Saturday, April 16, 2022

An underrated gem: WAY beyond conservation laws

I think the article Wigner-Araki-Yanase theorem beyond conservation laws by Mikko Tukiainen is an underrated gem (if we compare its content to the number of citations).

Here's why I think so.

First, I think the Wigner-Araki-Yanase theorem is underrated. It started with a paper by Wigner (here is an English translation.), who showed that you can't have an accurate ideal spin measurement which is also repeatable. By "repeatable" it's understood that, whatever result you get, by repeating the measurement you'll get the same result. In other words, accuracy requires that the measurement disturbs the system, so the spin is no longer what you measured it to be. You can avoid this by being satisfied with a less accurate result. Wigner also showed that repeatability can be obtained and the error can be made as small as wanted, if the measuring device is large enough so that the apparatus has large uncertainty for the conserved quantities.

Araki and Yanase generalized his result, and added some interesting observations, in particular that this limitation applies to the measurement device as well.

Wigner was brilliant enough to know how to give a more general proof, but he wanted the idea to be understood easily. He used the conservation of angular momentum along an axis to deduce the limitation of accuracy of spin measurement along an orthogonal axis. He only uses a conservation law, but all conservation laws contribute. He had to give a simple proof, without making too many assumptions about the evolution equation. So he probably thought, spin measurement is a simple example, and also entails the existence of other spin operators that are conserved by unitary evolution and don't commute with it.

While all conservation laws contribute limitations, on the one hand this is an expression of the symmetries, and on the other hand, in fact, they don't do anything. The limitation is in the transformation of the total state from the state before measurement into the state after the pre-measurement, that is, just before we invoke the collapse postulate (the collapse itself breaks the conservation laws). During pre-measurement the evolution is unitary, because collapse is invoked at the end. The evolution itself constraints the possible results of the measurement. Conservation laws were originally used as indications that we can use to find such limitations. A general proof in terms of general unitary transformations is very difficult, but you can look at a conserved quantity and deduce enough to know that the accuracy is limited if we want repeatability. So the conservation law was used to give a simple, although less general, proof. And to make it simpler, the conserved quantity had to be additive.
But these are just assumptions Wigner made to prove the result, and this made me initially think that there is nothing special or metaphysical about conservation laws in this context, despite Wigner's other very important realizations about the role of symmetry. But there is a very important lesson about symmetries (which, as we know from Emmy Noether, are the reason behind the conservation laws), as elucidated by the works of Ozawa, Loveridge, Busch, Miyadera and others.

Conservation laws are often used to deduce things without solving equations. But they don't constrain, they express the constraints of the system, since these constraints restrict the symmetries, and therefore the conservation laws. On the other hand, the symmetries of the system really capture an important aspect of the constraints, as explained in this wonderful article by Loveridge, Busch, and Miyadera.

The reason why I consider Mikko Tukiainen's paper important is that it seems to indicate another deeper aspect, that seems to go beyond that. He not only it gave a more general proof, but in that proof, conservation laws play no role (you can read it for free here). He used instead the idea of quantum incompatibility, which is a way to understand the major features of quantum mechanics that distinguish it from classical mechanics (although the most useful examples are still given by conservation laws). This is neat, complements the idea based on symmetry, and it's in some sense more general.

Both the symmetries and quantum incompatibility go deep, but maybe there is a deeper reason than both of these - the full range of such limitations of measurements is still unknown. And maybe there is no general characterization of this. But anyway, I think there's more to be learned about this.

Since both the WAY papers together have together a relatively small number of citations (hundreds), I consider them underrated too. This is another mystery to me.