Quantum Entanglement - What's the big deal?
I understand your confusion, but here's why people often feel that quantum entanglement is rather strange. Let's first consider the following statement you make:
2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties
A classical (non-quantum) version of this statement would go something like this. Imagine that you take two marbles and paint one of them black, and one of them white. Then, you put each in its own opaque box and to send the white marble to Los Angeles, and the black marble to New York. Next, you arrange for person L in Los Angeles and person N in New York to open each box at precisely 5:00 PM and record the color of the ball in his box. If you tell each of person L and person N how you have prepared the marbles, then they will know that when they open their respective boxes, there will be a 50% chance of having a white marble, and a 50% chance of having a black marble, but they don't know which is in the box until they make the measurement. Moreover, once they see what color they have, they know instantaneously what the other person must have measured because of the way the system of marbles was initially prepared.
However, because you painted the marbles, you know with certainty that person L will have the white marble, and person N will have the black marble.
In the case of quantum entanglement, the state preparation procedure is analogous. Instead of marbles, we imagine having electrons which have two possible spin states which we will call "up" denoted $|1\rangle$ and "down" denoted $|0\rangle$. We imagine preparing a two-electron system in such a way that the state $|\psi\rangle$ of the composite system is in what's called a superposition of the states "up-down" and "down-up" by which I mean $$ |\psi\rangle = \frac{1}{\sqrt 2}|1\rangle|0\rangle + \frac{1}{\sqrt{2}}|0\rangle|1\rangle $$ All this mathematical expression means is that if we were to make a measurement of the spin state of the composite system, then there is a 50% probability of finding electron A in the spin up state and electron B in the spin down state and a 50% probability of finding the reverse.
Now me imagine sending electron $A$ to Los Angeles and electron B to New York, and we tell people in Los Angeles and New York to measure and record the spin state of his electron at the same time and to record his measurement, just as in the case of the marbles. Then, just as in the case of the marbles, these observers will only know the probability (50%) of finding either a spin up or a spin down electron after the measurement. In addition, because of the state preparation procedure, the observers can be sure of what the other observer will record once he makes his own observation, but there is a crucial difference between this case and the marbles.
In electron case, even the person who prepared the state will not know what the outcome of the measurement will be. In fact, no one can know with certainty what the outcome will be; there is an inherent probabilistic nature to the outcome of the measurement that is built into the state of the system. It's not as though there is someone who can have some hidden knowledge, like in the case of the marbles, about what the spin states of the electrons "actually" are.
Given this fact, I think most people find it strange that once one observer makes his measurement, he knows with certainty what the other observer will measure. In the case of the marbles, there's no analogous strangeness because each marble was either white or black, and certainly no communication was necessary for each observed to know what the other would see upon measurement. But in the case of the electrons, there is a sort of intrinsic probability to the nature of the state of the electron. The electron truly has not "decided" on a state until right when the measurement happens, so how is it possible that the electrons always "choose" to be in opposite states given that they didn't make this "decision" right until the moment of measurement. How will they "know" what the other electron picked? Strangely enough, they do, in fact, somehow "know."
Addendum. Certainly, as Lubos points out in his comment, there is nothing actually physically paradoxical or contradictory in entanglement, and it is just a form of correlation, but I personally think it's fair to call it a "strange" or "unintuitive" form of correlation.
IMPORTANT DISCLAIMER I put a lot of things in quotes because I wanted to convey the intuition behind the strangeness of entanglement by using analogies; these descriptions are not meant to be scientifically precise. In particular, any anthropomorphisations of electrons should be taken with a large grain of conceptual salt.
Rather than repeat some very good standard answers, I want to discuss this issue from the perspective as to why classical systems should be viewed as strange.
If we accept quantum mechanics as being fundamental, then in some sense we shouldn't really find things like entanglement to be strange at all. As pointed out by the answer given by joshphysics, as well as the answer given by Lubos Motl in the similar question, entanglement is really just correlation. The strangeness enters because we are accustomed to the the idea of classical locality and separability of systems.
