Weak measurements in quantum mechanics and 2.6 children in an American family
Let me continue with my criticism and demystification of weak measurements.
In most cases, the essence of weak measurements is hidden behind relatively complicated cases considered in practice. So let us consider the simplest (and the most honnest) case in which no final experimental outcomes are discarded. In other words, let us consider the case in which the post-selected state is equal to the pre-selected state. In this case, the weak value is nothing but the well-understood average value
<psi|A|psi>
Is that a good representation of an actual value? As an example, consider a simple experiment with one 50:50 beam splitter and two standard particle detectors. Take A to be the position operator. One expects that the position of the photon should be either in the left arm or in the right arm of the experimental setting. Nevertheless, the weak position of the particle is in the middle between these two arms, where the particle will NEVER be found by a strong measurement.
If it is still not obvious to you that such a weak value should not be taken as an actual value, then the following should convince you. The weak value of the photon position above is completely analogous to the fact (that every American knows) that the average American family has 2.6 children. How can any family have 2.6 children? Of course it can't. This is just the average, that is the "weak value" of the number of children.
You can also make it "more complicated" by postselecting only those families that live in Manhattan, for example. In this case you will not get 2.6 but a smaller number. Nevertheless, it is still clear and trivial: the number you will get (say 1.7) is only an average and does not describe any real family.
An orthodox experimentalist may say: "But that is the number that I've measured, so I am obliged to take it seriously." But he is wrong, he has NOT measured this number. Instead, he has CALCULATED it. He has measured the total number of children Nc. He has also measured the the total number of families Nf. However, the number of children per family (nf) is a result of CALCULATION through a mysterious formula
nf=Nc/Nf
Mysterious? No, trivial! Silly? If you interpret the weak value as an actual value of an individual system, then it is more than silly.
To conclude, in a Ballentine style: A strong measurement reveals a property of an individual system, but a weak measurement only reveals a property of a large STATISTICAL ENSEMBLE of equally prepared systems. A weak measurement says nothing about properties of an individual system. All weirdness of weak values results from attempts to interpret properties of an ensemble (2.6 children) as properties of an individual system (a family).
Therefore, no weak measurement can be taken as an indication against the Bohmian interpretation or any other hidden-variable theory. Essentially, Bohmian interpretation says that children exist even when nobody watches them, and that the number of children in a family is allways an integer. The fact that the average American family has 2.6 children does not contradict the Bohmian interpretation.
In most cases, the essence of weak measurements is hidden behind relatively complicated cases considered in practice. So let us consider the simplest (and the most honnest) case in which no final experimental outcomes are discarded. In other words, let us consider the case in which the post-selected state is equal to the pre-selected state. In this case, the weak value is nothing but the well-understood average value
<psi|A|psi>
Is that a good representation of an actual value? As an example, consider a simple experiment with one 50:50 beam splitter and two standard particle detectors. Take A to be the position operator. One expects that the position of the photon should be either in the left arm or in the right arm of the experimental setting. Nevertheless, the weak position of the particle is in the middle between these two arms, where the particle will NEVER be found by a strong measurement.
If it is still not obvious to you that such a weak value should not be taken as an actual value, then the following should convince you. The weak value of the photon position above is completely analogous to the fact (that every American knows) that the average American family has 2.6 children. How can any family have 2.6 children? Of course it can't. This is just the average, that is the "weak value" of the number of children.
You can also make it "more complicated" by postselecting only those families that live in Manhattan, for example. In this case you will not get 2.6 but a smaller number. Nevertheless, it is still clear and trivial: the number you will get (say 1.7) is only an average and does not describe any real family.
An orthodox experimentalist may say: "But that is the number that I've measured, so I am obliged to take it seriously." But he is wrong, he has NOT measured this number. Instead, he has CALCULATED it. He has measured the total number of children Nc. He has also measured the the total number of families Nf. However, the number of children per family (nf) is a result of CALCULATION through a mysterious formula
nf=Nc/Nf
Mysterious? No, trivial! Silly? If you interpret the weak value as an actual value of an individual system, then it is more than silly.
To conclude, in a Ballentine style: A strong measurement reveals a property of an individual system, but a weak measurement only reveals a property of a large STATISTICAL ENSEMBLE of equally prepared systems. A weak measurement says nothing about properties of an individual system. All weirdness of weak values results from attempts to interpret properties of an ensemble (2.6 children) as properties of an individual system (a family).
Therefore, no weak measurement can be taken as an indication against the Bohmian interpretation or any other hidden-variable theory. Essentially, Bohmian interpretation says that children exist even when nobody watches them, and that the number of children in a family is allways an integer. The fact that the average American family has 2.6 children does not contradict the Bohmian interpretation.
Total Comments 3
Comments
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Nobody expects the expected value
As I have explained, a weak value is a kind of an average value. But in statistics, average value is also called "expected value". On the other hand, one does not need to know about QM to understand that the "expected value" may not be expected at all. For example, nobody expects to find 2.6 children in an American family.
Thus, in general, nobody expects the expected value. And consequently, in QM, nobody should expect the weak value.Posted Oct28-10 at 04:38 AM by Demystifier 
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Weak measurements
Dear members. Forgive if I am on the wrong forum but I have a related paper on Bell's theorem which I think is worth reading.
http://www.m-hikari.com/astp/astp201...17-20-2010.pdf.
In the paper I argue that Bell's theorem is incomplete. From this claim I think I may also state that Hardy's paradox, that is measurement after mutual annihilation, can be explained with classical probability measure theory.Posted Mar16-11 at 07:00 AM by hgeurdes
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A minor correction: It should be 2.4 children, not 2.6.
Posted Jun3-11 at 09:37 AM by Demystifier
