Here, from <http://www.scottaaronson.com/blog/?p=2464>:
A few weeks ago, Hensen et al., of the Delft University of Technology and Barcelona, Spain, put out a paper reporting the first experiment that violates the Bell inequality in a way that closes off the two main loopholes simultaneously: the locality and detection loopholes. Well, at least with ~96% confidence. This is big news, not only because of the result itself, but because of the advances in experimental technique needed to achieve it. Last Friday, two renowned experimentalists—Chris Monroe of U. of Maryland and Jungsang Kim of Duke—visited MIT, and in addition to talking about their own exciting ion-trap work, they did a huge amount to help me understand the new Bell test experiment. So OK, let me try to explain this.
While some people like to make it more complicated, the Bell inequality is the following statement. Alice and Bob are cooperating with each other to win a certain game (the “CHSH game“) with the highest possible probability. They can agree on a strategy and share information and particles in advance, but then they can’t communicate once the game starts. Alice gets a uniform random bit x, and Bob gets a uniform random bit y (independent of x). Their goal is to output bits, a and b respectively, such that a XOR b = x AND y: in other words, such that a and b are different if and only if x and y are both 1. The Bell inequality says that, in any universe that satisfies the property of local realism, no matter which strategy they use, Alice and Bob can win the game at most 75% of the time (for example, by always outputting a=b=0).
What does local realism mean? It means that, after she receives her input x, any experiment Alice can perform in her lab has a definite result that might depend on x, on the state of her lab, and on whatever information she pre-shared with Bob, but at any rate, not on Bob’s input y. If you like: a=a(x,w) is a function of x and of the information w available before the game started, but is not a function of y. Likewise, b=b(y,w) is a function of y and w, but not of x. Perhaps the best way to explain local realism is that it’s the thing you believe in, if you believe all the physicists babbling about “quantum entanglement” just missed something completely obvious. Clearly, at the moment two “entangled” particles are created, but before they separate, one of them flips a tiny coin and then says to the other, “listen, if anyone asks, I’ll be spinning up and you’ll be spinning down.” Then the naïve, doofus physicists measure one particle, find it spinning down, and wonder how the other particle instantly “knows” to be spinning up—oooh, spooky! mysterious! Anyway, if that’s how you think it has to work, then you believe in local realism, and you must predict that Alice and Bob can win the CHSH game with probability at most 3/4.
What Bell observed in 1964 is that, even though quantum mechanics doesn’t let Alice send a signal to Bob (or vice versa) faster than the speed of light, it still makes a prediction about the CHSH game that conflicts with local realism. (And thus, quantum mechanics exhibits what one might not have realized beforehand was even a logical possibility: it doesn’t allow communication faster than light, but simulating the predictions of quantum mechanics in a classical universe would require faster-than-light communication.) In particular, if Alice and Bob share entangled qubits, say $$\frac{\left| 00 \right\rangle + \left| 11 \right\rangle}{\sqrt{2}},$$ then there’s a simple protocol that lets them violate the Bell inequality, winning the CHSH game ~85% of the time (with probability (1+1/√2)/2 > 3/4). Starting in the 1970s, people did experiments that vindicated the prediction of quantum mechanics, and falsified local realism—or so the story goes.
Discussion in <https://news.ycombinator.com/item?id=10269297> OK, so local realism is dead. But what is it? An excerpt from above:
Perhaps the best way to explain local realism is that it’s the thing you believe in, if you believe all the physicists babbling about “quantum entanglement” just missed something completely obvious.
And here's the tl;dr:
(And thus, quantum mechanics exhibits what one might not have realized beforehand was even a logical possibility: it doesn’t allow communication faster than light, but simulating the predictions of quantum mechanics in a classical universe would require faster-than-light communication.)
Another chunk:
The violation of the Bell inequality has a schizophrenic status in physics. To many of the physicists I know, Nature’s violating the Bell inequality is so trivial and obvious that it’s barely even worth doing the experiment: if people had just understood and believed Bohr and Heisenberg back in 1925, there would’ve been no need for this whole tiresome discussion.
Me, I like the many worlds interpretation. But it's just an interpretation. What matters is the math.