Locality is best understood as the concept prohibiting action-at-a-distance, and is closely tied to Newton's Third Law of Motion. Newton's third law is the statement,
Every action has an equal and opposite reaction
which basically tells us that forces on an object are the result of the interaction by another object. Action-at-a-distance is a situation where two objects separated in space share perfect correlation in their motion, implying that one object is directly responsible for the other objects activities. In Newtonian mechanics, there is no limit on velocity, so action-at-a-distance, while seemingly unbelievable, is not prohibited.
This situation changed when it was realized that there is an ultimate speed limit to how fast two objects can communicate, or rather influence each other via the third law. This is the speed of light, as enshrined in the theories of special relativity and general relativity. This ultimate speed limit on the transfer of real information between two spatially separated regions is where our "classical intuition" fails (which is not a statement about human intuition, it is a statement about an apparent contradiction that arises in the logical statements one can make in the context of a particular theory).
So really the question isn't so much,
"Why is quantum mechanics weird?"
it's
"Why does our classical intuition fail?"
Much of this failure in our intuition is related to the separability of states which is an inherent feature of classical mechanics.
Separability of states is possible when one is able to describe composite states as direct products of subsystem state vectors.
To explain this a little better, there is a postulate of quantum mechanics that states
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems
This is written mathematically as $$\mathcal{H}_{AB} = \mathcal{H}_{A} \otimes \mathcal{H}_{B}$$ This can be imagined as just an abstract infinite dimensional space (just a really big space). The direct product $\otimes$ tells us to take every component of the each space times every component of the other space (e.g. if I can provide a basis for one space as $x$,$y$,$z$ and the basis for the second space as $a$,$b$,$c$; the direct product space would be $xa$,$xb$,$xc$,$ya$,$yb$,$yc$,$za$,$zb$,$zc$ )
As implied above, the component subspace can be given a basis that spans the space (span = provide a complete coordinate system that can describe every point):
$$\mathcal{H}_{A} \rightarrow \{ |a_i \rangle \}$$ and $$\mathcal{H}_{B} \rightarrow \{ |b_j \rangle \}$$
with our basis chosen, the pure state of the composite system can be defined as:
$$|\psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j \rangle $$
As discussed in the wikipedia article, if the state $$|\psi\rangle \in \mathcal{H}_{A} \otimes \mathcal{H}_{B}$$ can be written as $$|\psi_A\rangle \otimes |\psi_B\rangle$$ and $$|\psi_i\rangle$$ is a pure subsystem (e.g. also has an independent Hilbert space), then the system is described as separable. If it is not separable, it is entangled, and therefore:
$$|\psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j \rangle \neq |\psi_A\rangle \otimes |\psi_B\rangle$$
(Update Example borrowed from Marcini and Severini: Let $|\psi_{A1}\rangle$, $|\psi_{A\perp}\rangle$ be orthogonal states in $\mathcal{H_A}$, and $|\psi_{B1}\rangle$, $|\psi_{B\perp}\rangle$ be orthogonal states in $\mathcal{H_B}$. Then $$|\psi_{A1}\rangle \otimes |\psi_{B1}\rangle \in \mathcal{H_A} \otimes \mathcal{H_B}$$ as well as $$a|\psi_{A1}\rangle \otimes |\psi_{B1}\rangle + b|\psi_{A\perp}\rangle \otimes |\psi_{B\perp}\rangle \in \mathcal{H_A} \otimes \mathcal{H_B}$$ with $a$,$b$ $\in \mathbb{C}$. The first can be factorized into states of the subsystems, the second cannot. The existence of this second state would result in the above inequality.)
In our classical intuition, systems are separable, and it is only through some direct classical mechanical coupling that they show any correlation. So in the marble examples, there is some mechanical process that is involved in mixing marbles together. The marbles are still separable systems, and the correlation of one person finding a white marble, and one finding a black marble is still rooted in classical statistical mechanics, simply by the fact that the marbles have a definite color associated with them before they are put in the box. This means that the state of color for either marble is known and is not correlated with the state of the other marble. It makes sense for one to talk about the marbles being in a black or white state in classical mechanics. This is not a typical state in quantum mechanics, and systems having a definite state prior to observation is the root cause for the failure of our classical intuition
We must understand that the full state space in the entangled system is much larger than the space of separable systems. There is a good analogy in understanding the different size of state spaces in the context of the Born Oppenheimer approximation (and Emilio Pisanty does a good job explaining the derivation in his answer to this SE question). The Born Oppenheimer approximation provides a justification for allowing for the separation of the nuclear and electronic states of a molecular system:
$$\Psi_{Total} = \psi_{electronic} \times \psi_{nuclear}$$
This is possible by showing that one can ignore "vibronic coupling" associated with transitions of particles which would be represented by off-diagonal terms in the complete Hamiltonian matrix.
