Student grades and quantum measurements

Reading Scott Aaronson’s blog, I came across this series of lecture notes, and at the start of the first lecure there was this nice introduction to his grade system. Scott is really a great mind. His grade system reflects a deep understanding of quantum mechanics. I know, I know, grade systems talk about students, and those are complex macroscopic physical systems which interact strongly with their environment, yada yada yada, and therefore cannot manifest quantum effects. They are perfectly understandable within classical mechanics [enter caveat here about “in principle” or “given enough time and computational power”].

So I’m not saying that this understanding that Scott has about this system of human relationships is directly based on his knowledge of quantum information theory. But there is a mapping, an analogy, a metaphor. The metaphor is this: measuring the grade of a student is in some ways like performing a measurement in quantum mechanics. As Scott puts it in those notes:

I should tell you that I hate grades, I hate GPA’s. I’ll tell you why. Some people think grades are a bad idea because they reduce a complicated human being to a number. That’s not what I think. I think, if you’re going to reduce a human being to a number, at least do it in a statistically sensible way! “This person scores one standard deviation above the mean on this task within this population.” An A-: what does that mean? It means the teacher decided to you give you an A-. Because maybe he said, well, this student got a B+ according to my arbitrary made-up algorithm that’s different from everyone else’s, but B+ is really on the borderline of A-, so I’ll make it an A-. And these are math PhD’s. And they don’t notice the problem of infinite regress. And of course, for a less likable student, the B+ gets just as arbitrarily downgraded to a B.

The reason why Scott feels this way about student grades is perfectly understandable: we cannot put all students in a single ranking. This property of “good studentship” is a contextual property of the students. It manifests itself within a certain environment and cannot be dissociated from it. One reason is the one pointed out by Scott: grade measures are arbitrarily created by each teacher. Another is the interaction between students themselves. Alice and Bob go to the same university, become friends, fall in love. They study for their exams together, they help and support each other throughout the semesters. Alice and Bob get straight A’s throughout university. In a parallel universe, Alice and Bob go to different universities, in which they are seen as geeky socially-inept outcasts. They isolate from society, go into drugs and perform terribly in school. [Or, more plausibly: Alice and Bob go to the same university, become friends, fall in love. Their relationship is based around sex, drugs and rock’n’roll. Being too busy with more important things, they perform terribly at school. In a parallel universe, Alice and Bob go to different schools, in which they are seen as geeky socially-inept outcasts. They isolate from society, go into drugs and get straight A’s.]

In any case, we take the message that the “quality” of a student is not entirely an inherent property of a person, but something that is mediated by society, by the environment. Sure, the genes have a lot to do with it as well, but that doesn’t make it any less contextual. So when we want to compare different students, we need to take that context into account. So as Scott says, grades are nothing but a measure of how far from the average a student performs on a certain task, on a certain population. It is meaningless to talk about these properties without a means of measuring them, and any means of measuring them will be necessarily relative to a task and a population. You need to put some scale there. But scales are arbitrary, they are conventions, they are not fixed and eternal. So there’s no absolute meaning in abstracting the grades as numbers out of the context of a grading system.

And the same goes on in quantum mechanics. A physical property like position or spin means nothing outside of the context of an actual measurement which is carried out to determine that property. Now why is it that this seems obvious when referring to grades and students, but not at all obvious when referring to properties of particles and fields, of “real stuff”? The reason is that the analogous experiment in quantum mechanics takes things to an extreme which we are not used to in our human relationships. I’ll explain by extending the metaphor a bit further.

Suppose a teacher pulled a student out of a collection of classes, each of which with its own arbitrary scale, and asked for his or her grade. Of course, after all the discussion above, the grade of each student will have a specific measure. Alice went to Univ. of Z. and Bob went to X. College. So Alice can only be measured with respect to Univ. of Z’s measure. You can say Alice’s grade is 8 Z. And you can say Bob’s grade is 9 X. But we have no direct way of comparing these two scales. We can talk about averages with respect to other measures. Perhaps we find that 80% of students with a grade of 8 Z go on to achieve managerial jobs in multinational companies, as opposed to only 60% of students with a grade of 9 X, but the latter have a 10% better outcome on average in a standardised GRE test. But we can’t ask what grade Alice WOULD have had IF Alice had gone to X. College, because that question doesn’t even begin to make empirical sense. (Ok, that doesn’t stop philosophers from asking the question, and there’s a whole literature on the meaning and evaluation of counterfactuals as these, but most physicists are very pragmatic about these things — and for a good reason, as I’ll point out below).

So now imagine the teacher selects a student and puts him or her in a closed room. Then the university’s chancellor comes in and asks the student what is his or her grade. The chancellor doesn’t know what scale the student is gonna use for the answer. But of course, the student is already in the room, so the scale, whichever it is, is already defined. It will either be that of Univ. of Z, or X. College, or maybe the scale of the Y. Institute — whichever school the student happened to have gone to.

The surprising thing in QM, the thing which makes everyone puzzled, is not that the grades only make sense relative to which scale they are measured. That is obvious already in human relationships, and is therefore directly intuitive to us. The really puzzling thing about these quantum systems is — carrying the analogy for what it’s worth — that if the students were quantum systems the second teacher could pick and choose which scale he will measure each individual student by, even after they enter the room. The chancellor can’t choose her grades, but it’s as if he can decide which university Alice went to.


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