What is idealised in Wigner's analysis?

Wigner's analysis additionally assumes that the measurement is a so-called von Neumann measurement, that is, that a system that (in the case of spin) starts in a pure state |s> retains this state after the measurement.

This widespread assumption is, however, not justified in the Stern-Gerlach experiment.

A silver atom in a spin-up state |1> is located somewhere on the screen surface after the measurement, certainly no longer with spin-up, but entangled in a complicated way with the screen! There is therefore a big difference with the idealised von Neumann measurement!

The assumption of a von Neumann measurement is based on the idea that given a reliable result, a repetition of the measurement, even with a new measuring device, should reproduce the result. This is only the case with non-destructive measurements. Non-destructive measurements are, however, difficult, since they have to be measured indirectly, and could not be performed before 1980. In particular it is far from being the case that everything can be measured non-destructively.

Non-destructive measurements have been discussed in the literature, for example in Braginsky et al, Science 209 (1980), 547-557. They also explain why position measurements (pointer readings, pictures) can never be non-destructive.

Typical measurement procedures, in particular for the traditional prime examples (interference in the double-slit, Stern-Gerlach, photoelectric effect, Geiger counter, etc.) also fail to be von Neumann measurements.

A von Neumann measurement is highly idealised and actually describes no actual (irreversible) measurement, but only a further preparation. Thus one can also defer the reduction to an arbitrary place in the discussion.

In my opinion the traditional identification of measurement and von Neumann measurement is the result of conceptual sloppiness.

The passing-by of a magnet by a silver particle, or the passing-through of a double slit by a photon is not a measurement, but only the impact on the photographic plate. (If one actually tries to perform a measurement at the double slit, the interference pattern disappears.)

One is, however, happy to accept as measurements without comment the passage through filters (magnets, double slits, prisms, polarisation filters…) because one knows what the result would be if one made a measurement. Here lies the error.

Since QM makes a distinction (and falls into contradiction if it does not) as to whether the experimental setup actually loses information to the environment (which alone constitutes a measurement) or whether it only would do so, if… (this does not constitute a measurement).

Counterfactual, fictional considerations have no physical effects.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