Before a we suggest an experimental way to investigate "reality" it is necessary to have an objective (i.e. independently testable) definition of that term. We might call this "objective reality", except that science is limited to the objective, so here it is simply called "reality". The motivation for "reality" is that there is some substance or substances that exist apart from the processes of our observations, and these substances give rise to the phenomena we observe. Furthermore, note that our idea of such substances is based on the reliability of the observations we make. An independently testable notion of these substances can only be based on the results of measurement of physical attributes.
So the question of "reality" comes down to the question of whether physical attributes have defined values apart from our measurements of those attributes. For example, if your notion of objective reality involves a substance called "matter", then your belief in this substance will be based on observable physical attributes which are held to arise from matter. So, if matter exists independent of observation, then these attributes should have definite values even when matter is not being observed.
So, we may define "reality" (for the purposes of scientific inquiry) as the extent to which physical properties have defined values apart from observation.
One may object that we can never know whether physical properties have defined values apart from observation. It turns out, however, that we might be able to find some observable consequence of there being definite values before a measurement takes place—this is a tricky proposition, which is why it has only been done recently, and only with additional assumptions.
Locality
"Locality" is the idea that effects can only arise from local (i.e. nearby) causes. For example, you might react to something you see, but that is only because light has traveled from the remote object to your eye. The effect of your response arises from the local cause of this nearby light. Under "locality" if there were no mechanism for the remote information to be available to you, then you wouldn't be able to respond to it.
A premise of the (very successful) theories of Relativity is that there is a limit to how fast information can travel. The limit is exactly 299,792,458 meters per second, the length of the meter is actually defined in terms of the second so that this value is exact. It turns out that light, in a vacuum, travels at this speed, so the speed is usually called the speed of light, and given the symbol "c". When this is taken together with the principle of "locality", it implies that in order for a remote cause to have a local effect, there must have been sufficient time for the information to have traveled from the remote cause to the local effect.
The alternative to "locality" is sometimes called "non-locality", it is also known as "action at a distance".
Local realism
The combination of "locality" with "reality" is sometimes called "local realism".
The materialist view is that reality as we observe it is based on divisible substances which interact according to local causes. However, this view is inconsistent with Quantum Mechanics (in ways I'll discuss below). This led Einstein to be very skeptical that Quantum Mechanics would hold up to certain kinds of experiments.
"I cannot seriously believe in quantum theory because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance." - Albert Einstein
Uncertainty.
The "problem" really begins with Quantum Mechanics. After some refinement, Quantum Mechanics started to conform extremely well with some otherwise very puzzling experimental results. So it became hard to reject the theory even if it leads to very bizarre conclusions—the bizarre conclusions, where they could be tested, kept getting confirmed by the experimental evidence, which added to the list of things that Quantum Mechanics could predict but alternative theories could not. So part of the problem is that Quantum Mechanics is a very successful theory. The other part of the problem is that Quantum Mechanics leads to very bizarre conclusions.
For one thing (seemingly not so bizarre, at first), most of the predictions of Quantum Mechanics are probabilistic. There are properties whose values cannot be fully predicted by the theory—the theory just gives you a probability distribution so you know which values are likely and which are unlikely but there is no single, certain answer.
Let me get more specific. We might have something like an electron, but the theory may not be able to tell us exactly where it is, or how fast it is moving (it usually cannot answer either of these questions precisely and it can never answer both of those questions precisely at the same time). So we say there is uncertainty in its position (and also in its momentum).
When you measure these properties, you always get a definite answer, but the theory cannot predict exactly what that answer will be. The problem is uncertainty, not any inability to do a precise measurement.
(an aside: Note that this is the same kind of uncertainty that has been popularized in the famous Heisenberg Uncertainty Principle—that principle is a theorem that relates pairs of uncertainties in things called conjugate properties. But uncertainty is found in many places in Quantum Mechanics, not just in the Heisenberg Uncertainty Principle.)
Hidden variables.
Initially, there was a lot of concern that quantum mechanics seemed to be such a good (i.e. highly predictive) theory, but so incomplete. The great question was why the exact values of these particular properties should be uncertain within the theory. The exact values were assumed to exist in reality, but for some reason the theory could not precisely predict the result of a measurement.
They were thus called "hidden variables", and there was a lot of interest in why they were hidden, how they were hidden, and if there might be any indirect way to predict them.
