Quantum entanglement is considered mysterious because it defies classical intuitions and challenges our everyday understanding of how the world works. While your example with the two boxes and colored balls illustrates a correlation between the boxes, it doesn't capture the essence of quantum entanglement.
In quantum mechanics, entanglement refers to a phenomenon where the quantum states of two or more particles become intertwined in such a way that the state of one particle cannot be described independently of the other particles' states. The behavior of entangled particles cannot be explained by classical concepts, as their properties are interconnected in nontrivial ways.
Here's an example to highlight the mysterious nature of quantum entanglement: Consider a pair of entangled particles, such as electrons, that are created in a specific quantum state called a superposition. This means that until a measurement is made, the particles do not have definite properties like position, momentum, or spin. Instead, they exist in a combination of multiple states simultaneously.
When one of the entangled particles is measured, its quantum state collapses into a specific value. Remarkably, at the moment of measurement, the state of the other particle instantaneously changes, regardless of the distance between them. This change occurs instantaneously, even if the particles are separated by vast distances, violating the constraints of classical communication.
This peculiar feature of entanglement, known as "spooky action at a distance," is what makes it mysterious. It suggests that information or influences can propagate instantaneously between entangled particles, regardless of the space separating them. This contradicts our everyday experience of cause and effect, where information can only travel at or below the speed of light.
Furthermore, entanglement allows for correlations between measurements that cannot be explained by any classical means. These correlations persist even when the entangled particles are far apart, and their behavior cannot be accounted for by any local hidden variables theory—a concept proposed in classical physics.
It's worth noting that quantum entanglement has been experimentally verified through numerous tests, including the famous Bell's theorem experiments, which confirmed the existence of non-local correlations predicted by quantum mechanics.
While your example with the boxes and colored balls shows a correlation, it does not capture the non-local nature and the superposition of quantum entanglement, which is why quantum entanglement remains a fascinating and mysterious phenomenon in the realm of quantum physics.