Where does the concept of quantum entanglement originate? A video I watched from PBS spacetime talked about how the pilot wave theory provides an explanation for why quantum entanglement occurs, where it is a result of having global hidden variables. I just want to know if there is an explanation like this in the standard interpretation.
How does quantum mechanics predict quantum entanglement? Does the concept of entanglement come from any mathematical equations?
 A: Entanglement arises from the mathematical structure of quantum mechanical states. It is not exclusive to quantum mechanics - here is a non-quantum example of entanglement.
Suppose I have a bag containing $40$ balls. An experiment consists of drawing a ball at random from the bag and then putting it back again. After doing many experiments I know that half the balls in the bag are red and half are blue. I also know that half the balls in the bag are large and half are small. I draw a ball from the bag and it is a red ball. What is the probability that it is also a large ball ?
It is tempting to say $50\%$ since half the balls in the bag are large and half are small. And indeed this is possible; if the bag contains $10$ large red balls, $10$ large blue balls, $10$ small red balls and $10$ small blue balls, then the probability of drawing a red or a blue ball is independent of the probability of drawing a large or a small ball. We could say the bag is a "tensor" product of two separate probability distributions or "states":
$(0.5 \text{ red} + 0.5 \text{ blue}) \otimes (0.5 \text{ large} + 0.5 \text{ small})$
But suppose the bag contains $20$ large red balls and $20$ small blue balls. The probability of drawing a red or blue ball and the probability of drawing a large or a small ball are now interdependent - we could say they are "entangled". There are now no values $p$ and $q$ that will let us write the  contents of the bag as a tensor product
$(p \text{ red} + (1-p) \text{ blue}) \otimes (q \text{ large} + (1-q) \text{ small})$
Quantum entanglement arises in a similar way, although there are several important differences that mean the "bag of balls" analogy is not exact:

*

*The amplitudes in a quantum mechanical state are complex numbers, not real numbers.

*We may not be able to determine the colour and the size of a ball at the same time, because of Heisenberg's uncertainty principle.

*We think there is no exact equivalent of the bag of balls in quantum mechanics, because this would make quantum mechanics a hidden variable theory.

