Through my computer science “information is king” eyeglasses, there are really only two notions which thoroughly distinguish quantum theory from classical theories of how the world works: the nonlocal nature of quantum correlations as exemplified by Bell’s theorem and the much less well known contextual nature of quantum measurements as exemplified by the Bell-Kochen-Specker theorem. While the former is well known (and hence, to paraphrase Gell-Mann, what you’ve heard about it is mostly wrong), the later is less well known. Is that because it is a complicated idea? I don’t think so! Indeed I think you can learn what quantum contextuality means in less than ten minutes. Yep, it’s another edition of “explain quantum theory in ten minutes!”

Christmas where I grew up followed strange rules. Every Christmas morning, when I’d come down the stairs to see what Old Saint Nick had left for me, instead of finding my Christmas stocking hanging under the mantel, I would be confronted with a rectangular box (a square cuboid) left by Santa. This rectangular box was made of nine cubes pushed together. Here is a top down view of the rectangular box:

In addition to the cubes, on the ends of each of the rows of three cubes there was a button. Similarly on the ends of each column there was a button. These are the little knobby circles in the picture above.

Being confronted by such a contraption on Christmas morning, what would you do? Press one of the buttons, of course! And indeed this is what I have done every Christmas for an enumerable number year. The first time I woke up to this contraption, I pressed the button corresponding to the first row. And what happened? Well the top three boxes open, and out came colored elves (What else did you expect? Presents? You are such a tool of the religion of consumerism!):

After getting over the shock of there being small elves in the box, I had just enough time to note which box they were in and that two of them were blue and one of them was red. Then the elves disappeared along with the rectangular box.

That was what happened the first Christmas. The second Christmas, I decided that instead of pushing the button corresponding to the first row, I would push the button corresponding to the first column. And then bam, our popped another three elves before disappearing a few seconds later:

This time the elves were in the column of the button I had pushed. “Ah ha,” I remember thinking, “it must be that the button I push reveals the elves in the corresponding row or column! Strange contraption, Santa.” I also remember noting that just like before, the elf in the upper left hand box was colored blue.

Christmas number three. I decided that I would pick the row one button again, and see what happened:

Interesting! The third Christmas verified that indeed, when I push a button correspond to a row I get that row full of elves. And now something interesting happened. Instead of the upper left hand box containing a blue elf, now it contained a red elf. Okay, so apparently Santa wasn’t giving me a box with the same color elves under each box.

Now at about the fourth Christmas, I began to learn about science, and so I decided that I’d better keep track of the colors of all of the elves when I pushed different row and column buttons and see what color elves came out in which boxes. Here for example are the next sixteen Christmas box results:

Well by the nineteenth Christmas, it seems that I had collected a lot of data. But science isn’t just about collecting data, it is also about trying to understand this data.

So looking over the data from the nineteen Christmas boxes, I was struck by some regularities. The first was, that while a row or column might yield different blue or red elves, for all of the rows and for all of the columns except the last (farthest to the right) column, the number of elves which were blue was an even number. Go ahead, look at the data. Yep, every row and every column except the last (third) column have an even number of blue elves. And what about that last column? Well in that column it seems that there is always an odd number of blue elves.

I have now witnessed many more Christmas boxes. And indeed, at all of those Christmas times, the rules

- Every row revealed has an even number of blue elves
- The first and second column revealed an even number of blue elves
- The third column revealed an odd number of blue elves

held with one hundred percent accuracy. So now the question becomes, what could Santa be doing in order to produce this three by three rectangular box filled with elves?

Now Santa, we assume, doesn’t know what row or column we are going to choose to measure each Christmas. I mean he’s magical, but not *that* magical. So suppose that every Christmas he packs up nine red and blue elves into the nine boxes. He may choose these packings randomly, i.e. some years he may put a blue elf in the upper right corner and some years he may put a red elf in the upper right corner. That seems like a reasonable hypothesis. Then when, on Christmas day, we push a button the corresponding elves in those boxes are revealed. Could this possibly be an explanation for how the rectangular box works?

Well the answer, if you think about it, is no. Why? Well suppose that Santa packs the boxes with nine colored elves. If every row must contain an even number of blue elves, then the total number of blue elves in all nine boxes must be even (remember that if you have an even number and add an even number to it you get an even number.) Okay, fine, the total number of blue elves in the nine boxes is even. However, now think about the columns. In the first two columns Santa must put an even number of blue elves (again because even plus even equals even.) But now in that third column, Santa has to put an odd number of blue elves. But wait, combining this with the fact that the first two columns must have an even number of blue elves, this implies that all nine boxes must have an odd number of blue elves (because an even number plus and odd number gives and odd number.) So from obeying the row rules, we obtain that the total number of blue elves in the nine boxes must be an even number. But from obeying the column rules, we obtain that the total number of blue elves in the nine boxes must be an odd number. This is a contradiction!

