Donnerstag, 29. Dezember 2011

Untangling Entanglement

What to Feynman was interference (see the previous post), to Erwin Schrödinger (he of the cat) was the phenomenon known as entanglement: the 'essence' of quantum mechanics. Entanglement is often portrayed as one of the most outlandish features of quantum mechanics: the seemingly preposterous notion that the outcome of a measurement conducted over here can instantaneously influence the outcome of a measurement carried out way over there.
Indeed, Albert Einstein himself was so taken aback by this consequence of quantum mechanics (a theory which, after all, he helped to create), that he derided it as 'spooky' action at a distance, and never fully accepted it in his lifetime.
However, viewing quantum mechanics as a simple generalization of probability theory, which we adopt in order to deal with complementary propositions that arise when not all possible properties of a system are simultaneously decidable, quantum entanglement may be unmasked as not really that strange after all, but in fact a natural consequence of the limited information content of quantum systems. In brief, quantum entanglement does not qualitatively differ from classical correlation; however, the amount of information carried by the correlation exceeds the bounds imposed by classical probability theory.

Entanglement is deeply related to another, allegedly uniquely quantum property, known as superposition. Simply put, superposition is the capacity of a quantum system to exist in an arbitrary combination of states, rather than, like a classical system, in only one definite state at any given time. Superposition, in fact, follows very directly from the picture of quantum-mechanics-as-probability-theory, and it's there that we start today's foray into quantum weirdness.
Afterwards, we're going to think about correlations -- the question of when and why knowledge about one thing or event carries with itself knowledge about another thing or event. These can be explained fully within classical probability theory, which has the advantage of allowing intuitive examples; as already hinted at, the quantum case will not introduce any great conceptual revolutions here. Together with the concept of superposition, this will allow us to arrive at a simple, yet powerful picture of quantum entanglement, which essentially will amount to realizing that in the quantum case, unlike the classical one, all the information in a system may be contained entirely in the system's internal correlations.

Superposition: Indecision in the Quantum World
As usual, we will start by focusing our attention on the simplest system imaginable: a bit, which can be in either of two classical states. If the precise state of the bit is unknown, we can consider it to be in a state of 'classical superposition': a measurement of the bit might yield either state with some probability. However, note that the bit is, 'underneath' our ignorance, all the time really in a definite state, which we just don't happen to know! This is one crucial difference to quantum theory, where, as we have already seen, probabilities are irreducible (to briefly recap, this is due to the fact that any system with a finite information content, once one has exhausted the number of simultaneously decidable propositions, i.e. once one has asked it as many questions as it can answer at any one time, must answer perfectly randomly to each further question, as anything else would amount to extracting extra information -- which, however, just isn't there).
The other main difference is related to the fact that in the classical theory, probabilities are real, positive numbers, obeying the constraint that their sum must equal one -- i.e. if the bit is in the state 0 with a probability of 70%, it follows that it must be in the state 1 with a probability of 30% -- while the 'probabilities', i.e. the amplitudes, of the quantum theory are complex numbers, whose squares sum to one.
This means that a quantum bit has a state space that is much greater than just the two possibilities the classical theory offers: the set of all pairs of complex numbers α and β such that |α|2 + |β|2 = 1, which can be represented as a three dimensional unit sphere. (If you recall, in the last post, I introduced complex numbers as rotations, so if one complex number is a rotation around one axis, two complex numbers suffice to characterize rotations around two axes, and any given point on the surface of a three dimensional sphere is related to any other by two rotations -- around latitude and longitude, for example.)
This representation of the state space of a qubit is known as the Bloch sphere, and you'll recognize it from the top of this and prior posts:

Fig. 1: The Bloch sphere. (Image credit: wikipedia.)

Perhaps this is easier understood by considering first the state space of a 'real' bit, whose 'probabilities' are irreducible, can adopt arbitrary real values between -1 and 1 (negative numbers to allow for interference, see the previous post), and obey the constraint a2 + b2 = 1. This is nothing but a unit circle:

Fig. 2: State space of a 'real bit'.

