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I'm struggling to understand the concept of quantum entanglement. I've distilled my understanding into an analogy, and I need your help to validate it. Here it is:

Let's say I receive two envelopes. Both envelopes have this written on them:

Hello! Open this letter...
If the paper inside is red, then other envelope has blue paper.
If the paper inside is blue, then other envelope has red paper.

So, if I open one letter, and the paper inside is red, then I know the other letter contains the blue paper.

Is this equivalent to quantum entanglement? The fact that I will know the color of the other paper when I open (i.e. observe) one envelope?

In theory, if one of the letters "appeared" in a different galaxy, I would still know the color of the paper inside that letter, just by opening my letter, instantly, correct?

Before I open the letter, is the paper inside each envelope in superposition (??), i.e. 50/50, red or blue?

In this analogy, I am assuming that what's written on the letters is true. Is this a correct assumption in quantum physics? I think so yes?

In real quantum physics, would I be able to change the color of one paper, so that I change the color of the other one? No right?

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If you want to turn this into an analogue of quantum entanglement, you need to assume this: You and I each carry one of the envelopes to a different location. Then we open them. Then:

  • If we both choose to open our envelopes from the top, we always find papers of opposite colors.
  • If either of us chooses to open his envelope from the top while the other chooses to open his envelope from the bottom, we still always (or nearly always) find papers of opposite colors.
  • If we both choose to open our envelopes from the bottom, we always (or nearly always) find papers of identical colors.

Now try telling a story like yours --- where the envelopes carry true information about what's "really" in the other envelope --- that fits these facts. Good luck.

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No, that is not an analogy for quantum entanglement. From a quantum mechanics perspective, what you are describing is called a local hidden variables theory (though people often use socks instead of envelopes). Bell's theorem tells us that such LHV theories cannot account for all the possibilities allowed by the quantum mechanics of entangled states - and experiment sides with quantum mechanics over LHV theories.

For an example of the kind of thing you can do with entanglement that you can't do with LHV theories, see WillO's fantastic answer to this thread.

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Your analogy is not equivalent to entanglement. It's best not to make analogies, since most of them are misleading.

In an entanglement experiment you have to be able to measure at least two different physical quantities on each system, e.g. - the spin of an electron in the x direction and the spin in the z direction. Your analogy only has one measurement: is the letter red or blue?

In an entanglement experiment if you measure the same quantity for each half of the entangled pair (z spin for both electrons, say), when you compare the results you find that they are correlated. The correlation diminishes for measurements that don't match. So whether you find a correlation depends on whether the same measurement was conducted on each system. The dependence of the correlation on the difference between the measurements is such that it can't be explained by a local theory in which physical quantities are represented by stochastic variables (single numbers picked randomly). Quantum mechanics describes systems in terms of more complicated mathematical objects called observables, not in terms of stochastic variables. Those observables change locally, so quantum mechanics is local, proclamations to the contrary notwithstanding:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

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