Describe the fundamental difference in the correlation properties between classically correlated systems and quantum entangled systems, especially regarding measurements on spatially separated parts.

Classically correlated systems exhibit correlations that arise from shared past events or pre-existing, locally defined properties. If you have two coins that are rigged to always land on the same side, they are classically correlated: knowing the outcome of one coin immediately tells you the outcome of the other. The correlation is based on prior knowledge and doesn't violate locality – the state of one coin doesn't instantly affect the other. Quantum entangled systems, in contrast, exhibit correlations that cannot be explained by classical physics and violate locality. When two or more particles are entangled, their quantum states are linked regardless of the distance separating them. If you measure a property of one entangled particle, you instantaneously know the corresponding property of the other particle, even if they are light-years apart. This instantaneous correlation isn't due to any signal passing between the particles; rather, it's an inherent property of the shared quantum state. The difference is highlighted by Bell's theorem, which provides mathematical inequalities that classical correlations must obey. Quantum entangled systems can violate these inequalities, demonstrating that their correlations are stronger than any possible classical correlation. Furthermore, classical correlations can be explained by local hidden variables, meaning that each particle had a pre-determined state before the measurement. Entanglement, however, cannot be explained by local hidden variables. Measuring one entangled particle fundamentally affects the state of the other, regardless of distance, demonstrating a non-classical, non-local connection.