Complex numbers, despite in a sense being equal to two real numbers each, only add one more degree of freedom to the state space, making it into a two-dimensional surface, because, while each complex number comes with its own phase, only relative phases are physically relevant, eliminating one degree of freedom.
The quantum probability theory thus leads to a much richer description than the classical one does. While the classical bit can only ever be in either of the states denoted |0⟩ and |1⟩, the qubit can be in arbitrary combinations with complex coefficients α and β, so that the general qubit state has to be written as |ψ⟩ = α|0⟩ + β|1⟩. Again, while a similar notation in the classical case only denotes uncertainty about the true state, in the quantum case, because of the irreducibility of quantum probabilities, there is no more fundamental 'true' state.
To drive this point home, consider that every qubit can only ever yield one classical bit of information upon measurement; that's all there is. Now say we have 'exhausted' this bit by a measurement along the x-axis in the picture, and have obtained the state |0⟩. If we now make another measurement along either of the other axes, there is no information 'left' in the qubit to determine the measurement outcome -- it must be perfectly random, i.e. both possible measurement outcomes must be equally likely. With respect to the other two directions, the qubit thus must be in an equal superposition of both possible outcomes, i.e. it must be the case that
|α|2 = |β|2 = 0.5: the superposition is 'real' in this sense, as opposed to the classical case, where it can only ever be apparent, the bit having a 'real' classical state underneath, that just happens to be unknown.
One thing is common to both the classical and the quantum case: whenever we perform a measurement, we will only find a definite state, both for the classical and the quantum bit. It is as if the quantum state suddenly forgets about all its beautiful extra structure in order to give us something our classical brains can more easily deal with. This appearance has led to the notion of 'wave-function collapse', and its occurrence (or the appearance thereof) is known as the measurement problem, which we will not yet tackle, but file under 'uncomfortable and unresolved' for the time being.
To sum up, superposition is thus a feature of the quantum probability theory, in which the system does not contain enough information to decide which of a number of possible states it is in, and thus, must be considered to not be definitely in either; only upon measurement does this superposition 'collapse', and a definite state emerge (at the expense of indefiniteness elsewhere). This straightforwardly generalizes from qubits to more complicated systems that have more states available to them.

