What is the significance of measuring in the plus/minus basis in the second step of the quantum teleportation protocol?
In the quantum teleportation protocol, measuring in the plus/minus basis in the second step holds significant importance. To understand this significance, let us first consider the basics of the protocol and the properties of the Bell state circuit.
The quantum teleportation protocol allows for the transfer of quantum information from one location to another without physically moving the quantum state itself. It relies on the phenomenon of quantum entanglement, where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the state of the other particles. This entanglement forms the basis of the Bell state circuit, which is an essential component of the teleportation protocol.
The Bell state circuit is a quantum circuit that prepares a pair of qubits in one of four Bell states: |Φ⁺⟩, |Φ⁻⟩, |Ψ⁺⟩, and |Ψ⁻⟩. These Bell states are maximally entangled and possess unique properties. For instance, the |Φ⁺⟩ state can be expressed as (|00⟩ + |11⟩)/√2, where |0⟩ and |1⟩ represent the computational basis states. Similarly, the other Bell states have their own unique expressions.
Now, coming back to the second step of the teleportation protocol, after the sender (Alice) entangles her qubit with the qubit to be teleported, they share a Bell state. Alice then performs a joint measurement on her two qubits, followed by a classical communication of the measurement outcomes to the receiver (Bob). The significance lies in the specific type of measurement performed by Alice, which is the measurement in the plus/minus basis.
The plus/minus basis refers to the eigenstates of the Pauli-X operator, which is a fundamental quantum gate that performs a bit-flip operation on a qubit. The eigenstates of Pauli-X are given by |+⟩ = (|0⟩ + |1⟩)/√2 and |-⟩ = (|0⟩ – |1⟩)/√2. These states form an orthonormal basis in the two-dimensional Hilbert space of a qubit.
By measuring in the plus/minus basis, Alice obtains one of the four possible outcomes: |+⟩, |-⟩, |+i⟩, or |-i⟩. These outcomes correspond to the classical bits that Alice communicates to Bob. The choice of the plus/minus basis is important because it allows Alice to extract the necessary information about the quantum state she is trying to teleport.
For example, let's consider the case where Alice's entangled qubit is in the |Φ⁺⟩ state and the qubit to be teleported is in an arbitrary state |ψ⟩. When Alice measures in the plus/minus basis, she has an equal probability of obtaining either |+⟩ or |-⟩. If she measures |+⟩, Bob's qubit will be projected into the state |ψ⟩, and if she measures |-⟩, Bob's qubit will be projected into the state -|ψ⟩. Thus, by performing the plus/minus basis measurement and communicating the measurement outcome to Bob, Alice effectively transfers the quantum state |ψ⟩ to Bob.
Measuring in the plus/minus basis in the second step of the quantum teleportation protocol is significant because it allows for the extraction of the necessary information about the quantum state being teleported. This measurement, combined with classical communication, enables the successful transfer of quantum information from one location to another.
Other recent questions and answers regarding Bell state circuit:
- If measure the 1st qubit of the Bell state in a certain basis and then measure the 2nd qubit in a basis rotated by a certain angle theta, the probability that you will obtain projection to the corresponding vector is equal to the square of sine of theta?
- How is the violation of the Bell inequality related with quantum entanglement?
- How does Alice choose which quantum gate to apply to Bob's qubit in the quantum teleportation protocol?
- In the quantum teleportation protocol, what information does Alice communicate to Bob?
- How does the quantum teleportation protocol rely on entanglement?
- What is the purpose of the quantum teleportation protocol?