized. Commented Jan 25 at 7:05
• – glS
Commented Jan 25 at 9:57

I am assuming that $$|j|=1$$, i.e. $$j$$ is just a phase factor. Otherwise, the state would not be normalized (as already mentioned in a comment above) and the state would also not be maximally entangled (as required according to the question) because it cannot be created using local unitaries acting on the maximally entangled state $$\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$.
That being clarified, assume that Alice performs the standard teleportation protocol and sends her outcome to Bob. This can be done regardless of the phase factor $$j$$. Next, Bob needs to get rid of the phase by performing a phase gate $$$$T = \begin{pmatrix} 1 & 0 \\ 0 & j^* \end{pmatrix} \;,$$$$ on his side of the state. Clearly, if Bob would act with $$T$$ on his side of the maximally entangled pair before Alice does anything, they would recover the maximally entangled state $$\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$ and the teleportation would be the standard one. The reason why Bob can do this only after Alice has performed her measurements is because Bob only acts on his side (''his Hilbertspace'') with $$T$$ while Alice performs her measurements on her side of the system, thus the actions commute. Lastly, Bob can proceed to follow the standard teleportation protocol and recover the teleported state $$|\psi\rangle$$ with perfect fidelity. (All of this of course assumes that the phase $$j$$ is known.)