Correlation: Information-Jelly Smeared on Spacetime-Bread
Two events A and B are correlated if, knowing that A occurred, you assign to the occurrence of B a different probability than if A hadn't occurred. So if I have two balls, one red, and one green, and two boxes, and put either ball into a box, shuffle them, and hand you one of the boxes, if you open your box, and find the red ball, you will assign to the possibility that I find the green ball once I open my box a probability of 100% -- the two events are perfectly correlated.
This already allows us to head off one frequent misconception about quantum entanglement, which is that it somehow may be used to transmit information faster than light. But ultimately, entanglement is just a correlation like the one above -- so while you instantly know what will happen when I open the box, even if you are so far away from me that a signal, travelling at the speed of light, never could transfer this information to you in time, it is nevertheless clear that this scheme can't be used to transfer any information. In order to do so, you would have to be able to determine in advance what ball you will find in your box, and thus, what ball I will find -- but, barring supernatural abilities, the possession of which probably would mean that you wouldn't have to rely on such a clumsy scheme to communicate instantaneously across great distances anyway, this is of course impossible.
But let's return to the issue of correlations. The example I have presented is one of perfect correlation, but less than perfect correlations are possible, as well. For instance, consider a case in which there are three balls, two of which are green, one of which is red. I randomly select two, and put them in one box each. The probability that in my box, there is a green ball, is about 67%; the probability of my box containing a red ball is 33%. Now, if you open your box, and find a red ball, you know that my box must contain a green ball; but if you find a green ball, you know that my box contains a red or a green ball with equal likelihood. Thus, you finding a green ball makes me finding a red ball more likely -- the probability jumps from 33% to 50 %. The two events are correlated, but it's not the case that whenever you find a green ball, I find a red one; but if you find a green ball, I find a red one more often than if you don't find a green ball.
Probabilities change with the acquisition of new information. Thus, in order for the probability you assign to the content of my box to change, opening your box must have given you information about my box. But how can there be information about my box in your box? Your box contains one bit of information: the color of your ball is either green or red, and with finding out its color, that one bit is exhausted, and it hasn't told you anything about my box. In principle, any color of your ball is compatible with any color of my ball; there's no fundamental law of nature that prohibits arrangements of two red, or two green balls.
But, in the case of perfect correlation, there is an extra bit of information in a proposition that does not pertain to either of the two boxes, but to the system of both boxes as a whole: the color of my ball is the opposite of your ball's color. This is a correlation, a bit of information not contained in either box, and it together with the color of your ball determines the state of the system completely: 'my ball is red' and 'your ball is green' is equivalent to 'my ball is the opposite color of your ball' and 'your ball is green', even though neither proposition pertains to the color of my ball alone. In the first case, the information is completely local; in the second case, the information 'in your box' is local, as well, but one bit of information is 'shared' between both your ball and mine, correlating their color; this bit thus does not encode a property of either box, but expresses a relation between them.
This does not surprise us: we have already learned that information is a relational entity. A sea of indistinguishable red balls can't be used to encode any information; only if one ball differs from the other, if a distinction is introduced (say, one ball is painted green), can the system be used to represent information, and then as many bits as there are distinctions. Properties that are the same across the 'sea' are invisible in a sense: there is no observable difference between the indistinguishable red balls, and a similar sea of red sports cars -- though the latter have more 'structure', they nevertheless don't contain more information. The extra structure forms an unobservable background, until, through the introduction of a distinction, a property is made observable. An object thus only has properties with respect to other objects that lack them; thus, its information does not inhere in itself, but in the relations between it and its environment. From this point of view, as we will see, quantum entanglement is natural; it is in fact its absence in the classical theory that will need explanation!
One question that still needs to be addressed is: when are two systems correlated? In the thought experiment, the correlation arises through the existence of two balls of different color, and through me putting each in one of the boxes. That is, the balls' colors have a common origin. In general, two systems, in order to become correlated, must have interacted in some way -- and conversely, generally, when two systems have interacted with one another, they will be correlated. This only makes sense: if the systems had never interacted, neither system would have knowledge of the other; thus, it can't be possible to extract information about one system through observations made on the other, since it simply does not contain such information. But correlations between the systems would allow this -- thus, there can't be any correlations.
A word of caution here: even though one ball's color allows us to infer the other's, it's strictly speaking wrong to say that because your ball is, say, green, mine is red -- both have their respective colors because I put them into the boxes, and arranged things that way. This is the essence of the old (but true) adage that 'correlation does not equal causation'. Just because the occurrence of event A means that one should expect event B to occur with greater likelihood, does not mean that A causes B in any way; there could, for instance, be a third event, C, which is the cause for both A and B -- in the thought experiment, that would be me putting the balls into the boxes.
A classic example for such a faulty inference was the conclusion that the fields emitted by power lines cause poor health: it was observed that, in the neighborhood of power lines, the general health was lower than the population average. Thus, 'living near power lines' and 'having poor health' are correlated. However, the reason is simply that the cost of living near power lines is lower -- power lines are ugly things, and thus, their presence devalues property. So people living near power lines tend to be somewhat poorer, simply because they can more easily afford living there. But less wealth means less health care, which means poorer health on average. Thus, 'lack of money' is correlated with both 'living near power lines' and 'having poor health'; and in this case, the correlation is actually a causative one.