The definition of "hidden variables" is equivalent to the definition of "reality" we gave above.
Maybe there are no hidden variables.
This is going to seem weird, and the only reason I bring it up is that this turns out to be the correct view (in mainstream physics).
Actually, it was worse than I put it above. Quantum mechanics isn't simply unable to predict the values of certain properties, it actually turns out to be inconsistent with these properties having precise values in reality, apart from a measurement. So either Quantum Mechanics is wrong, or there are no hidden variables. I'll get to why hidden variables are inconsistent with Quantum Mechanics below.
But in Quantum Mechanics, it has to be that a property (such as the position of the electron), under most circumstances, simply does not have an exact value until it is measured (even though the measurement can be very precise).
If you are having trouble picturing this, you are in good company. It is hard to picture it. It is even hard to know fully what it means for something like the position of the electron to not have an exact value. How to picture it is something that nobody has a really good answer to, as far as I know. There are ways to picture some of these things, which are helpful, but there is always something wrong with the picture.
Conservation laws.
Some properties in physics act as though they are substances in the sense that the total of those properties remains constant in any interaction. Such properties may be thought of as "quantities" and we say that those quantities are "conserved".
It isn't that there are necessarily any substances involved, it is more by analogy that the term arises. What it means for a quantity to be conserved is that you always end up with the same total that you started with.
For example, momentum is a conserved quantity (and not generally thought of intuitively as as substance). If you push on a freely movable object (no friction) with a given force for a given amount of time, you will give it a certain momentum in the direction of your push. To stop the object you will have to cancel that momentum by pushing in the other direction. Momentum is always conserved, even in the case with friction, but if there are other forces, such as frictional forces, the change in momentum is determined by the net force rather than just the force you applied with your push.
If you try to create momentum by pushing something in one direction, it will push back with an equal and opposite force, so all the momentum you create in the object will be matched by an equal and opposite momentum in the other direction (on you, and on the Earth if you are standing on the Earth). Usually you don't notice this other momentum because it takes a lot of momentum to produce a noticeable change in the velocity of the Earth. But when a car accelerates from a stop light, the car is pushed in one direction and the surface of the Earth is pushed in the other. Equal and opposite forces. Equal and opposite changes in momentum. The change in velocity of the car is much greater because it is much less massive.
In any case, the important thing here to keep in mind about the strict conservation laws, such as conservation of momentum, is that they are exact. Never is there the slightest observed violation.
So one of the very confusing things about uncertainty is that, even though the values of certain properties of individual particles is uncertain, the total of all of these values may be precisely known, if there is a conservation law involved.
Let me restate this in its full oddness, just to make sure it is clear how strange this is. If we have two electrons, their individual momentums (momenta, I guess) may be uncertain, and yet we know that if we do measure the momenta they will add to an exactly known value. Uncertainty in the parts, but not in the sum of the parts.
Measurement error versus fundamental uncertainty
Because of the conservation laws, there is a problem with considering the uncertainty in quantum mechanics to be caused by the process of making the measurement. That is, if the problem is that the measurement itself is disturbing the system, in a random way, then why does the total of these measurements on an ensemble of particles add up to a precisely predictable value?
In this kind of case, where the properties of multiple particles is individually uncertain, but correlated (usually by a conservation law), the particles are said to be "entangled".
Einstein, along with Poldosky and Rosen proposed an experiment, called the EPR Paradox, which was intended to show that Quantum Mechanics was wrong because the idea of maintaining conservation laws through entanglement was inconsistent with local realism. Unfortunately for Einstein, experiments showed that the conservation laws were upheld, seemingly in violation of local realism. This is an example of why Quantum Mechanics is such a strong theory. Here Einstein proposed a result that was predicted by Quantum Mechanics but which was "obviously" untenable. And experiment confirmed the surprising predictions of Quantum Mechanics.
So there was something of a search for a way to reconcile local realism with the observed results.
Spin (not the political kind).
Many of the experiments involving this kind of subject (that I know of) involve a property called spin. Imagine the way the earth spins. The name "spin" is kind of by analogy with that, except that there might not be any actual spinning going on in the property of that name. But there is a property of spinning objects called "angular momentum" (which I will discuss below), and the property called "spin" refers to an intrinsic angular momentum.
