Do all well-measured observables effectively commute with each other?
The rest of this long post clarifies what I mean by that simple-looking question.
Consider quantum field theory in Minkowski space-time, and let $A$ denote the algebra generated by all observables, using the Heisenberg picture. In any specific model, we have a collection of subalgebras $A(O)\subset A$ associated with open subsets $O$ of Minkowski space-time. "Local observables" are elements of these subalgebras.
One of the familiar postulates of relativistic quantum field theory (called "microcausality" or "Einstein causality") says that if $O_1$ and $O_2$ are two spacelike-seaprated regions, meaning that no timelike world-line passes through both of them, then everything in $A(O_1)$ commutes with everything in $A(O_2)$. In particular, it says that spacelike-separated local observables commute with each other, whether or not they have been measured.
Another postulate (sometimes called "local primitive causality", a refinement of the "time-slice" axiom) says that if $O_1$ and $O_2$ are two regions such that every timelike world-line through $O_1$ also passes through $O_2$, then $A(O_1)\subseteq A(O_2)$. In particular, this says that observables in $A(O_1)$ are also members of the algebra $A(O_2)$, even if the region $O_2$ is far in the future (or past) of $O_1$. Again, I am using the Heisenberg picture. The key message here is that two local observables separated by a timelike interval typically do not commute with each other.
My question refers to "measured observables". Of course, the concept of "measurement" is inherently ambiguous: there is no clear line between "has been measured" and "has not been measured". However, we are still able to use quantum theory very effectively because we can empirically recognize whether or not an observable has been measured, despite the technical ambiguity in exactly where the line should be drawn. Since laboratories are made of molecules (etc), I infer that we should also be able to recognize what has been measured on paper, using a model that includes all those molecules (etc) and that respects the principles of quantum field theory. The ambiguity inherent in the definition of "measurement" should not be any more of an obstacle on paper than it is in the usual empirical approach. The required calculations may be prohibitively difficult in practice, but the question is still meaningful in principle, and I wonder if it might be answerable by some kind of general theorem even if the details in specific examples are intractible.
My question says "well-measured" instead of just "measured". This is meant to exclude "weak measurements" in which the state of the thing being measured does not strongly influence the state of the rest of the system (the laboratory). An example of a "weak measurement" would be trying to use radio waves with a wavelength of 100 km to monitor the location of an electron in a typical laboratory-scale double-slit experiment. In that case, we would not say that the electron's location has been "well-measured" by the influence it exerts on the radio waves. But if the radio waves are replaced by gamma-rays, then the electron's location can be "well-measured." The key here is that my question refers only to situations in which the thing being measured strongly influences the rest of the system, in a way that clearly distinguishes between the possible outcomes.
My question uses the phrase "effectively commute with each other." To define this, let $|\psi\rangle$ be a state-vector representing the whole system (the thing being measured, the laboratory, etc), and let $X$ and $Y$ be two observables. When I say that $X$ and $Y$ "effectively commute with each other", I mean that the commutator $[X,Y]$ approximately annihilates the given state-vector $|\psi\rangle$, in the sense that the norm of $[X,Y]|\psi\rangle$ is negligible compared to the norms of $X$ and $Y$ and $|\psi\rangle$.
If $X$ and $Y$ are separated from each other by a spacelike interval, then $[X,Y]|\psi\rangle=0$ for all state-vectors $|\psi\rangle$, whether or not anything has been measured. However, if $X$ and $Y$ are separated by a timelike interval, then they typically don't commute with each other, even if both are measured. My question is, if $X$ and $Y$ are both are well-measured, then do they effectively commute with each other in the sense defined above, so that $[X,Y]|\psi\rangle\approx 0$?
If we tried to ask this question in the context of, say, single-particle quantum mechanics, then the question would be empty. A model like single-particle quantum mechanics cannot describe the microscopic state of anything like a whole laboratory. Therefore, no matter what state-vector $|\psi\rangle$ we choose, an observable in that model cannot be recognized as having been measured solely on the basis of what the model says on paper. This question becomes non-empty only in the context of a model that includes the laboratory (etc) as part of the quantum system, so that the occurrence of measurement can be diagnosed entirely on paper (in principle) just as reliably as we normally diagnose it empirically.
