The idea of ‘weak measurements’ has been much discussed recently in the popular science presses as well as in physics journals. This post aims to demystify some of the claims made about quantum systems based on weak measurements.

As illustration of the basic idea of weak measurement, consider the following analogy. A quantum shoe factory makes 2 models of shoe: a casual shoe ‘C’ and a dress shoe ‘D’. But since these are quantum shoes, the factory churns them out in a superposition of both models–call that ‘B.’ (For physicists, B is analogous to ‘spin up along x’, and C and D are spin up and down along z, respectively). It is only when Fred, the shoe checker, inspects each shoe that their nature as C or D is clearly ‘collapsed’ and thereby established. At this point, Fred places each kind of shoe in its respective bin for shipment to two different stores–one that only orders C and the other only orders D. (This is analogous to a sharp measurement that destroys the interference patern in the electron two-slit experiment.)

Now suppose it is Saturday morning, and Fred had one too many Happy Hour drinks the night before. As he ‘measures’ each shoe in its B state, his sloppiness results in some C shoes erroneously being placed in the D bin, and vice versa. If enough of these errors are made, such that Fred is just as likely to place a shoe in the wrong bin as he is to put it in the right bin, then each bin contains equal amounts of C and D, and the shoes have just had their initial combined state B confirmed. (This is analogous to having retained the interference pattern in the electron two-slit experiment.)

Suppose that hungover Fred has to sort 100 shoes. He has just enough of his faculties left to put ever-so-slightly more shoes in the correct bin than in the incorrect bin. This is the basic ‘weak measurement’. He has *almost* retained the original shoe state, but not quite–the shoes in each box have gotten a bit ’tilted’ more toward C or D than they originally were.That is, each in the C bin is slightly more likely to be found in the state C than in the state D, and vice versa.

Now, to make contact with some of the claims in the literature concerning ‘weak measurements’, we have to add one more step: a follow-up careful (‘sharp’) measurement of every shoe in each of the bins. (For physicists, this is the post-selection measurement of z spin). Suppose the person carrying out this measurement is Gretchen, who unlike her co-worker Fred, did not attend Happy Hour the previous evening. Gretchen first takes bin C, and with coffee in hand, carefully measures each shoe in the box. She finds that (say) 52 of the shoes have come out (correctly) C and 48 of the shoes have come out (incorrectly) D. Then she takes the other bin D, and finds that (say) 53 of the shoes have come out (correctly) D and 47 have come out (incorrectly) C. At this point, only Gretchen knows and has written down which bin each shoe came from and whether Fred correctly sorted it or not. But the probability of a shoe’s having been placed in Fred’s bin C is slightly higher if it was found by Gretchen to be C.

Now, some researchers have made the following claim based on this procedure: because of Gretchen’s final measurement, each shoe somehow ‘knows’ *before* Fred’s sloppy sorting which of his bins it’s going to end up in. That is, Gretchen’s measurement is claimed to act rather like Merlin the Magician, who travels from the future into the past and helps beings to fulfill their destiny. The idea is that each shoe is* retroactiviely* steered by Gretchen’s final measurement toward its respective bin placement by the hungover Fred.

But this is incorrect, which we can see as follows. Suppose now a shoe store representative, Helen, comes to the factory just after Gretchen’s quality control. Helen decides to play a guessing game with her, as follows. She picks up each C or D shoe and tries to guess in which bin Fred had put it. For a shoe that ended up C, she has slightly better luck guessing that it came from Fred’s bin C, and similarly with D.Why is this? Simply because the result that Gretchen found was more likely *to have come from a state* favoring that outcome (a state created by Fred’s sloppy measurement) than from a state inhibiting that outcome. We don’t in fact need Merlin the Magician to explain any of this. The situation is no different conceptually from being able to predict that a person coming to the U.S. from Poland is more likely to be of Polish ancestry than, say, Japanese ancestry. The fact that we now see that the person came from Poland does not retroactively cause the person to have been born of Polish parents!

So the next time you see claims such as “future measurements affect past measurement results,” be wary. None of the quantum shoes sorted by Fred needed a Merlin-like retrocausal influence from Gretchen for these correlations to arise. Fred simply tilted the shoes to states more likely to end up with one property, upon measurement, than the other. And this is all that is demonstrated by these kinds of experiments: standard quantum mechanics.

“since these are quantum shoes, the factory churns them out in a superposition of both models–call that ‘B”

“If enough of these errors are made, such that Fred is just as likely to place a shoe in the wrong bin as he is to put it in the right bin, then each bin contains equal amounts of C and D, and the shoes have just had their initial combined state B confirmed”

Seems to me these statemnts are incompatible.

Thanks Marvin — the measured statistics for C and D outcomes when Fred makes maximal errors are the same as what one would get when measuring C vs D for the initial state B. So I see no incompatibility here.

I find the ‘quantum shoe’ notion uncomfortable; its properties are not clear, like a fractional integer. But if, in many trials, I discover half up-spins and half down-spins coming out of a Stern-Gerlach apparatus, I cannot conclude that the original beam consisted of horizontal spins. ( “the shoes have just had their initial combined state B confirmed”)

This blog post uses everyday analogies to try to get the point across in a non-technical way. The basic point is that you get no effective ‘steering’ away from the statistics that would be yielded by the initial state if the measurement is maximally weak.

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