Angular momentum, is intuitively something like regular momentum but it involves spinning rather than moving in straight lines. If you start spinning a bicycle wheel you will have to cancel that by applying a twisting force (called a torque) in the other direction. Like regular momentum, angular momentum is also conserved (when you apply a torque to spin the bicycle tire, the tire also pushes on you with an equal and opposite torque).
Electrons have a small amount of intrinsic angular momentum, as though they were spinning. You might visualize this as though they were tiny, spinning balls. It isn't right to visualize them that way, but it might be helpful for now (there really isn't a good way to visualize them, but it is hard to think about them without some kind of visualization unless you are comfortable thinking of them as abstract manifestations of the equations that describe them).
One more thing you need to know about angular momentum. You know that it describes a property of something that is moving around an axis (think of a spinning ball or bicycle tire). So how do you describe the direction of motion of something moving around an axis? You might think you would just say "clockwise" or "counter-clockwise", but you also have to say which direction the axis is pointing. Since the axis is a line, we just consider the direction as being along that line, and we point along that line toward one end or the other, depending on whether the rotation is clockwise or counter-clockwise.
Again, it may help you visualize this by considering a spinning ball. If you spin the ball about a vertical axis so that when you look down on the ball, it is spinning counter-clockwise, then the angular momentum is considered to be "up". If, when you look down on the ball, it is spinning clockwise, the angular momentum is considered to be "down". This is called the "right hand rule", if you point your right thumb in the direction of the angular momentum, the spinning motion is in the direction that the relaxed fingers on that hand curl. There isn't a lot of "why" about this, it is mostly just a way of talking about it that physicists find convenient (it turns out you can do math on rotations conveniently if you adopt this convention). With this way of speaking about the direction of angular momentum, there is a direction that you can point in which will let people know which direction a spinning object is spinning (i.e. the direction of the axis of rotation and which way the object is spinning about that axis).
An odd thing about this is that every electron has the same magnitude of intrinsic angular momentum. The amount is the value of Planck's constant (usually written as h) divided by $4\pi$. Only the direction can be different. Note that the $\pi$ in that expression is just the ordinary $\pi$ that is the ratio between the circumference and diameter of a circle, about $3.14$.
Some people get overwhelmed when named things start getting thrown into the description, things like Planck's constant. Don't get overwhelmed. Think about the kilogram. It is defined by an actual block of something kept in a vault in France. So when you say that you are buying two kilograms of flour you are just buying two of that defined amount. Planck's constant is just a defined amount of angular momentum. It turns out that this particular defined amount is oddly significant, unlike the kilogram which was chosen somewhat arbitrarily. It is so significant that we like to honor the person who first described it. But it is really just a particular, defined amount of angular momentum, just in the same way that a kilogram is a particular, defined amount of mass.
The value of Planck's constant is a very small amount of angular momentum. I mean *tiny*. If you had a mass of 1 gram, and spun it around an axis at a distance of 1 centimeter, and at a rate of 1 revolution per second, it would have an angular momentum about 1 billion times 1 billion times 1 billion times greater than Planck's constant, (i.e. about 1,000,000,000,000,000,000,000,000,000 times Planck's constant.) I think I have that right. Certainly Planck's constant represents a small, but measurable, amount of angular momentum.
The fundamental unit of angular momentum is actually $\overtwopi{h}$, and electrons have half that spin, so they are called "spin $\overtwo{1}$" particles. The "$\overtwo{1}$" means half of $\overtwopi{h}$. Because, the way modern Physics is done, Plank's constant almost always has to be divided by $2\pi$, Physicists have a special symbol for that. They put a little bar through the "h" (almost as though they were crossing a lower-case "t"), and call it "h bar". I think I can do that—"$\hbar$"—if that doesn't look like an h with a bar through it, let me know.
Now here is the *really* weird part. You can measure the intrinsic angular momentum of the electron about any axis you like and you will always get the same magnitude, the only difference will be which way it points along that axis (e.g. up or down).
I am talking about this concretely, in terms of $\hbar$, just because I want you to be able to see that we are talking about concrete concepts. Electrons intrinsically have angular momentum, angular momentum is something you experience in ordinary life (e.g. the spinning tire), the amount of intrinsic angular momentum that electrons have is very small, but it is conceptually just some value that can be measured. It has been measured, and it always has this one particular very small value, only the direction varies.
