How do Alice and Bob use their shared entangled state to generate non-local correlations in the CHSH game?
In the field of Quantum Information, the concept of entanglement plays a important role in understanding the phenomenon of non-local correlations. Alice and Bob, two distant parties, can utilize their shared entangled state to generate these correlations in a game known as the CHSH game, which stands for Clauser-Horne-Shimony-Holt inequality. This game serves as a test to demonstrate the violation of local realism, a principle that assumes the existence of hidden variables governing the behavior of quantum systems.
To consider the process of generating non-local correlations in the CHSH game, we first need to understand the basics of entanglement. In quantum mechanics, entanglement refers to the strong correlation between the states of two or more particles, even when they are physically separated. These entangled states cannot be described independently but must be considered as a whole system. When a measurement is performed on one of the entangled particles, it instantaneously affects the state of the other, regardless of the distance between them. This instantaneous correlation is what allows Alice and Bob to achieve non-local correlations in the CHSH game.
The CHSH game involves two players, Alice and Bob, who each possess one particle from an entangled pair. The goal of the game is for Alice and Bob to generate correlations that violate the CHSH inequality, thereby proving the existence of non-local correlations. The CHSH inequality is a mathematical expression that bounds the correlations achievable by local hidden variable theories.
To begin the game, Alice and Bob must share an entangled state. One commonly used example of an entangled state is the singlet state, also known as the maximally entangled state or the Bell state:
|Ψ⟩ = (|01⟩ – |10⟩)/√2
In this state, the first qubit belongs to Alice, and the second qubit belongs to Bob. The state is written in the computational basis, where |0⟩ and |1⟩ represent the two orthogonal states of a qubit.
Next, Alice and Bob perform measurements on their respective qubits. In the CHSH game, they have two measurement options, labeled as A0, A1 for Alice, and B0, B1 for Bob. Each measurement corresponds to a specific observable, such as spin in a particular direction.
The outcome of each measurement is a binary value, either 0 or 1. The correlation between Alice and Bob's measurement outcomes is then calculated using the following formula:
E = P(A0,B0) + P(A0,B1) + P(A1,B0) – P(A1,B1)
Here, P(Ai,Bj) represents the joint probability of Alice obtaining outcome Ai and Bob obtaining outcome Bj.
To violate the CHSH inequality, Alice and Bob must generate correlations that yield a value of E greater than 2. According to local realism, the maximum value of E is 2, but in the quantum realm, entangled states can lead to correlations that exceed this limit.
To achieve the violation, Alice and Bob choose their measurement settings, A0, A1, B0, and B1, in a specific way. By selecting the appropriate combination of measurement settings, they can maximize the correlations and obtain a value of E greater than 2.
For example, Alice and Bob could agree to use the following measurement settings:
A0: Measure spin along the x-axis
A1: Measure spin along the z-axis
B0: Measure spin along a different axis, such as (x + z)/√2
B1: Measure spin along a different axis, such as (x – z)/√2
By using these settings, Alice and Bob can generate correlations that violate the CHSH inequality, providing evidence against local realism and demonstrating the presence of non-local correlations.
Alice and Bob utilize their shared entangled state to generate non-local correlations in the CHSH game by performing measurements on their respective particles. By selecting specific measurement settings, they can achieve correlations that violate the CHSH inequality and provide evidence for the existence of non-locality. This phenomenon highlights the unique properties of entanglement in quantum information.
Other recent questions and answers regarding CHSH inequality:
- Does testing of Bell or CHSH inequalities show that it is possible that quantum mechanics is local but violates the realism postulate?
- Describe the ongoing efforts to design experiments that can eliminate all the loopholes simultaneously and provide even stronger evidence against local realism.
- What are the loopholes that have been addressed in experiments testing the CHSH inequality, and why are they important to eliminate?
- Explain the CHSH inequality and its significance in testing the predictions of quantum mechanics against local realism.
- What is quantum entanglement and how does it differ from classical correlations?