Entanglement: Show Us Your Bits
We have seen that the system made from your box and mine, each with their respective balls in them, is a system that contains two bits of (relevant) information; these two bits determine the color of each ball. They can be 'distributed' in two different ways: they can either be both local, if the system is described by the two propositions 'your ball is green' and 'my ball is red', or one can be non-local, distributed between both boxes in the form of a correlation, corresponding to propositions such as 'your ball is green' and 'my ball is the opposite color of yours'. Clearly, both descriptions are completely equivalent: they uniquely determine the color of each ball.
But, there seems to be a third option looming, which is anticipated by the relational nature of information: can both bits be distributed non-locally?
Classically, this is not possible. We will discuss why in more detail later, but for the moment, think of it in the following way: a classical system contains, for all practical purposes, infinitely much information. So if we tried to 'remove' the one local bit that determines the truth value of the proposition 'your ball is green', we would find that this proposition is still true -- the bit is still there, because the classical ball contains an infinite reservoir. And the bit we've removed then doesn't help if we put it into the correlation, as the whole system is already exactly determined; it can't tell us anything we don't already know.
In quantum mechanics, however, the amount of information in any given system is finite, and a system of two 'quantum balls' ('quballs') would actually only contain two bits of information. So, what would happen if we removed the one remaining local bit in this case?
Well, as we have discussed above, this means that the state of your quball would then no longer be determined -- there would be no information 'left' to determine the truth value of the proposition 'your quball is green'. It would thus enter a superposed state, red and green with equal probability. But if your quball is in a superposed state, and my quball's color is determined by the proposition 'my quball is the opposite color of your quball', then this must mean that my quball is in a superposition, as well.
And this is then the origin of the basic phenomenology of entanglement: once you open your box, you will observe a ball of a definite color, as the superposition 'collapses'; instantaneously, you know the color of my ball, as well, thanks to the correlation existing between both. Given the relational nature of information, together with the realization that there is only a finite amount of information in any given quantum system, there is nothing mysterious about this; the correlation between both balls, in particular, is not qualitatively different from the correlation in the classical case.
However, there is an added snag in the quantum case, which stems from the fact that the qubit has such a large state space. What I've called 'quballs' aren't really qubits -- they're a sort of hybrid quantum/classical chimera, which has the characteristically finite information, but otherwise has been treated using ordinary probability theory, essentially. Concretely, quballs were assumed to only have one property with regard to which they may differ: their color. Hence, the bit we 'pulled out' of the local description of your quball essentially was left dangling, and did not contribute to the phenomenology anymore; the situation was completely described by the correlation and the random color you observe once you open the box.
However, essentially because of the complex nature of quantum probability amplitudes, things are different if you consider proper qubits. The value of a qubit can be measured along three orthogonal axes, the x-, y-, and z-directions in the Bloch sphere picture above. Nevertheless, the qubit only contains one bit of information; as you recall, this means that if, say, its value along the z-axis (for historical reasons, one often uses the word 'spin' in this context, because electrons having a certain spin were among the first examples of two-level quantum systems, i.e. qubits; but all such systems are isomorphic, i.e. have the same description, so one can talk 'as if' one were considering electron spins for concreteness) is absolutely determined, its spin along the x- and y-axes must be completely undetermined, i.e. it is in a superposed state with respect to these directions, these particular properties.
Thus, if, in the quantum case, only one bit were contained in the correlations, the behavior of the second qubit would still be insufficiently determined -- only one direction of spin would be correlated. So if you 'opened your box', i.e. did a measurement in order to determine the value of the spin of your qubit along, say, the x-axis, it may be the case that my qubit's spin is perfectly correlated, while, if you chose to measure along the y-axis, my qubit's spin along that axis might not be correlated with yours at all. But luckily, we still have that second bit left over, which is sufficient to fix this problem: both bits now encode the truth values of the propositions 'my qubit's spin along the x-axis is opposite to your qubit's spin along the x-axis' and 'my qubit's spin along the y-axis is opposite to your qubit's spin along the y-axis'. Note also that the third direction, the spin along the z-axis, is not independent of the spins along the x- and y-axes; this can be understood by realizing that, in the case of a single qubit, the indefiniteness of the spin along these axes implies the definiteness of the spin along the z-axis -- the bit of information associated with the qubit has to be somewhere. In short, the two bits of correlations between the two qubits suffice to fix my measurement outcome, if the outcome of your measurement is known. This is discussed in more detail and rigor in the paper The Essence of Entanglement by Caslav Brukner, Marek Zukowski, and Anton Zeilinger, which was the major inspiration for the point of view I have so far presented.
So we see that the difference between the quantum and classical case is merely that in the quantum case, all the information about the system can be in the correlations between the parts of the system, while in the classical case, every single part of the system contains at least the information necessary to define its own state.
It's perhaps useful to recapitulate the possibilities of correlation between two systems. To this end, let's look at some pictures. This is the case of two uncorrelated one-bit systems:

Fig. 3: Uncorrelated systems.

The dots stand for the balls, quballs, or qubits, and the lines signify correlations. As you can see, in this case, every system is correlated only with itself; thus, every system only encodes the truth value of one proposition, i.e. one bit of information. For instance, if the left one stands for 'your ball', the line might be the proposition 'your ball is green', and if the right one is 'my ball', the line might signify 'my ball is red'.
Now, we introduce a correlation between the two parts.

Fig. 4: Classical correlation.

