# Is it possible to use Bayesian method in improving the measuring conditions and accuracy of an electron's (or photon) momentum and position?

for you who need a definition for what a "Bayesian method" is, and as per wiki's easy definition. it's a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability".

i would like to hint that me using the term "Bayesian" does not mean literal implication to this definition as in the realm of classical measurements, but rather, the most Bayesian way to measure in a quantum world. in other words, As Bayesian as a quantum measurement method can get

the question says it all really, but those who are interested in details i might want to say that my question does not aim to measure electron's momentum and position simultaneously, but rather improve the conditions by some sort of a Bayesian method. was there any attempt to research that? if so is there any papers that discusses that?

more on that, can it be really applicable on a single electron measurement, without the need to make a conclusion based on studying, say, a beam of electrons or so.

Also, what will be the upper limit on this Bayesian process?

it's worth noting that the tag section does not let me tag anything related to Bayes or Bayes theorem.

thanks in advance!

## 2 Answers

I don't think a Baysian method can help with quantum mechanics because the Shrödinger uncertainty relation is physical, if you know observable $$A$$ your state collapses to it's eigenfunction and you know about $$B$$ with an uncertainty dictated by the relation. A great example of using this is in squeezed coherent states where you exploit this relation to get a state with your desired properties.

I don't know enough about measurement techniques but I'm sure using the Baysian method is useful there to overcome classical measurement uncertainties and if it's useful it is probably used already.

This is indeed a very good idea to include Bayesian inference to the process of quantum measurement processes and this is actually what is done when looking at conditional expectation values. In particular when you are interested in the outcome of a specific measurement which is conditioned on the outcome of a subsequent measurement.

In this paper (Aharonov, Bergmann, and Lebowitz, "Time symmetry in the quantum process of measurement." Physical Review (1964)) you can find an analysis of conditional expectation values during a history of measurements. Even though it is not stated explicitly, they use Bayes's theorem in order to get these conditional probabilties from concentional quantum theory.

But I don't think that you can improve the conventional method in the sense that a measurement on a single electron is sufficient to obtain significant conlusions out of it. Since quantum theory remains an ensemble theory this conditional probabilies from Bayes' theorem make in particular use of the concept of ensembles by using only a sub-ensemble of the original prepared ensemble in order to satisfy the final boundary conditions on the last measurement in this measurement sequence. Thus a measurement on a single electron, which is still probabilistic, can result in an outcome which is not the condition you impose on your measurement scheme (like the condition in Bayes' theorem) so can not draw any conclusions on this null-event.