Describe the scenario involving Alice and Bob and their random bit values in the CHSH inequality.
In the scenario involving Alice and Bob and their random bit values in the CHSH inequality, we are examining the concept of quantum entanglement and its implications on local realism. The CHSH inequality, named after Clauser, Horne, Shimony, and Holt, is a fundamental test used to investigate the violation of local realism in quantum systems.
To understand the scenario, let's first establish the concept of quantum entanglement. In quantum mechanics, entanglement refers to the phenomenon 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 correlation persists even if the particles are separated by vast distances.
Now, let's consider Alice and Bob, who each possess a qubit, the basic unit of quantum information. The state of Alice's qubit can be represented as |ψ⟩_A = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes, and |0⟩ and |1⟩ represent the two possible states of a qubit. Similarly, Bob's qubit state can be represented as |ψ⟩_B = γ|0⟩ + δ|1⟩.
In the CHSH inequality scenario, Alice and Bob are physically separated, and each performs measurements on their respective qubits simultaneously. They have a choice between two measurement settings, conventionally labeled as 0 and 1. Each measurement setting corresponds to a specific basis in which the qubit state is measured.
Let's denote Alice's measurement settings as A0 and A1, and Bob's measurement settings as B0 and B1. The outcome of Alice's measurement in setting A0 is denoted as a, and the outcome in setting A1 is denoted as a'. Similarly, the outcomes of Bob's measurements in settings B0 and B1 are denoted as b and b', respectively.
To analyze the scenario using the CHSH inequality, we consider the correlation between the measurement outcomes. The CHSH inequality is given by:
S = E(a, b) + E(a, b') + E(a', b) – E(a', b') ≤ 2,
where E(a, b) represents the correlation between Alice's outcome a and Bob's outcome b, and similarly for the other terms.
In the case of local realism, the expectation value S should be less than or equal to 2. However, in quantum mechanics, entangled states can violate this inequality, indicating the presence of non-local correlations that cannot be explained by local realism.
To see this violation, let's consider an example where Alice and Bob share an entangled state known as the Bell state. The Bell state can be represented as |Φ+⟩ = (|00⟩ + |11⟩)/√2, where |00⟩ and |11⟩ are tensor product states of the qubits.
When Alice and Bob measure their qubits in the same basis (A0 = B0 and A1 = B1), they will obtain correlated outcomes. For example, if Alice measures her qubit and gets outcome a = 0, then Bob's outcome b will also be 0. Similarly, if Alice measures her qubit and gets outcome a = 1, then Bob's outcome b will also be 1. In this case, the correlation E(a, b) will be 1.
However, when Alice and Bob measure their qubits in different bases (A0 ≠ B0 or A1 ≠ B1), they will obtain anti-correlated outcomes. For instance, if Alice measures her qubit and gets outcome a = 0, then Bob's outcome b will be 1, and vice versa. In this case, the correlation E(a, b) will be -1.
By calculating the expectation value S using these correlations, we find that S = 2√2, which violates the CHSH inequality (S ≤ 2). This violation demonstrates the presence of non-local correlations and the failure of local realism.
The scenario involving Alice and Bob and their random bit values in the CHSH inequality is a fundamental demonstration of the violation of local realism in quantum systems. By using entangled states, such as the Bell state, Alice and Bob can achieve correlations that cannot be explained by local realism. This violation has profound implications for our understanding of the nature of reality at the quantum level.
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