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Here I criticize the whole concept of weak measurements and weak values in quantum mechanics, first proposed by Aharonov, Albert and Vaidman:
http://prola.aps.org/abstract/PRL/v60/i14/p1351_1
1. By a suitable choice of the post-selected state, the weak value can be made ARBITRARILY large. In particular, the title of the paper above reads:
"How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100"
I don't think that such arbitrariness (made by experimentalists!) should be viewed as a measurement of a true value existing out there in nature itself.
If this is not convincing enough, then the following should:
2. It is often stressed that weak measurements represent a solution of the Hardy "paradox" (which in the original formulation involves electrons and positrons). This solution of the paradox is based on a NEGATIVE number of particles obtained by a weak measurement. What, for the God sake, a negative number of particles means? It is nothing but a number obtained when some POSITIVE numbers directly obtained by measurements are put into a weird MATHEMATICAL FORMULA supposed to represent a "physical" quantity called "weak value".
Perhaps weak values can be best understood through an every-day analogy in the classical world. The physical (i.e., strong) amount of money cannot be negative. Yet, a weak value of the amount of money can be negative. Namely, there is a reasonable MATHEMATICAL formula that in some cases attributes a negative amount of money in certain situations in which the amount of money is not strongly measured. Indeed, we have a standard word for such a negative amount of money - DEBT.
Let me push the analogy further. There is a guy called Bohm who proposed that money exists even when nobody observes it. It is called the Bohmian interpretation of economics. According to this interpretation, the amount of money is allways non-negative. However, weak measurements of money show that the amount of money can be negative. Some economists take this as an indication that the Bohmian theory of money is wrong, or at least that it is highly unnatural.
Nevertheless, the Bohmian theory is very simple and natural. It proposes a double ontology, according to which both money (particles) and human rules of behavior (wave function) separately exist. A "negative amount of money", that is debt, is not really an amount of money, but a part of human rules which say that some humans should give money to some other humans whenever certain circumstances take place. We see that money (which is a hidden variable which exists even when nobody observes it) is guided by the pilot human rules. We also see that this hidden variable is highly contextual (although local, or course).
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.
http://prola.aps.org/abstract/PRL/v60/i14/p1351_1
1. By a suitable choice of the post-selected state, the weak value can be made ARBITRARILY large. In particular, the title of the paper above reads:
"How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100"
I don't think that such arbitrariness (made by experimentalists!) should be viewed as a measurement of a true value existing out there in nature itself.
If this is not convincing enough, then the following should:
2. It is often stressed that weak measurements represent a solution of the Hardy "paradox" (which in the original formulation involves electrons and positrons). This solution of the paradox is based on a NEGATIVE number of particles obtained by a weak measurement. What, for the God sake, a negative number of particles means? It is nothing but a number obtained when some POSITIVE numbers directly obtained by measurements are put into a weird MATHEMATICAL FORMULA supposed to represent a "physical" quantity called "weak value".
Perhaps weak values can be best understood through an every-day analogy in the classical world. The physical (i.e., strong) amount of money cannot be negative. Yet, a weak value of the amount of money can be negative. Namely, there is a reasonable MATHEMATICAL formula that in some cases attributes a negative amount of money in certain situations in which the amount of money is not strongly measured. Indeed, we have a standard word for such a negative amount of money - DEBT.
Let me push the analogy further. There is a guy called Bohm who proposed that money exists even when nobody observes it. It is called the Bohmian interpretation of economics. According to this interpretation, the amount of money is allways non-negative. However, weak measurements of money show that the amount of money can be negative. Some economists take this as an indication that the Bohmian theory of money is wrong, or at least that it is highly unnatural.
Nevertheless, the Bohmian theory is very simple and natural. It proposes a double ontology, according to which both money (particles) and human rules of behavior (wave function) separately exist. A "negative amount of money", that is debt, is not really an amount of money, but a part of human rules which say that some humans should give money to some other humans whenever certain circumstances take place. We see that money (which is a hidden variable which exists even when nobody observes it) is guided by the pilot human rules. We also see that this hidden variable is highly contextual (although local, or course).
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.