This intrinsic angular momentum is the property of electrons known as "spin". Sometimes you might see it as a value in units of angular momentum, or sometimes it will just be "up" or "down" because the magnitude is known. Remember, that it applies to any axis, so it might really be "left" or "right" instead of "up" or "down". Sometimes people measure with respect to a particular direction (e.g. "up"), so that positive values are considered "up" and negative values are considered "down". Sometimes you will see it given the value $\onehalf$, but that means $\onehalf$ of $\hbar$.
Just to be sure you have the full oddness of this, suppose you measure the vertical spin—you get a value of $\halfhbar$ up or down. Then if you measure the horizontal spin, you will get a value of $\halfhbar$ left or right. (I warned you it was not really right to imagine it as a spinning ball.)
Other particles also have "spin". A photon is a particle of light. All photons have an equal magnitude of angular momentum, in this case it is $\hbar$ (not $\halfhbar$), and it always points either along their direction of motion or in the opposite direction. The article discusses photon spin, but I think electrons are easier to think about. So I am hoping that the discussion of electrons helped you visualize this, I thought it might have been harder if we started with the angular momentum of a little massless ball of light. As I said, the reality of these things is always hard to visualize, if you really think about them correctly.
(An aside. Every electron has the same magnitude of spin, and every photon has the same magnitude of spin, but this is not a general property of particles. There are some particles that have more choices, but there are always a limited number of choices.)
(Another aside: The conjugate properties referred to in the Heisenberg Uncertainty Principle are always related to each other in the way that the product of their units will have the units of angular momentum. There is a certain statistical measure of the uncertainty of prediction that is called the "root mean square deviation" or "RMS deviation". It is found by taking measurements, squaring the deviation of those measurements from the predicted mean (i.e. average) value, averaging the value of those squared deviations, and then taking the square root of the result. Yes, it is tedious work. The Heisenberg Uncertainty Principle states that, if you do this over a large enough number of measurements, the product of the RMS deviations found in each of the conjugate properties will not be less than $\halfhbar$. That is, it isn't really a statement about a limitation on measurement, as is widely believed, rather it is a statement about uncertainty. The measurements come into it as a way to measure the uncertainty.)
Successive measurements of spin.
Suppose you measure the spin of an electron in a vertical axis and happen to find it is "up", or $\plushalfhbar$. If you measure in a second direction you will find that the value is always $\plushalfhbar$ or $\minushalfhbar$ but usually not with equal probability. If that second direction is also vertical, you will always measure "up" (two successive measurements along the same axis will match). If it is nearly vertical then you will be very likely to measure $\plushalfhbar$, but occasionally you will measure $\minushalfhbar$. If the second direction is horizontal, then the second measurement will be uncorrelated with the first, you will be equally likely to measure $\plushalfhbar$ as $\minushalfhbar$. (This is a crucial point.)
Spin uncertainty and quantum entanglement.
If all particles of some kind, such as an electron, have the same magnitude of spin, then how can the value be uncertain? Because the direction might not be known.
For example, suppose a process creates two photons, moving in opposite directions, one to the left and the other to the right. Suppose it is known, because angular momentum is conserved, that the total angular momentum must be zero. In that case you might either measure the spin of both photons pointing along their individual direction of travel, or both pointing opposite to their direction of travel, but you wouldn't know which of these two situations you were going to get until you had measured the spin of at least one of the photons.
So now we can go back to the question of hidden variables, but in this case the hidden variable is the spin of an individual particle, before the spin of either entangled particle has been measured.
Bell's theorem
So far, we know that Quantum Mechanics requires that uncertainties be, well, uncertain, but common sense says that these things have definite values (hidden variables) and so we might suspect that there is some flaw in the theory. The last hope for the common sense view was laid to rest after experimental verification that followed a theorem by John Bell, which he published in 1964.
What Bell was able to show (when you put his theory together with the experimental results of others) was that if you looked at a large number of entangled pairs of particles, you could decide whether there were hidden variables on the basis of a statistical analysis. That is, you could decide whether a particle had its own definite value of spin, before it was measured.
