# What is Bell’s inequality and how is it violated in entangled systems? Quantum entanglement is a phenomenon that occurs when two or more particles interact in such a way that their properties become correlated, regardless of the distance between them. This correlation is often referred to as “spooky action at a distance” and is one of the most intriguing features of quantum mechanics. The study of quantum entanglement has led to many important insights into the nature of reality at the smallest scales of the universe. One of the most significant contributions to this field was made by John Bell, who developed an inequality that allows us to test whether or not a given system is entangled. In this essay, we will explore Bell’s inequality and how it is violated in entangled systems.

Bell’s Inequality:

Bell’s inequality is a mathematical expression that puts a limit on the amount of correlation that can exist between two particles in a local hidden variable theory (LHV). An LHV theory assumes that particles have predetermined properties that are hidden from us, and that their behavior is deterministic. In other words, the outcome of any experiment can be predicted with certainty if we have complete knowledge of the system.

Bell’s inequality takes the form of an inequality that relates the correlation between two particles to certain measurable quantities. In its simplest form, it states that the correlation between two particles should be less than or equal to a certain value, which is derived from the measured values of the particles’ properties. If the correlation exceeds this value, then it is said that the particles violate Bell’s inequality, which implies that the system cannot be explained by a local hidden variable theory.

The Bell test experiment:

To test Bell’s inequality, scientists perform what is known as a Bell test experiment. In a Bell test experiment, two particles are prepared in an entangled state and then separated by a distance. The particles are then measured along one of two possible axes, which can be at any angle relative to each other. The correlation between the particles is then measured, and this data is used to calculate whether or not Bell’s inequality is violated.

When the experiment is performed, the correlation between the particles is found to exceed the value predicted by Bell’s inequality. This violation of Bell’s inequality is strong evidence that the particles are entangled and cannot be described by a local hidden variable theory.

The concept of locality:

One of the key assumptions of Bell’s inequality is locality, which means that the outcome of any experiment should be independent of the distance between the particles. In other words, the correlation between two particles should not depend on the distance between them.

However, quantum mechanics suggests that this assumption may be incorrect. According to quantum mechanics, particles can be in a superposition of states until they are observed, at which point their wave function collapses and their properties become definite. This means that the outcome of an experiment can be influenced by the observer, which implies a non-locality.

Non-locality and entanglement:

Non-locality is an important concept in quantum mechanics, and it is closely related to the phenomenon of entanglement. In entangled systems, particles can become correlated in such a way that their properties are intertwined, regardless of the distance between them. This means that the outcome of an experiment performed on one particle can be correlated with the outcome of an experiment performed on the other particle, even if they are separated by a large distance.

This correlation violates the assumption of locality, and this violation is the key to understanding why Bell’s inequality is violated in entangled systems. When particles are entangled, their correlation is so strong that it exceeds the limit predicted by Bell’s inequality, which implies that the system cannot be described by a local hidden variable theory.

Conclusion:

Bell’s inequality is a mathematical expression that puts a limit on the amount of correlation that can exist between two particles in a local hidden variable theory.