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Is there a simple way to explain the difference between a classical and truly quantum correlation to a non-quantum person who has basic understanding classical correlation?

I mean without invoking quantum mechanics, a simple CHSH type Bell inequity can be explained without it--perhaps with a little more math than I'm looking for, but more importantly non-classical correlation does not necessarily imply Bell violation.

Has anyone found a successful approach to explaining quantum correlation to a non-expert audience? Even partially successful approaches would be helpful.

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    $\begingroup$ I usually describe hidden variables (particle A and particle B decide how they will respond to various measurements, etc.) then say that quantum goes beyond that. Then they say "can you give a concrete example?" and I explain CHSH. This line of conversation usually does not turn out in a good way. So I too am interested in a better way to explain this. $\endgroup$ – Dan Stahlke Apr 2 '14 at 13:09
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I particularly like the Mermin paradox (also known as GHZ paradox), because you can understand it without knowing anything about probability theory.

Basically it goes as follows:

You have a specific entangled state of three particles, called GHZ state. On each of the particles you can do one of two measurements, $X$ and $Y$, and in both cases, you may get either $+1$ or $-1$ as result. Of course, you can choose independently on each single particle of the state whether you measure $X$ or $Y$ on it.

If you look at the measurement results of any single particle of the state, you find that you have randomly either $+1$ or $-1$. However if you look at a full set of measurements, you'll notice a pattern:

Whenever you measure $X$ on exactly one of the entangled particles, and $Y$ on the other two, you'll notice that the product of all three measurement results is always $-1$. This is the case no matter for which of the three particles you measure $X$.

Now this would not yet be a problem: It is compatible with the assumption that each measurement value is predetermined. Be $x_i$ the measurement result of measuring $X$ on particle $i$, and $y_i$ the measurement result of measuring $Y$ on particle $i$. Then the above fact means that we have the three equations $x_1 y_2 y_3 = -1$, $y_1 x_2 y_3 = -1$ and $y_1 y_2 x_3 = -1$.

Now we can just multiply those three terms together and using the fact that each $y_i$ is either $+1$ or $-1$, and thus $y_i^2 = 1$, we get: $$x_1 x_2 x_3 = x_1 y_2 y_3 \cdot y_1 x_2 y_3 \cdot y_1 y_2 x_3 = (-1) \cdot (-1) \cdot (-1) = -1$$

Therefore we would expect that if we measure $X$ on all three particles, we also find that the product of the three values is $-1$.

However, quantum mechanics tells us, and experiment confirms (within measurement error), that if we measure $X$ on all three particles, the product of the three measurement results always is $+1$.

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You could try something like this. Suppose you have two electrons in the singlet state. You can measure the electron's spin along three directions x,y,z. Regardless of the direction you measure the spin the probability of getting up or down is 1/2. If you measure the x spin of both electrons when you compare the results you find that they are opposite: if the spin of one of the electrons is up the spin of the other will be down. If you measure the electrons in different directions (e.g. - you measure the spin in the x direction on on electron and measure the other in the z direction) the probability that they match will be 1/2. So if the spins were decided in advance how could it be the case that they match when you do some measurements but not when you do others?

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