Can entanglement be explained as a consequence of conservation laws?
"What's so strange about it? " This was essentially the point of the EPR argument in 1935, which says that it's silly to postulate all sorts of quantum weirdness when correlations could be explained simply by assuming that particles had definite properties. If I put each of a pair of socks in two boxes and move the boxes far apart, then it is no surprise that the socks match when the boxes are opened - it doesn't involve any weirdness or superluminal speeds.
The EPR argument was largely ignored, not so much it seems because people had any good counter-argument, but because people really, really like quantum weirdness. In the early 1960's John Bell took up the EPR argument again. What he found was that although the EPR argument worked for simple cases, like the socks, for slightly more complicated setups it ran into problems.
Two entangled particles are emitted by a central source, to be measured by Alice an Bob, who are some distance apart, and who each have a measuring device with a knob with three settings A, B and C and which gives a zero or a one as a result of the measurement.
When comparing there results for long runs of join measurements, Alice and Bob find that if the settings are the same then the results are always the same. If the settings differ by one position (so one has A and the other B, or one has B and the other C) then the results differ 1/7 of the time. A bit of thinking then leads to the result if one has setting A and the other setting C, then the results can differ at most 2/7 of the time.
The trouble is that this isn't what happens in reality. A quantum experiment can be set up such that the results differ half of the time. It is this result that requires quantum weirdness as an explanation
Maybe a good demonstration of why entanglement is so puzzling is the Mermin-Peres magic square game. There are three players, two of whom (A and B, say) have entangled states.
A and B are allowed to communicate and arrange their strategy in advance, but they cannot communicate once the game is in progress.
There is a $3 \times 3$ grid. You can ask A for any column of the grid (but just one), and you can ask B for any row of the grid (but just one). The rules are that A and B must assign 0s and 1s for their cells in the grid, they must agree on the cell where the row and column intersect, and the number of 1s in a column is always odd, but the number of 1s in a row is always even.
For example, if you ask for column 1 and row 2, they might return:
A B
1xx xxx
0xx 011
0xx xxx
If they have entangled states, A and B can always win.
"Simple," you say, "A and B have agreed on which cells have 0s and 1s in advance, and they just return those values."
But if that's their strategy, does the master grid have an even or an odd number of 1's?
For my money, the easiest-to-understand example of why entanglement is weird is Mermin's version of Bell's theorem. It's written out in a nice non-technical way in
N. David Mermin, "Quantum Mysteries for Anyone". The Journal of Philosophy, Vol. 78, No. 7. (1981), 397-408.
Here's the basic idea: We have an experimental setup consisting of three parts. Two of the parts are "detectors"; they each consist of a switch that can be put into one of three positions (A, B, or C) and a display that can either light up "YES" or "NO". The third part is a "transmitter", which is stationed between the two detectors. Whenever we press a button on the transmitter, it sends two particles out, one to each of the detectors. Each detector then lights up with "YES" or "NO".
If we run this experiment many times, varying the settings of the switches on both detectors, we find the following:
Each individual detector flashes "YES" 50% of the time, and "NO" 50% of the time. This is independent of the setting of the detector's switch (A, B, or C).
When the switches on the two detectors are both set to the same setting (A, B, C), the results of the two detectors always agree: they both flash "YES" or both flash "NO".
When the switches on the detectors are set to different settings, the results of the two detectors agree $\frac{1}{4}$ of the time (i.e., both "YES" or both "NO".) They disagree $\frac{3}{4}$ of the time.
How are we to explain these results? Result #2 could be explained pretty easily. The transmitter could simply be emitting two particles with a set of "instructions" that tell it what to do when it gets to each detector. For example, the particles could carry the instructions "make the detector flash 'NO' if the switch is in position A or B, and make it flash 'YES' if it's in position C." In other words, the particles have definite properties when they're emitted, and the detectors are simply measuring these properties.
But let's think about Result #3. If we buy that the particles both have definite "instructions" attached to them, then there are eight possible instruction sets they can be given:
A B C
-------
Y Y Y
N N N
Y Y N
Y N Y
N Y Y
Y N N
N Y N
N N Y
For example, the instructions I mentioned above (make the detector flash 'NO' if the switch is in position A or B, and make it flash 'YES' if it's in position C) correspond to the last row of the table.
For the first two sets of instructions, the detectors will always agree when their switches are set to different settings. For the other six sets of instructions, the detectors will agree 1/3 of the time, and disagree 2/3 of the time. The particles being sent off by the transmitter can't have the same instructions each time, because of Result #1; rather, it must be picking a different set of instructions each time. Still, no matter how it picks these "instructions", we would expect that the detectors would agree at least 1/3 of the time; any one instruction set either leads to 100% agreement or 33% agreement.
But Result #3 says that the detectors agree $\frac{1}{4}$ of the time, and $\frac{1}{4} < \frac{1}{3}$. So where has our argument gone wrong?
SIDEBAR: The devices can be constructed (in principle) as follows: the transmitter creates two electrons in an entangled state $|\uparrow \rangle |\downarrow \rangle - |\downarrow \rangle |\uparrow \rangle $, and sends one electron to each detector. The detectors consist of a Stern-Gerlach apparatus that can be rotated about the direction of travel of the electrons; the settings A, B, and C set this angle to either 0°, 120°, or 240°. One of the detectors flashes "YES" when an electron is deflected towards the north pole of its magnet, and "NO" when an electron is deflected towards the south pole. The other detector uses the opposite convention. Standard quantum mechanics predicts that the two devices will agree a fraction $\cos^2 \theta$ of the time, where $\theta$ is the angle between their magnets. In this case, $\cos^2\theta = \frac{1}{4}$ if the detectors are set to different angles, and $\cos^2 \theta = 1$ if they are set to the same angle.