# I don't understand the paradox of entanglement

## Main Question or Discussion Point

People talk about entanglement with such passion and they tell me how big paradox it is, and yet I dont see why. I think maybe I just dont understand quantum mechanics enough? I dont know...

Basically you fire two electrons different directions, and then you can measure 50% chance spin up or down of one of them, and you instantly know the other one must be down or up, respectively. Action at a distance? Why :s There is no action on distance... When you fired the electron's of each other in opposite directions, there was 50% chance that the spin up electron went, say, right. So if you measure it being up later on, you know that the spin down had to go left... Its not like the electron suddenly becomes spin up while the other electron suddenly becomes spin down when you measure them... The probability 50% must refer to the actual event of firing them of, and the determination of which one goes left or right by some funky laws... but once that is done, the rest should be determined.

i guess im just confused... i dont see any paradoxes in there

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Hurkyl
Staff Emeritus
Gold Member
The short answer is that you haven't learned enough quantum mechanics to understand what the issue is; you first need to learn the basics of a quantum state and uncertainty.

You are imagining a classical picture where each electron is certainly in one of those two states, and the probability distribution reflects our lack of knowledge of which actually happens. That is a wrong picture of quantum mechanics. Quantum mechanically, the probability distribution on the possible outcomes of measurement is the most fundamental description of the state of the electron.

JesseM
The key to seeing why you can't explain the results by just imagining the electrons had preexisting spins on each axis is to look at what happens when the two experimenters pick different axes to measure. Here's an analogy I came up with on another thread (for more info, google 'Bell's inequality'):

Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get opposite results--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a lemon.

Classically, we might explain this by supposing that there is definitely either a cherry or a lemon in each box, even though we don't reveal it until we scratch it, and that the machine prints pairs of cards in such a way that the "hidden" fruit in a given box of one card is always the opposite of the hidden fruit in the same box of the other card. If we represent cherries as + and lemons as -, so that a B+ card would represent one where box B's hidden fruit is a cherry, then the classical assumption is that each card's +'s and -'s are the opposite of the other--if the first card was created with hidden fruits A+,B+,C-, then the other card must have been created with the hidden fruits A-,B-,C+.

The problem is that if this were true, it would force you to the conclusion that on those trials where Alice and Bob picked different boxes to scratch, they should find opposite fruits on at least 1/3 of the trials. For example, if we imagine Bob's card has the hidden fruits A+,B-,C+ and Alice's card has the hidden fruits A-,B+,C-, then we can look at each possible way that Alice and Bob can randomly choose different boxes to scratch, and what the results would be:

Bob picks A, Alice picks B: same result (Bob gets a cherry, Alice gets a cherry)

Bob picks A, Alice picks C: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks B, Alice picks A: same result (Bob gets a lemon, Alice gets a lemon)

Bob picks B, Alice picks C: same result (Bob gets a lemon, Alice gets a lemon)

Bob picks C, Alice picks A: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks C, Alice picks picks B: same result (Bob gets a cherry, Alice gets a cherry)

In this case, you can see that in 1/3 of trials where they pick different boxes, they should get opposite results. You'd get the same answer if you assumed any other preexisting state where there are two fruits of one type and one of the other, like A+,B+,C-/A-,B-,C+ or A+,B-,C-/A-,B+,C+. On the other hand, if you assume a state where each card has the same fruit behind all three boxes, like A+,B+,C+/A-,B-,C-, then of course even if Alice and Bob pick different boxes to scratch they're guaranteed to get opposite fruits with probability 1. So if you imagine that when multiple pairs of cards are generated by the machine, some fraction of pairs are created in inhomogoneous preexisting states like A+,B-,C-/A-,B+,C+ while other pairs are created in homogoneous preexisting states like A+,B+,C+/A-,B-,C-, then the probability of getting opposite fruits when you scratch different boxes should be somewhere between 1/3 and 1. 1/3 is the lower bound, though--even if 100% of all the pairs were created in inhomogoneous preexisting states, it wouldn't make sense for you to get opposite answers in less than 1/3 of trials where you scratch different boxes, provided you assume that each card has such a preexisting state with "hidden fruits" in each box.