The line connecting both parts now encodes a proposition that does not refer to either part alone, but to the system as a whole; thus, if again the left dot is your ball, the line starting and ending on it signifies 'your ball is green', while the line connecting both dots means 'my ball is the opposite color from your ball'. Note that the color of my ball on its own is now not completely determined; the color of your ball is a necessary input in order to fix it uniquely. This is as far as classical correlations go.

Fig. 5: Entanglement.

Going beyond this, we get a picture of quantum entanglement: each dot now represents a qubit, and the entire information about the system is contained in the correlations. Indeed, as we will see, this can be thought of as a characteristic of quantum systems. To describe this, David Mermin coined the phrase 'correlation without correlata', and based his 'Ithaca interpretation' of quantum mechanics on this point of view.
So again, we see that the only difference between the quantum and the classical case is the amount of correlation, not the kind; and because all (or most) of the information about the system is contained in the correlations, there is no information left to determine uniquely the state of each of the system's parts -- they must be in superposition. As Schrödinger put it in his essay The Present Situation in Quantum Mechanics:
"Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all."

Entanglement and Entropy
Entropy, as we have learned, can be thought of as a measure for the information one lacks about a system. This suggests an immediate connection to entanglement: if there is information about the system contained in the correlations, then only having access to a part of the system may not allow you to access the full information about even that part -- thus, if you only have access to one of the dots in the pictures above, you may not have full information about even the one dot (cf. Schrödinger's quote above), meaning that it must be in a state of nonzero entropy (remember how the color of my ball above was only described by 'the color of my ball is the opposite of the color of yours'; without access to your ball, this obviously fails to uniquely specify the color of my ball).
However, the total system, composed of both dots and the correlations between them, may be in a completely known state, and thus, have zero entropy! Indeed, this is a characteristic of von Neumann entropy, which is the direct generalization of the concept of entropy in classical probability theory to quantum probability theory.
Due to quantum entanglement between two subsystems, blocking off access to one of them, even if they are spatially distant, may lead to a description of the other in which it has a certain entropy -- the reason for that simply being that the information about the subsystem is not entirely contained in the subsystem, but also in the correlations between both systems. Entropy, in this sense, can be seen as a measure of entanglement; from this point of view, it is easy to see that, if I 'hide' the left dot, the right dot must have the same entropy as the left dot would have, if the right dot was hidden -- as both are entangled to the same degree.
In quantum mechanics, a state of zero entropy is called a pure state, because it describes a quantum system which is known to be in some state with certainty. Conversely, a state with nonzero entropy is called mixed, as it can be regarded as a statistical mixture of pure states -- i.e. there is a set of states, each of which a quantum system might be in with a certain (classical!) probability. It is a feature of quantum mechanics that pure states only ever evolve into pure states, as quantum evolution is reversible, i.e. every process is possible to occur in a time-reversed version. This is the case because quantum evolution conserves information: if the state at a given time t contains a certain amount of information, at a later time t' it must still contain the same amount; but this means that knowing the state at time t is equivalent to knowing the state at time t', and vice versa.
More generally, quantum evolution always leaves the total amount of entropy constant. But in our everyday world, entropy rises constantly! Indeed, I have previously argued that this rise of entropy is one of the most fundamental laws of the universe, if not the most fundamental one. How can this apparent incompatibility be reconciled?
This is, in fact, the already mentioned measurement problem in disguise. The classic answer is that, during the measurement process, the wave-function collapses, which is a process that does not conserve information, and hence, leads to rising entropy and is not time-reversible. The reason is simply that before the collapse, there are many different states that may collapse to one and the same observed state; thus, knowing the observed state, it is not possible to reconstruct the state pre-collapse. We have lost information, and entropy has risen. The acceptance of this as a genuine, random process (which state to collapse into is chosen at random, which introduces the oft-quoted indeterminism into quantum mechanics) is essentially what the so-called Copenhagen interpretation of quantum mechanics is all about.
But it is possible, using our picture of entanglement as correlations, and of entropy as a measure of the correlatedness between two systems, to attack this issue from a different angle. For what the above discussion neglects is the observer: essentially, things are portrayed as if they could be viewed without interacting with the system, as if observers were non-corporeal spirits that had direct knowledge of physical reality. But this isn't so; in order to observe, we must interact, and the interaction between observers and physical systems is of exactly the same kind as the interaction between all other physical systems -- this is just the recognition that observers are physical, too.
