Suppose Alice has three qubits in the state α|000>+β|111>. If she sends these qubits to Bob by applying the teleportation protocol to each of them separately, what is the state that Bob receives?
To determine the state that Bob receives after Alice applies the teleportation protocol to each qubit separately, let's understand the steps involved in the teleportation protocol.
The teleportation protocol is used to transmit the state of a qubit from one location to another without physically moving the qubit. It involves the use of entanglement and quantum measurement.
In this case, Alice has three qubits in the state α|000> + β|111>. To send each qubit to Bob separately, Alice will perform the following steps for each qubit:
1. Alice and Bob need to have two additional qubits, usually referred to as the "entangled pair" or the "Bell pair." Let's call them qubit A and qubit B.
2. Alice applies a controlled-Hadamard gate (C-Hadamard) on the qubit in her possession (one from α|000> and β|111>) and the qubit A from the entangled pair.
3. Alice then applies a controlled-Not gate (C-Not) with the qubit in her possession as the control and qubit B from the entangled pair as the target.
4. After that, she applies a Hadamard gate on the qubit in her possession.
5. Next, Alice performs a measurement on the two qubits she has, which are the qubit in her possession and qubit A from the entangled pair. This measurement results in two classical bits (0 or 1) being obtained.
6. Alice communicates the values of these two classical bits to Bob using classical communication.
7. Based on the received classical information, Bob applies a specific sequence of gates to his qubit, which could be either the identity gate (I), the Pauli-X gate (X), the Pauli-Y gate (Y), or the Pauli-Z gate (Z). The gates Bob applies depend on the values of the two classical bits communicated by Alice.
Now, going back to the state α|000> + β|111>:
For the first qubit:
- Alice entangles it with qubit A.
- Alice applies the controlled-Hadamard (C-Hadamard) gate to the qubit in her possession and qubit A.
- Alice applies the controlled-Not (C-Not) gate with the qubit in her possession as the control and qubit B as the target.
- Alice applies the Hadamard gate to the qubit in her possession.
- Alice performs a measurement on the qubit in her possession and qubit A.
Following these steps for the other two qubits, Alice will communicate the classical information obtained from the measurement to Bob for each qubit.
Based on the classical information received, Bob will perform the corresponding gate sequence on his qubit. Thus, the state Bob receives will depend on the specific classical information communicated.
In summary, to determine the state that Bob receives after applying the teleportation protocol to each of Alice's qubits separately, we need to know the specific values of the classical information obtained from the measurements performed by Alice for each qubit, and then determine the gate sequence Bob applies to his qubit based on that information.