How does Alice choose which quantum gate to apply to Bob's qubit in the quantum teleportation protocol?
In the quantum teleportation protocol, Alice chooses which quantum gate to apply to Bob's qubit based on the measurement outcomes of two entangled qubits, known as the Bell state circuit. The Bell state circuit is a fundamental component of quantum information processing, and it plays a important role in achieving quantum teleportation.
To understand how Alice chooses the quantum gate, let's first review the steps involved in the quantum teleportation protocol. The protocol involves three parties: Alice, Bob, and a shared entangled pair of qubits. The goal is to teleport the quantum state of a qubit from Alice to Bob.
1. Initialization: Initially, Alice and Bob share an entangled pair of qubits, typically in the Bell state. The Bell state is a maximally entangled state that can be represented as:
|Φ⁺⟩ = 1/√2(|00⟩ + |11⟩)
|Φ⁻⟩ = 1/√2(|00⟩ – |11⟩)
|Ψ⁺⟩ = 1/√2(|01⟩ + |10⟩)
|Ψ⁻⟩ = 1/√2(|01⟩ – |10⟩)
2. State Preparation: Alice prepares the qubit she wants to teleport (let's call it qubit A) in an arbitrary state |ψ⟩, which can be represented as:
|ψ⟩ = α|0⟩ + β|1⟩
Here, α and β are complex amplitudes.
3. Entanglement and Measurement: Alice performs a joint measurement on qubits A and one of the entangled qubits (let's call it qubit B) using the Bell state circuit. The Bell state circuit consists of two CNOT gates and a Hadamard gate, applied in a specific order. The measurement outcomes determine the classical information that Alice communicates to Bob.
The Bell state circuit can be represented as follows:
—[Hadamard]—[CNOT]—[CNOT]—
Initially, the two qubits are in the state |ψ⟩⊗|Φ⁺⟩.
The Hadamard gate (H) is applied to qubit A, resulting in:
(H⊗I)|ψ⟩⊗|Φ⁺⟩ = (α|0⟩ + β|1⟩)⊗(1/√2)(|00⟩ + |11⟩)
Next, a CNOT gate is applied with qubit A as the control and qubit B as the target. This results in:
(CNOT)|ψ⟩⊗|Φ⁺⟩ = α|0⟩⊗(1/√2)(|00⟩ + |11⟩) + β|1⟩⊗(1/√2)(|01⟩ + |10⟩)
Finally, another CNOT gate is applied with qubit B as the control and qubit A as the target. This yields:
(CNOT)|ψ⟩⊗|Φ⁺⟩ = α|0⟩⊗(1/√2)(|00⟩ + |11⟩) + β|1⟩⊗(1/√2)(|01⟩ + |10⟩)
The measurement outcomes of the Bell state circuit are determined by measuring qubits A and B in the computational basis (|0⟩ and |1⟩). Depending on the measurement outcomes, Alice obtains one of the four possible results: 00, 01, 10, or 11.
4. Communication: Alice communicates the measurement outcomes to Bob using classical communication channels. This requires two classical bits of information.
5. Gate Application: Based on the measurement outcomes received from Alice, Bob applies a specific quantum gate to his qubit (qubit B). The gate selection is determined by the measurement outcomes and follows a predefined protocol:
– If the measurement outcome is 00, Bob does nothing.
– If the measurement outcome is 01, Bob applies the X gate (bit-flip gate) to his qubit.
– If the measurement outcome is 10, Bob applies the Z gate (phase-flip gate) to his qubit.
– If the measurement outcome is 11, Bob applies the ZX gate (a combination of the X and Z gates) to his qubit.
6. Teleportation: After applying the appropriate quantum gate, Bob's qubit (qubit B) now holds the teleported state |ψ⟩.
Alice chooses which quantum gate to apply to Bob's qubit in the quantum teleportation protocol based on the measurement outcomes obtained from the Bell state circuit. The measurement outcomes determine the classical information that Alice communicates to Bob, who then applies the corresponding quantum gate to his qubit.
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?
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