But now suppose Alice and Bob look at all the trials where they picked different boxes, and found that they only got opposite fruits 1/4 of the time! That would be the violation of Bell's inequality, and something equivalent actually can happen when you measure the spin of entangled photons along one of three different possible axes. So in this example, it seems we can't resolve the mystery by just assuming the machine creates two cards with definite "hidden fruits" behind each box, such that the two cards always have opposite fruits in a given box.

thanks for both your replies they both made me see my error - there is more to it than i previously though. I actually did take Introductory course to quantum mechanics, so I am familiar with some of the concepts, but I cannot accept the fact that QM is actually a valid theory describing how our world actually works... so my brain keeps rejecting it :)
..anyways, thanks. I'll have to read up more on this QM non-sense :)

J.S. Bell gives an excellent overview from the perspective you describe in his "Speakable and Unspeakable in Quantum Mechanics."

Hello Mephisto,

you may have a look at the paper "The mystery of the quantum cakes" by Kwiat and Hardy http://hexagon.physics.wisc.edu/teaching/2006f_ph448/interesting%20papers/kwiat%20quantum%20cakes%20ajp%202000.pdf [Broken]

From this paper I learned the first time that the paradox is about whether values for measurements preexist or not.

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... I cannot accept the fact that QM is actually a valid theory describing how our world actually works ...
You don't have to ... because quantum theory (in its design and application) doesn't involve describing how our world actually works (whatever that might mean).

There can be action-at-a-distance (quantum nonlocality) in quantum mechanics because the evolutions and transitions described by the theory happen in an imaginary, mathematical space.

Quantum entanglement itself isn't a paradox. It's an experimental fact. And you can take comfort in the fact that nobody really understands it (the physical experimental phenomena, that is).

Entanglement is itself a pretty large field within quantum theory. Lots of new technical stuff getting generated all the time.

Anyway, take the advice of the mentors and give yourself at least a few years of daily study before you even begin to understand the difficulties presented by the results of quantum experiments. And, as I mentioned, you can rest assured (for the time being anyway) that nobody really understands those results.

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JesseM
You don't have to ... because quantum theory (in its design and application) doesn't involve describing how our world actually works (whatever that might mean).

There can be action-at-a-distance (quantum nonlocality) in quantum mechanics because the evolutions and transitions described by the theory happen in an imaginary, mathematical space.
But I thought when people talked about "quantum nonlocality" they were referring to the violation of Bell inequalities, which depend only on observed experimental results and not on the mathematical structure of QM used to make predictions.

But I thought when people talked about "quantum nonlocality" they were referring to the violation of Bell inequalities, which depend only on observed experimental results and not on the mathematical structure of QM used to make predictions.
Maybe you're right. Here's my take on it.
Violations of Bell inequalities would imply nonlocality (the kind that would conflict with special relativity) if (and only if) the locality condition employed in whatever representation is being used is actually a locality condition. It usually boils down to the nonfactorability of, say, a biparticle quantum state representation. This nonfactorability is the general definition of quantum entanglement, so experimental violations of Bell inequalities can usually be taken as indicators of the presence of entanglement. Whether they can be taken as indicators of the presence of nonlocality (the kind that would conflict with special relativity) in nature is arguable (and ultimately unprovable). Such violations can be taken as indicating the presence of quantum nonlocality which has everything to do with the formal quantum theory and quite possibly nothing to do with nature.

The term quantum nonlocality can be used to characterize a certain aspect (and consequence) of quantum theory (involving projection along a certain axis following a qualitative result), which is part of the theory whether violations of Bell inequalities are considered or not. It's one of the things about quantum mechanics which has prompted some to say that maybe it should be called quantum non-mechanics.

This sort of nonlocality (that is, quantum nonlocality) is always instantaneous, acausal, and nonphysical.