Imagine a device that emits two entangled electrons at a time, one to the left and the other to the right. Suppose it is known (from the conservation laws) that their total spin is zero, along any axis. That is, if we measure the spin of the left-going and right-going electron along a similar axis (e.g. both vertical, or both horizontal), the sum of the spins will be zero.
As noted above, a measurement along one axis introduces randomness the measurement along a different axis. But we can use entanglement to effectively take a measurement of the spin along two independent directions, under the assumption of local realism. That is, we might measure the left-going electron in a vertical axis (having satisfied ourselves what this would imply for a vertical spin measurement of the right-going electron) and we could measure the horizontal spin of the right-going electron. If the result of the measurement depended on hidden variables, then these independent measurements let us know the result of the value of these variables, for any particular pair, in their spin in two independent directions.
Bell's Theorem is, however, based on an inequality involving three criteria, not just two. What Bell's inequality says (and this can sometimes be a useful fact to know in ordinary life) is that "the number of items that possess property A and not property B, plus the number of items that posses property B and not property C, must always be greater than or equal to the number of items that possess property A and not property C". An example might be that, in any group of people, considering left-handed vs right-handed, tall vs short, men vs women (that is A = right-handed, B = short, C = male), the number of tall right-handers plus the number of short women cannot be less than the number of right-handed women.
So this is just a mathematical fact. But the problem is that Bell's inequality involves three attributes, and we can only measure electron spin, even using entanglement, in two different orientations.
What Bell proposed was to choose randomly selected orientations of the spin detectors along three different axes and look at the correlations among large numbers of pairs to see if the inequality held in the aggregate.
So, if the spin orientations along the three axes were decided by hidden variables, when the electrons were first emitted, then in the aggregate Bell's inequality should hold. Note that this doesn't involve Quantum Mechanics at all, all it involves is the idea that the total spin is a conserved quantity and that measurements of spin along any axis always give $\plushalfhbar$ or $\minushalfhbar$. These are observed results, confirmed by experiment, independent of any theoretical basis.
It turned out, however, that Bell's Inequality was violated. Thus, either the electron spin is not determined when the electron is emitted or the measurement of the spin of one electron affects the hidden variables of the other electron (e.g. measuring the vertical spin of the right-moving electron changes the horizontal spin of the left-moving electron). You cannot have both reality (i.e. hidden variables) and locality.
Essentially, what it seemed to show is that if you measure the vertical spin of one of the electrons, it instantly randomizes the horizontal spin of the other electron (and vice-versa).
That is, Bell's theorem (together with the confirming experimental results of Aspect and others) showed that local, hidden variables are not only inconsistent with Quantum Mechanics, they are much more generally inconsistent with the experimental results.
Locality vs. "Spooky action at a distance."
Bell's theorem showed that, if you assume reality, then the information that arises from measuring one of the entangled particles is instantly available to a measurement of the spin of the other particle (so that the measurements can retain correlation and yet violate the inequality). This kind of "action at a distance", the kind which Einstein called "spooky", violates locality.
So, somehow the conservation laws are satisfied and yet the local values of these properties is not determined until a measurement is made. So there is some kind of action at a distance that happens.
The kind of interaction that happens in "spooky action at a distance" is limited. It doesn't allow people to send other information between the entangled pairs, it only sends this specific information that is involved in maintaining the correlation.
The new results.
People who have been trying to resurrect some kind of materialist world-view from the wreckage of the Aspect experiments and Bell's theorem have thus taken to considering non-local models, in which there is still a material objective reality, but it has some non-local hidden variables whose values somehow are available throughout the universe.
The new experiments (if I am interpreting the article correctly) rule out additional large classes of hidden variables (i.e. objective reality existing apart from measurement) even of the non-local kind.
Leggett's paper doesn't rule out all forms of nonlocal reality, but rather a large class of nonlocal reality, which he calls "crypto-nonlocal". The kind of locality he has relaxed is a possible dependence on the non-local experimental set-up. That is, he has shown the inconsistency is a matter of holding both to reality and to locality with respect to the measured outcomes alone. The non-locality with respect to apparatus does not manifest itself with a non-locality with respect to the results of measurements—it remains hidden.
In other words, the "action at a distance" implied by the new experiment must involve sending information about the results of the measurement and not merely sending information about the settings on the equipment.
Leggett's paper is, A.J. Leggett (2003) Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem Foundations of Physics, 33 (10), 1469-1493.