What are Bell's inequalities and how do they quantify the correlations between measurements in Bell's experiment?
Bell's inequalities are a set of mathematical inequalities that were derived by physicist John Bell in 1964. They provide a way to quantify the correlations between measurements in Bell's experiment, which is designed to test the concept of quantum entanglement. Quantum 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.
In Bell's experiment, two particles, typically referred to as "Alice" and "Bob," are prepared in an entangled state and then sent to separate measurement devices. Each measurement device has a setting that can be adjusted, and when a measurement is made on the particle, it will yield a certain outcome. The goal of Bell's experiment is to determine whether the outcomes of the measurements on Alice and Bob's particles are correlated in a way that cannot be explained by classical physics.
To quantify the correlations between the measurements, Bell introduced the concept of Bell's inequalities. These inequalities are derived based on certain assumptions about the nature of the physical world. The most well-known of these assumptions is called "local realism," which states that physical properties of objects exist independently of measurements and that information cannot be transmitted faster than the speed of light.
Bell's inequalities involve statistical correlations between the outcomes of measurements made on Alice and Bob's particles at different settings. By comparing these correlations to the values predicted by local realism, one can determine whether the observed correlations violate the inequalities. If the inequalities are violated, it implies that the measurements are not explained by local realism and that the particles are indeed entangled.
The violation of Bell's inequalities has been experimentally observed in numerous experiments, providing strong evidence for the existence of quantum entanglement. These violations demonstrate that entangled particles can exhibit correlations that are stronger than what can be explained by classical physics.
To illustrate this concept, consider a scenario where Alice and Bob each have a measurement device with two possible settings: A or B. Each setting corresponds to a different property of the particles they are measuring. For simplicity, let's assume that the possible outcomes of the measurements are +1 or -1.
If the particles were not entangled and the measurements were explained by local realism, the correlations between Alice and Bob's measurements would follow certain limits. These limits are defined by Bell's inequalities. For example, one such inequality, known as the CHSH inequality, states that the absolute value of the correlation between Alice's measurement at setting A and Bob's measurement at setting B, plus the absolute value of the correlation between Alice's measurement at setting B and Bob's measurement at setting A, must be less than or equal to 2.
However, if the particles are entangled, the correlations between Alice and Bob's measurements can violate these limits. For instance, if the particles are in a maximally entangled state called a Bell state, the correlations can reach a value of 2√2, which exceeds the limit imposed by the CHSH inequality. This violation demonstrates the non-classical nature of the correlations and provides evidence for the existence of quantum entanglement.
Bell's inequalities are mathematical expressions that quantify the correlations between measurements in Bell's experiment. They provide a means to test whether the observed correlations violate the limits imposed by local realism. Violations of these inequalities indicate the presence of quantum entanglement and challenge the classical understanding of physical reality.
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