Alice and Bob share a state a| + +> + b |- -> , where the first qubit is Alice's and the second qubit is Bob's. Alice measures her qubit in the standard basis and sends the measurement outcome to Bob. If Bob wants his qubit to end up in the state a |0> + b |1>, what gates does he apply to his qubit in the following two cases? Write your answer in terms of X ZZ H I using comma as a delimiter. (For example, you may write X if the answer is a single gate X , or you can write H,X,Z if the answer is to apply H first, and then Y , and then Z .)

To determine the gates Bob needs to apply to his qubit, we can look at the measurement outcomes that Alice can obtain and apply the appropriate transformations to Bob's qubit accordingly.

Case 1: Alice measures |+> state

When Alice measures her qubit in the standard basis and obtains the outcome |+>, it means that the state collapses to the +1 eigenstate of the Pauli-X operator. To bring Bob's qubit to the desired state a|0> + b|1>, Bob needs to apply the Pauli-Z operator (Z).

Therefore, for this case, Bob needs to apply the gate Z.

Case 2: Alice measures |-> state

When Alice measures her qubit in the standard basis and obtains the outcome |->, it means that the state collapses to the -1 eigenstate of the Pauli-X operator. To bring Bob's qubit to the desired state a|0> + b|1>, Bob needs to apply the Hadamard gate (H) followed by the Pauli-Z operator (Z).

Therefore, for this case, Bob needs to apply the gates H, Z.

In summary:
- If Alice measures |+>: Bob applies Z
- If Alice measures |->: Bob applies H, Z

Please note that the Pauli-X operator is represented by X and the Pauli-Z operator is represented by Z.