How does the CHSH inequality specifically test the violation of local realism?
The CHSH inequality, named after its discoverers Clauser, Horne, Shimony, and Holt, is a important tool in testing the violation of local realism in the context of quantum entanglement. Local realism refers to the idea that physical systems have pre-existing properties that determine the outcomes of measurements made on them, and that these properties are independent of any measurement choices and are not influenced by distant events. On the other hand, quantum entanglement is a 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(s).
To understand how the CHSH inequality tests the violation of local realism, let's first consider the concept of Bell's inequalities. Bell's inequalities are mathematical expressions that impose constraints on the correlations that can exist between the measurement outcomes of entangled particles under the assumption of local realism. Violation of these inequalities implies that local realism is not a valid description of nature and that the behavior of entangled particles cannot be explained by pre-existing properties.
The CHSH inequality is a specific form of Bell's inequality that is particularly useful in experimental tests. It involves four measurements, denoted by A, A', B, and B', that can be performed on two entangled particles, typically referred to as Alice and Bob. Each measurement has two possible outcomes, usually labeled as +1 and -1. The CHSH inequality is given by the following expression:
|E(A, B) + E(A, B') + E(A', B) – E(A', B')| ≤ 2
where E(A, B) represents the correlation between the outcomes of measurements A and B, and so on. The correlation is calculated as the average product of the measurement outcomes.
In a local realistic scenario, the correlation between the outcomes of different measurements is expected to be limited by the inequality, with the absolute value of the left-hand side being less than or equal to 2. However, quantum mechanics predicts that entangled particles can exhibit correlations that violate this inequality.
To see why this is the case, let's consider an example using the singlet state of two spin-1/2 particles, which is a maximally entangled state. Alice and Bob each choose one of two possible measurement directions for their particles, which we can represent as unit vectors in three-dimensional space. The measurement outcomes are determined by the projections of the spin of each particle onto the chosen measurement axis.
If Alice and Bob choose measurement directions that are parallel, the correlation between their outcomes is always -1. If they choose measurement directions that are anti-parallel, the correlation is always +1. However, if they choose measurement directions that are at an angle of 45 degrees with respect to each other, the correlation is given by the cosine of the angle between their measurement axes. By appropriately choosing the angles, it is possible to achieve a correlation of -sqrt(2), which violates the CHSH inequality.
Experimental tests of the CHSH inequality have been performed using entangled particles such as photons, electrons, and ions. These experiments involve measuring the correlations between the outcomes of different measurement settings and comparing them to the predictions of local realism. If the measured correlations violate the CHSH inequality, it provides strong evidence against the existence of pre-existing properties that determine the outcomes of measurements and supports the non-local behavior of entangled particles.
The CHSH inequality is a specific form of Bell's inequality that is used to test the violation of local realism in the context of quantum entanglement. Violation of this inequality implies that the behavior of entangled particles cannot be explained by pre-existing properties and supports the non-local correlations predicted by quantum mechanics.
Other recent questions and answers regarding Bell and local realism:
- Locality limits interaction between two spatially separated systems by the velocity of light?
- What does it mean for two spatially separated systems to be inside the locality limits?
- What does the violation of the CHSH inequality imply about the relationship between locality and realism in quantum systems?
- Describe the scenario involving Alice and Bob and their random bit values in the CHSH inequality.
- Explain the concept of Bell's inequality and its role in testing local realism.
- What is quantum entanglement and how does it relate to the state of particles?