Finding a unitary operator for quantum non-locality.

[koroshii]

Homework Statement

Recall that in the Quantum Teleportation protocol, Alice and Bob start withthe state $(\alpha\ket{0}+\beta\ket{1})\otimes\left(\frac{\ket{00}+\ket{11}}{\sqrt{2}}\right)$, and end in one of the four following states probabilistically, after Alice’s measurement: $$\ket{00}\otimes(\alpha\ket{0}+\beta\ket{1})$$ $$\ket{10}\otimes(\alpha\ket{0}-\beta\ket{1})$$ $$\ket{01}\otimes(\beta\ket{0}+\alpha\ket{1})$$ $$|11〉⊗(-\beta\ket{0}+\alpha\ket{1})$$


Give the 2×2 matrix representation of the unitary operator Bob must apply to turn his state into the desired state $\alpha\ket{0}+\beta\ket{1}$ for each of the four possibilities when Alice communicates the result of her measurement (00, 01, 10, or 11).

Relevant Equations

This appears to be the last step of the Quantum Teleportation protocol in my notes


"Alice sends a 2-bit classical message to Bob telling him her measurement result. He applies a local unitary; i.e., $I_{A_1, A_2}\otimes U_B$, to transform his state to $\ket{\phi} = \alpha\ket{0}+\beta\ket{1}$"

My trouble might be from how I interpret the problem. Alice and Bob are entangled. After Alice makes the measurement both of their states should collapse to one of these states with a certain probability. (Unless my understand of how entanglement is wrong.) The way I am understand the question is to find an operator $U_B$ (using the first one as an example) such that

$$I_{A_1, A_2}\otimes U_B\left(\ket{00}\otimes(\alpha\ket{0}+\beta\ket{1})\right) = \alpha\ket{0}+\beta\ket{1}$$
$$= I_{A_1, A_2}\ket{00}\otimes U_B(\alpha\ket{0}+\beta\ket{1}) = \alpha\ket{0}+\beta\ket{1}$$

My professor doesn't explicitly say it but I'm assuming $I_{A_1, A_2}$ is an identity, so the first term is unaffected. That already seems to make this equation impossible. If the first term can somehow be ignored, $U_B = I$ and I could pretty easily find the other unitary operators.

This whole section on entanglement and non-locality really hit me by surprise and I am still trying to wrap my head around it. Could someone point me in the right direction?

Thank you.

 

[member 553083]

Your interpretation of the problem is correct. The task is to find a unitary operator $U_B$ such that, when applied to Alice and Bob's entangled state, it collapses both of their states to one of the given states with the appropriate probability.The key here is that the unitary operator $U_B$ only acts on Bob's qubits. The identity operator $I_{A_1, A_2}$ just ensures that Alice's qubits remain unchanged. In other words, $I_{A_1, A_2} \otimes U_B$ is equivalent to just $U_B$.Therefore, your equation should read $\left(\alpha\ket{0}+\beta\ket{1}\right) = U_B\left(\alpha\ket{0}+\beta\ket{1}\right)$. This equation has a solution: $U_B = I$, the identity operator. To find the other unitary operators, you can use the fact that any unitary operator can be written as a product of unitary matrices. You can also use the results from the projective measurements you made in the first part of the problem to determine the form of the unitary operator.I hope this helps!