Similarly in our "classical intuition" we can ignore many terms that describe the state of the system simply because their effects are too small to be considered. As systems become smaller, these effects are harder to ignore, and the notion of a quantum object being able to have a definite state (e.g. being a definitely black or white marble) prior to our observation is not possible. However, the correlation of the outcomes is not removable from the system, in this sense the correlation must be viewed as more fundamental then the definiteness of the state. This is a very different state of affairs than what we find in classical mechanics, where definiteness of state is viewed as more fundamental.
So hopefully this gives a little more clarity as to why we think quantum entanglement is a "big deal". It requires a fundamental change in our understanding and approach to physics.
Here is the answer that made me realise what the big deal is. The description below is basically an expanded version of this blog post, which I came across a long time ago.
Imagine we are going to play a game. It's a cooperative game, so we'll either both win or both lose. If we win, we get lots of money, but if we lose we both die, so we should do our best to win.
The game is played as followed: you will be taken on a spacecraft to Pluto, whereas I will stay here on Earth. When you arrive at Pluto, someone will flip a fair coin. Depending on its result, they will ask one of the following two questions:
- Do you like dogs?
- Do you like cats?
You will then have to answer "yes" or "no". At the same moment, someone on Earth will flip a different fair coin and ask me one of the same two questions based on its result.
The rules of the game are slightly strange. They are as follows: we win the game if we each give a different answer from the other, unless we're both asked about cats, in which case we have to give the same answer as each other in order to avoid losing.
Since we're several light-hours apart there's no way we can communicate with each other during the game, but we can spend as long as we like discussing strategies before we go, and each of us can take anything we want along with us to help us answer the questions.
Now, with a little bit of thought you should be able to convince yourself that in a classical world, the best we can do is to have a $75\%$ chance of winning the game. To do this, we just agree that no matter which question we're asked, you'll say "yes" and I'll say "no". If we do this, we'll win unless we both get asked about cats, and the probability of that happening is 1 in 4. It doesn't matter what we take with us - as long as it behaves according to the familiar rules of classical mechanics, it can't help us do any better than this simple strategy. In particular, it doesn't make any difference if we each take a hidden object with us, which we later measure in some way.
However, in a quantum world things are slightly different: we can win the game $85.3\%$ of the time. I'm not going to go into the details of exactly how we achieve this, but it involves creating an entangled pair of particles, of which you take one and I take the other. Depending on whether you're asked about cats or dogs, you make one of two different measurements on your particle, and I do something similar. It just works out according to the rules of quantum mechanics that if we follow this procedure correctly, we'll win this game with a probability of $\cos^2(\pi/8)$ , or $85.3\%$. Many experiments that are equivalent to this game have been performed (they're called Bell test experiments) and the game is indeed won $85\%$ of the time.
There are other games that can be constructed, which are slightly more complicated to explain, where using entanglement allows you to win $100\%$ of the time, even though in the classical world you can't avoid losing some of the time. A paper describing such a game (among other examples of such quantum games) can be found here.
This is why entanglement is a big deal. It allows us to make things be correlated in this sort of way slightly more than they can be correlated in the classical world. It allows us to do something that wouldn't possible if entanglement didn't exist.
As an aside, there's another reason why entanglement is a bit weird: in the cats and dogs game, why does entanglement only allow us to win $85\%$ of the time and not $100\%$? It turns out that you can invent universes with "alternative physics" in which this game can be won $100\%$ of the time, while still not letting information be transmitted faster than light, but in our universe, $85.3\%$ is the maximum possible score. The reason why entanglement should be limited in this way is an open question in the foundations of quantum mechanics.