Classical entanglement occurs for instance when I know there is two balls in a box: a red one and a black one. If I take one ball from the box, then as soon as I see its color, I know what is the color of the other one still in the box.
Quantum entanglement has something more than that: I could also take a red "plus" black ball. Then at that moment I would know that what remains in the box is a red "minus" black ball. One then speaks about a superposition of two states.

Balls make no sense in quantum mechanics, however a quantum particle can be in a superposition of two localized wave paquets (in the pictures the two red discs stand for a non zero probability distribution in 2D position space x,y).

1 - In the following situation, the two coordinates x and y of the particle are separable: if I measure y I don't know about x, i.e. I have no correlation between x and y.

2 - In the following one, coordinates x and y are entangled: if I measure y I know about x, x and y are correlated.

Geometrically both above situations look pretty much the same, it is just the orientation of the observer in the room which is different.
Entanglement is not a geometrically intrinsic property but something related to what we can observe in a specific configuration.


The more famous situation of two EPR particles, with entangled spins, is formally indentical, except that spin replaces the position:
one would have particle A with spin x1 or x2 and particle B with spin y0, y1 or y2.
- In the first case one particle has a well defined spin (y0), so measures of spin are uncorrelated.
- In the second situation the spin of one particle is correlated to the spin of the other: both particles are then entangled through their spin.

Analogy in day to day life

If two people are called Bob, whenever I call "Bob!" they don't know who I am calling.
If they are called Bob and Paul, then I can say "Bob!" when I want to call him.

Entanglement has something to do with language adequacy.

Defenders of hidden variables theory would argue that when two people are called Bob they are still two different persons, and I can distinguish them by other means.
However in strict Quantum Mechanics, if two people are called Bob they are the same person; and this results with such strange things as entanglement.

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