How does the state of the three qubits change after the CNOT gate is applied in the teleportation protocol?
In the context of quantum teleportation using the CNOT gate, the state of the three qubits undergoes a transformation after the application of the CNOT gate. To understand this transformation, let's first review the basics of quantum teleportation and the role of the CNOT gate in the protocol.
Quantum teleportation is a fundamental concept in quantum information theory that allows the transfer of quantum states from one location to another without physically moving the qubits themselves. The protocol involves three qubits: the sender's qubit (A), the entangled pair of qubits (B and C), and the receiver's qubit (D).
The CNOT gate, short for Controlled-NOT gate, is a two-qubit gate that performs a NOT operation on the target qubit (C) if and only if the control qubit (B) is in the state |1⟩. In the teleportation protocol, the sender (Alice) applies the CNOT gate to her qubit (A) and the entangled pair (B and C), with her qubit (A) acting as the control qubit and one of the entangled pair (B) acting as the target qubit.
Now, let's examine the state of the three qubits before and after the application of the CNOT gate. Initially, the three qubits are in the following state:
|Ψ⟩ = α|0⟩A ⊗ (|00⟩BC + |11⟩BC) ⊗ |0⟩D
Here, α represents the unknown quantum state of the sender's qubit (A), and the tensor symbol (⊗) denotes the tensor product between qubits.
When the CNOT gate is applied to the sender's qubit (A) and the entangled pair (B and C), the state of the three qubits evolves as follows:
|Ψ'⟩ = α|0⟩A ⊗ (|00⟩BC ⊗ |0⟩D) + α|1⟩A ⊗ (|11⟩BC ⊗ |1⟩D)
The CNOT gate flips the state of the target qubit (C) if and only if the control qubit (B) is in the state |1⟩. Consequently, the state of the three qubits after the CNOT gate is a superposition of two terms. The first term corresponds to the case where the control qubit (B) is in the state |0⟩, and the second term corresponds to the case where the control qubit (B) is in the state |1⟩.
Now, the sender (Alice) performs a measurement on her two qubits (A and B) in the Bell basis, which consists of four orthogonal states: |Φ+⟩, |Φ-⟩, |Ψ+⟩, and |Ψ-⟩. The measurement outcomes determine the state of the receiver's qubit (D). Depending on the measurement results, the receiver's qubit (D) can be in one of the four possible states:
1. If the measurement outcome is |Φ+⟩, the state of the receiver's qubit (D) is unchanged.
2. If the measurement outcome is |Φ-⟩, the state of the receiver's qubit (D) is flipped.
3. If the measurement outcome is |Ψ+⟩, the state of the receiver's qubit (D) is rotated counterclockwise by 90 degrees.
4. If the measurement outcome is |Ψ-⟩, the state of the receiver's qubit (D) is rotated clockwise by 90 degrees.
After the application of the CNOT gate in the teleportation protocol, the state of the three qubits becomes a superposition of different terms, and the measurement outcomes on the sender's qubits determine the state of the receiver's qubit.
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