Now, the observer in quantum mechanics is unfortunately often regarded with a certain mystical air, which leads to things like Deepak Chopra, What The Bleep Do We Know?!, and nonsense about how each observer chooses their own reality, and so on. In fact, observer effects are not magic -- they exist in a classical context just as well. If you want to know the voltage in a circuit, you have no choice but to wire an appropriate instrument into the circuit; this act, however, will influence the actual voltage. Similarly, observer-dependent 'realities' are quite possible without invoking quantum magic: when you look at a rainbow, you will see a different rainbow than I do -- you will observe it in a slightly different place, for one. But this is just simple optics.
It is in this spirit that we will treat observer effects in quantum mechanics. Thus, the observer is a physical system like any other, that interacts with the system she observes. And, as we have learned, interaction will in general lead to correlation, and moreover, to entanglement. But since the observer observes a system, which has now become a subsystem of an entangled observer-observed system, she will miss the correlations that exist between her and the system -- she observes only part of the system, after all, which, as we now know, may not contain the complete information about itself. Thus, the observer, having become entangled with the system she observes, will observe it in a state carrying nonzero entropy, even if it was in a pure state before.
This is not a contradiction with the information-conserving (which physicists call 'unitary') evolution of quantum mechanics: the complete system of observer and observed is still in a pure state; before the act of observation, such was each of its subsystems, but afterwards, that's no longer the case, as both subsystems have become entangled. So we see that the apparent non-unitarity is merely an artifact of looking only at a part of the system: the complete system, consisting of observer and observed, evolves unitarily. Nevertheless, the observer sees entropy production -- without the need of postulating any wave-function collapse, or similar hacks.
Note, however, that this does not solve the measurement problem completely. For one, while it shows a phenomenology that is certainly similar to that of wave-function collapse, it could be an entirely coincidental similarity -- collapse might need to occur anyway, perhaps because the scheme proposed here is unable to account for all the entropy production. Also, there is no way given to determine which state the wave function collapses to. These and other problems will be considered in a future post about the interpretation of quantum mechanics.
This point of view of the apparently non-unitary dynamics of measurement is also proposed by Carlo Rovelli in his relational interpretation of quantum mechanics. He further justifies the 'incomplete' picture the observer has of the dynamics of her interaction with the measured system in part through an appeal to arguments by Marisa Dalla Chiara and Thomas Breuer, which appeals to the problems inherent in logical self-reference in order to demonstrate that complete quantum mechanical self-measurement is impossible. See also the Stanford Encyclopedia of Philosophy article on relational quantum mechanics, sects. 4.1-2.
We are now in a position to answer the question, left over from above, of why classical objects can't become entangled (or are at least very hard to entangle), while for quantum objects, it is natural. First, let's review the reason of why it is natural for quantum objects to entangle easily. In the quantum realm, we approach the limit of information and interactions -- a typical quantum system contains only a few bits of information, and can be regarded as isolated from other quantum systems. We can thus see the relational nature of information more easily: a quantum object has its properties with respect to some other quantum object. In the case of entanglement, one qubit may have its properties relatively to another. But what about the case of a single qubit? Well, the view I want to propose is that this qubit has its properties relatively to the observer.
This goes as follows: a qubit in an indeterminate state is measured, and becomes through this act correlated with the observer. The observer now 'knows' the qubit's state; that knowledge is expressed by the correlation between the observer and the qubit. Hence, I put 'know' in apostrophes: the observer need not be human, or indeed a conscious being, in order for this description to hold -- indeed, this observer might be some mechanical measuring apparatus, indicating his 'observation' via the position of some pointer, for instance.
This correlation has the consequence that observing the observer -- the measuring apparatus -- tells you something about the state of the qubit; observing this meta-observer -- say, seeing you scribble down the value of the measurement -- in tuen tells me something about the qubit, and so on. The information about the qubit's state is distributed through a web of correlations. It 'leaks into the environment', as it is sometimes phrased.
But this web has the effect of 'fixing' the state; in order to bring the qubit again into an indeterminate state, it has to be 'untangled', isolated from the environment. This is possible in the quantum case, but classical systems are big -- they contain lots of information, and thus are correlated with the environment in lots of different ways. It is practically hopeless to undo all these correlations and perfectly isolate a classical system from the environment. Hence, it is not the case that the classical system consisting of the balls in the boxes is only a two-bit system -- for all practical purposes, both balls contain infinitely much information, and thus, undoing one correlation can't put a classical system into a superposed state; there are still an untold amount of correlations fixing the property 'my ball is green'.
In summary, we can say that quantum mechanics is really all about correlations -- because it is all about information, and information one system has about another is a correlation between those systems. The higher the correlations (with the environment or any other observer), the more information the system carries, the more its behavior will tend towards classicality. The picture is simple, but its explanatory power is huge; indeed, it goes far beyond what we have touched upon up to now.

