The unsolvable problem

Even though what I am concerned with in this chapter is called the measurement problem, it is almost a consensus that the term ‘measurement’ should not appear as a primitive unanalysable term in QM. As Bell (1987, 1990) explains, first of all it is unclear what the term ‘measurement’ means. (What constitutes a measurement? Is it complete when an experiment gives a certain result or when the physicist is conscious of the result?) Second, if physics is a precise description of the microphysical world and an explanation of the observable phenomena, then it is unacceptable to have measurements as a black box within the theory. Leaving measurements unanalysed by physics not only neglects that the measured object and the measurement apparatus are made up of elementary particles, it amounts to acknowledging that there is something in nature of which we do not know what it is. This stands in extreme contrast to the methodological doctrine of physics to give a complete description of the physical world. Bub & Pitowsky (2010) critically call this analysability criterion for measurements a “dogma of quantum mechanics”. However, I do not agree with the ‘dogma charge’, since, as explained, the criterion rests on at least two good motivations. But how far can the term ‘measurement’ be reduced and clarified in

QM?

To begin with, a measurement can roughly be defined as follows: A measurement is an interaction between object and apparatus by which information about the object before the interaction can be obtained from information about the apparatus after the interaction (cf. Wigner 1983, p. 329; Fine 1969, p. 111). To make this more precise, let us see how an interaction that can count as a measurement is described in QM. Suppose we have an object in state | ni that is an eigenstate of

observable ˆA. Second, there is an apparatus that initially is in a state | ri, where it

is ready to measure the object’s state. The apparatus does so by taking on one of its pointer states | ni that are correlated to the possible states of the object and

eigenstates of observable ˆB. To let the systems interact with each other, they have to be coupled to | i = | ni | ri. Now a unitary operator ˆO can be found that

makes the transformation ˆO(_| ni | ri) = | ni | ni. This state is an eigenstate of the

operator ˆI⌦ ˆB, with ˆI being the identity-operator acting on | ni, and the apparatus

can be understood as measuring a property of the object (cf. von Neumann 1932, p. 235 f.).

So far there is no problem in this analysis. The measurement problem arises when the initial state of the object is not an eigenstate but a superposition of eigenstates | i = Pncn| ni, with n 2. The coupled system of object and apparatus will

then evolve according to ˆO[(P_ncn| ni) | ri] = P_ncn| ni | ni, which is not an

eigenstate of ˆI⌦ ˆB. The apparatus is now in a superposed state and the superposition has ‘spread’ from the object onto the apparatus. There is no unitary operator that would continue to evolve the superposition in such a way that the apparatus changes into a definite state | ni. Furthermore, simply postulating that the superposition

collapses into a definite state is not only ad hoc and in contradiction with the Schrödinger equation, but it is also highly unclear at what point in time this collapse should happen. Unfortunately, all this is contrary to our experience that macro physical objects always are in a definite state and not a superposition. With this background, the measurement problem can be formulated.

I want to distinguish between two versions of the measurement problem, a weak and a strong one. The weak version goes as follows (cf. Lyre 2010, sec. 2):

Weak measurement problem: How is a superposition of object states connected to the definite state of the apparatus that is observed?

This captures already the situation sketched in the previous paragraph, namely, that there has to be an explanation of why measurement devices always are in a definite state even though QM predicts that they are not. It seems, however, that the measurement problem can be formulated in a way that proves its insolubility by QM alone. This will be the strong measurement problem. It is one of the aims of this chapter to show that at the moment the weak version is preferable.

Two versions of the insolubility proof can be distinguished, which differ in respect of one of their premises. In one version the measurement problem is solved, if the final apparatus state is a mixed state constituted of eigenstates each weighted with

a certain probability p.10 _{For instance, the final apparatus state of a Stern-Gerlach}

experiment could be a mixture of pointer-state spin-up and spin-down, each with p = 1/2. This amounts solely to a disappearance of interference terms of the object-apparatus state. I do not hold this premise to be strong enough to solve the measurement problem for the following reason. One might argue in favour of an ignorance interpretation towards mixed states. That is, a mixed state merely shows our incomplete knowledge of reality, while the system actually is in only one of the components of the mixed state. However, this stance is already highly problematic concerning microphysical objects and much more so concerning a macro physical measurement apparatus. It is unclear what an apparatus in a mixed state would look like, since an experimental physicist in principal always knows the state of the apparatus. Additionally, “if the state of the apparatus which carried out the primary measurement is just as difficult to ascertain as the state of the object, it is not very realistic to say that the establishment of a correlation between its and the object’s state is a fully completed measurement.” (Wigner, 1983, p. 332) Therefore, the measurement problem is not solved if the apparatus remains in a mixed state and I opt for the other version of insolubility proofs. These have the premise that the measurement problem is only solved if the final state of the apparatus is a pure eigenstate of the measured observable.11 _{In a non technical formulation the}

insolubility proof, which I call the strong measurement problem, goes as follows (cf. Albert 1994, pp. 73-79; Maudlin 1995, p. 7; Busch et al. 1996, pp. 91-93):

Strong measurement problem:

Premise 1 : QM is complete and in particular the state vector is a complete description of a system.

Premise 2 : The state vector always evolves according to linear and unitary dynam- ics.

Conclusion: A collapse of a superposition into a definite state cannot be described within QM.

It is worth pointing out that in some formulations (e.g. Maudlin 1995) premise two is stated in terms somewhat like this: ‘The state vector always evolves according to the Schrödinger equation.’ This, even though it is correct, is not helpful, since it conceals the real source of the measurement problem. This lesson can be learned from the insolubility proofs: Due to the linearity of the Schrödinger equation superpositions are possible. However, the main problem is that as a consequence of the unitarity of the Schrödinger equation a superposition will not evolve into an eigenstate. If the

10_{This argument can be found, e.g., in D’Espagnat (1966); Earman & Shimony (1968); Fine (1969,}

1970) and Brown (1986). It is clear, though, that these authors are not claiming to solve the measurement problem, but merely use mixed final states as a premise to show the insolubility of the measurement problem.

premises are valid, the conclusion is that the measurement problem is unsolvable by QM alone.12

There is a second problem that has to be solved for a complete description of how measurement results come about. It has never been in the focus of research and I will only mention it briefly as well. It is the problem of the preferred basis: Every state | i in QM can be represented by using different basis vectors, i.e., | i =Pci| ii.

For example, a state that represents spin in x-direction can also be expressed using the spin in y-direction. Since both states have the same measurement probabilities, it is unclear which property actually is measured in an experiment (cf. Redhead 1987, p. 57; Schlosshauer 2004, p. 1270; Breuer & Petruccione 2002, p. 271 f.). It also follows that any apparatus that can measure one observable can in fact measure every observable. This is an absurd consequence given the technical limits of real measurement devices (cf. Breuer & Petruccione 2002, p. 273). However, the strong measurement problem can be considered to be more severe and I will concentrate on it. In the remainder of this chapter, I will not follow one of the common roads (i.e. GRW, Decoherence-Everett or De Broglie-Bohm), instead, I will attack premise two of the strong measurement problem.