What is the purpose of applying a CNOT gate in the quantum teleportation protocol?
The purpose of applying a Controlled-NOT (CNOT) gate in the quantum teleportation protocol is to enable the transfer of an unknown quantum state from one qubit to another. The CNOT gate plays a important role in the entanglement-based teleportation scheme, allowing for the faithful transmission of quantum information.
In the quantum teleportation protocol, there are three qubits involved: the sender's qubit (A), the entangled qubit pair (B and C), and the receiver's qubit (D). The goal is to teleport the state of qubit A to qubit D. To achieve this, the CNOT gate is applied to qubit B, acting as the control qubit, and qubit A, acting as the target qubit.
The CNOT gate is a two-qubit gate that flips the target qubit (qubit A) if and only if the control qubit (qubit B) is in the state |1⟩. This gate is represented by the following matrix:
CNOT = |1 0 0 0|
|0 1 0 0|
|0 0 0 1|
|0 0 1 0|
Initially, qubit B is entangled with qubit C in a Bell state, such as the maximally entangled state |Φ+⟩ = (|00⟩ + |11⟩)/√2. This entanglement is established beforehand and shared between the sender and the receiver. The application of the CNOT gate on qubits B and A entangles qubit A with qubit C, while preserving the state of qubit B.
Next, a Hadamard gate (H gate) is applied to qubit A, followed by a measurement in the Bell basis, which consists of the two Bell states |Φ+⟩ and |Φ-⟩ = (|01⟩ + |10⟩)/√2. The measurement outcomes are two classical bits, which are transmitted to the receiver.
Upon receiving the measurement outcomes, the receiver performs operations based on the measurement results and the state of qubit D. If the measurement outcome is |Φ+⟩, no further operations are necessary. If the outcome is |Φ-⟩, the receiver applies a Pauli-X gate (X gate) to qubit D. If the outcome is |Ψ+⟩ = (|01⟩ – |10⟩)/√2, the receiver applies a Pauli-Z gate (Z gate) to qubit D. Finally, if the outcome is |Ψ-⟩ = (|00⟩ – |11⟩)/√2, the receiver applies both the X gate and the Z gate to qubit D.
By applying these operations, the receiver successfully recreates the unknown quantum state of qubit A on qubit D. Thus, the CNOT gate, along with other gates and measurements, allows for the teleportation of quantum information from one qubit to another.
To summarize, the purpose of applying a CNOT gate in the quantum teleportation protocol is to establish entanglement between the sender's qubit and the entangled pair, enabling the faithful transfer of the unknown quantum state to the receiver's qubit. The CNOT gate plays a pivotal role in creating this entanglement and forms an essential component of the teleportation process.
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