A Bit of Holography
The concept of the holographic principle was proposed in response to Jacob Bekenstein's discovery that, contrary to prior belief, black holes must be extremely high-entropy objects; in fact, in conjunction with work done by Stephen Hawking, he could show that the entropy of a black hole is the maximum entropy any object of the same size can have, and that this entropy is proportional to the area of the black hole's horizon (the 'point of no return', beyond which nothing, not even light, can escape the black hole's gravitational pull). Because of the connection between entropy and information, this is interpreted as the information of everything that falls into a black hole being 'stored' on its horizon -- since the horizon is a two dimensional surface, but the things that had the misfortune of crossing it are three dimensional, this means that three dimensional information is stored on a two dimensional medium, whence the phenomenon got the name 'holography'.
What does this have to do with entanglement? Well, perhaps nothing. But perhaps, rather a lot: consider that what a horizon does is effectively hiding things from view. However, as we have seen, hiding things -- subsystems of an entangled system, for instance -- generically leads to entropy production. One can thus imagine a region of space enclosed in a sphere, which has the property of hiding everything behind its radius, and then compute the entropy 'generated' by this hiding of information. And indeed, the calculation has been done in 1993 by Mark Srednicki, who found that the thus produced entropy actually does scale with the area of the sphere, as is the case with black holes!
Actually, a simple argument will make this heuristically plausible. Consider a universe, filled with some gas, the atoms of which are randomly entangled among each other. Hiding a spherical volume behind a horizon produces entropy. Generally, one might expect this entropy to scale with the volume of the sphere. But recall that the situation is symmetrical -- the gas inside the sphere is entangled with the rest of the gas as much as the gas outside is with the gas inside; thus, if I were 'inside' the sphere, and hid the outside gas, it must have the same entropy. However, the gas inside the sphere obviously has a different volume from the gas outside the sphere. In fact, the only quantity the two have in common is the area of their boundary.
A picture will make this even clearer.

Fig. 6: Entanglement entropy.

In this picture, for simplicity, only one 'object' is assumed to be inside the sphere, which however is heavily correlated with its surroundings. For some correlations, I have indicated the correlated systems, while others I have just left dangling, assuming they go off somewhere into the environment. Also, I have shown the 'punctures' where a correlation crosses the imaginary horizon. The number of punctures indicates the number of correlations, so to somebody who can't see beyond the horizon, it appears as an object correlated with the environment in a way that depends on the number of links puncturing the horizon's surface.
This does not straightforwardly imply holography: it is clear that, in the picture, I could vary the size of the horizon, without varying the number of correlations, at least to some degree. Only if there were some connection between correlations and area could the proportionality be guaranteed -- this is a hint we should keep in mind. However, for a black hole, the size of the horizon is fixed by the amount of mass inside, such that the only way to have it grow is to throw more mass into it -- and the mass, being approximately classical, will itself be highly correlated with the environment, and thus, entail an increase in correlations.
Some more interesting hints come from string theory, out of the so-called black hole/qubit correspondence, due to Mike Duff and collaborators: there seems to exist a connection, at least in the mathematics, between entangled systems of qubits and so-called 'extremal' black holes, where extremal means that they have the property that their mass is equal to their charge (in appropriate units) (and since we're talking string theory here, there isn't just the one electric charge we're familiar with, but there's also magnetic charge, and due to supersymmetry, there are actually multiple different ones of each kind). There is much that is interesting about this unexpected relation -- in particular, for a certain such black hole, its entropy is given by a measure of entanglement between three qubits, the so-called 3-tangle!
But still, the identification of entanglement entropy and Bekenstein-Hawking entropy is somewhat controversial; nevertheless, the idea should be enough to make one wonder.

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