So it must be that Santa, assuming he doesn’t know which row or column we will measure before hand, is not preparing the boxes such that behind every single box there is a red or blue elf. Whether a red or blue elf appears behind the row or column of boxes depends on which row or column we choose, and the boxes cannot be filled with preset values (even if they are randomly chosen) which are then revealed by our uncovering a row or column. This mysterious property, that the boxes do not have a preset value, but have a value which depends on which row or column we choose to measure, we will call *contextuality*.

What does all of this have to do with quantum theory?

Well suppose you have a quantum system. One thing you can do to this system is perform a measurement on this system. As I explained before, quantum theory predicts the probabilities of the different measurement outcomes for this system given a particular experiment we perform on this system. In particular we might talk about measurements that have two outcomes. I.e. that when you perform the measurement the outcome is either a “turkey” or a “duck”. Or for this discussion the outcome is either “red elf” or “blue elf.” So now you can guess what we are going to do! Suppose that every box corresponds to a particular experiment we can perform which has one of two outcomes “red elf” or “blue elf.” Now another property of quantum theory is that you can’t always make all measurements you want at the same time. Only certain sets of measurements are allowed “at the same time.” If you prefer not to think about measurements at the same time, then you can think about the measurements being performed in a sequence. Only certain sets of measurements are guaranteed to yield results which do not depend on the order in which you performed these measurements (for generic preparations of a quantum system), and these are sets of “simultaneous” measurements.

So now you can begin to see how the elves are connected to quantum theory. There are a set of quantum measurements which have two outcomes “red elf” or “blue elf.” Each box above will correspond to one of these measurements. In particular, if you want to see how quantum theory people might right this for the particular case relevant to the Santa box, these boxes will be labeled as

For example, the upper left box corresponds to measuring something called X_{1}. Now the rules of quantum theory tell us that we can simultaneously (or in order without the order mattering) measuring any three of these measurements if these measurements are taken from a single row or a single column (other combinations are possible, but we will only need these six combinations.) Thus, for example, we can measure X_{1}, X_{2}, and X_{1}X_{2} at the same time (and obtain three colored elves as a consequence.)

Now back to the main point. Suppose that you believe that when you perform the measurement of a quantum system corresponding to say X_{1} there is some color of elf preset which you are revealing when you make this measurement. Indeed suppose that you believed this for all nine boxes: that the result of a measurement performed in a particular way was set before hand. But if I now tell you that the results of performing measurements on a quantum system yield results exactly like those of the Christmas box (that is they follow the rules listed above for Christmas boxes), then, as we have reasoned above, it *cannot* be that quantum measurements have preset values for all nine boxes. In other words which set of three measurements (corresponding to a row or column in the Christmas box picture) makes a difference in the outcome you get for measurements on quantum systems. We thus say that quantum theory is a *contextual* theory.

So what should we make of all this? Well, the first thing we should make of this is that quantum measurements should not be thought of as revealed data which is just hidden away from us, but whose outcome is independent of the context of this measurement. Now should we be surprised by this? Well, John Bell thought otherwise (and I tend to agree with him.) Suppose you think about a system and a device measuring it as people who are having a conversation. Now suppose the person representing the measuring device asks the first person a series of questions. Would it be surprising that which set of questions you ask would result in different answers? Well no: certainly a person can respond in any way they’d like, given the set of questions asked. Where you get into trouble is if you believed that all the questions you could ask have answers which don’t depend on this context. If you think about quantum systems as flip-flopping politicians, where their answers aren’t just set in stone, but change depending on what other questions you ask them, then you’ve got a decent idea of contextuality in quantum theory.

Okay, has it been ten minutes? Elves, Santa, boxes, and quantum measurements. That’s the Bell-Kochen-Specker theorem: quantum measurements are contextual. That wasn’t so painful was it? Okay, well talking about colored elves blinking into and out of existence at Christmas time may have been a bit much.

**A postscript for those who want to read more:** the example described above is taken from David Mermin’s article, “Hidden variables and the two theorems of John Bell” Rev. Mod. Phys. 65, 803 – 815 (1993). Unfortunately this may be unaccessible to those outside of academia. A nice summary with links to more information is The Kochen-Specker Theorem at the Standford Encyclopedia of Philosophy.

**As a postscript for those with more experience in quantum theory:** If you read the comments to the last explain quantum theory post you will see an argument about whether by only allowing “Turkey” and “Duck” in the measurement outcomes was too limiting. As other commentors pointed out, this problem depends first upon the physical system you are measuring. But I also tried to point out that the two level quantum system, the qubit is a special quantum system. In particular it is the system for which (projective) measurements have a non-contextual hidden variable theory. That is for qubits one can pretend that measurements have set outcomes independent of the context of the measurement (for qubit measurements this follows directly from the fact that a projective measurement along one direction can have only one other context: a measurement along a perpendicular direction.) So thinking about qubits is really really bad, from a foundational perspective: it lets you believe that the qubit is just an arrow (or more properly a ball attached by strings to its surroundings a la Dirac), where the value of whether the qubit lies along this arrow or against this arrow is a present value. But when you move to larger systems than a qubit this type of reasoning fails miserably. It thus seems to me that the particular case of the qubit is a bad example and in particular introducing quantum theory with qubits should avoid leading to reasoning that yields non-contextual hidden variable theories.