Describe the Einstein-Podolsky-Rosen (EPR) paradox and explain how Bell's theorem challenges classical assumptions about locality and realism.

The Einstein-Podolsky-Rosen (EPR) paradox, proposed in 1935, highlights a conflict between quantum mechanics and our intuitive understanding of locality and realism. Realism, in this context, asserts that physical properties of a system have definite values, regardless of whether or not they are measured. Locality asserts that an object is only directly influenced by its immediate surroundings; in other words, no influence can travel faster than the speed of light. EPR considered a pair of entangled particles, such as two photons with correlated polarization. According to quantum mechanics, the polarization of each photon is undefined until measured. However, if the photons are entangled, measuring the polarization of one photon instantly determines the polarization of the other, no matter how far apart they are. EPR argued that if quantum mechanics is complete, then these properties must have been determined from the beginning; otherwise, quantum mechanics would be incomplete. They posited the existence of 'hidden variables' that predetermine the outcome of any measurement, restoring determinism and locality. Bell's theorem, developed by John Bell in 1964, provides a mathematical framework to test whether local hidden variable theories can explain the correlations observed in entangled systems. Bell derived inequalities that place limits on the correlations that can be observed if both realism and locality hold true. Experiments have repeatedly shown that these inequalities are violated by entangled systems. This violation implies that at least one of the assumptions – locality or realism – must be false. Since experiments confirm the predictions of quantum mechanics and violate Bell's inequalities, it is generally accepted that either locality or realism (or both) must be abandoned. The common interpretation is that quantum mechanics is non-local, meaning that entangled particles can influence each other instantaneously, regardless of the distance separating them, and that properties are not defined until measurement.