# Why is there no unique "canonical" sense in which we can talk of simultaneous measurement of $\hat{x}$ and $\hat{p}$?

Why is it impossible to measure position and momentum at the same time with arbitrary precision?

and in the most voted answer, it is talked about how that there are senses in which we can measure the position $$\hat{x}$$ and momentum $$\hat{p}$$ of a quantum particle simultaneously, but that there is not a unique sense in which we can do so. It is, however, mentioned in the context that there is also not a unique "sense of measurement" of $$\hat{x}$$ or $$\hat{p}$$ by themselves either, and so supposedly this shouldn't be a surprise.

However, for me, it is: even if we can think of multiple measurement methods for $$\hat{x}$$ and $$\hat{p}$$ individually, we can still define a "canonical" (even if practically impossible) theoretical perfect measurement of them, which is what is meant in your typical first-course quantum textbook when we talk about "measuring $$\hat{x}$$ or $$\hat{p}$$" and is the measurement whose probability distribution is given by the positional or momental wave function and collapses the wave function to an eigenstate of one or the other.

Yet it seems what the article above is suggesting is no such canonical measurement choice exists for the joint case. Is that correct? If so, why?

To put it another way, and perhaps what I'm really closer to going after is, am I right to believe there is no canonical way to create a joint probability distribution

$$f_{XP}(x, p)$$

on both $$x$$ and $$p$$ simultaneously, even while there is a perfectly canonical way to do so for $$x$$ and $$p$$ individually? Moreover, if this is so, how then are we to understand, without making arbitrary choices, the "measurement" going on when we look at, say, a car driving down the highway and note both where it is and its speed, which is a simultaneous measurement of $$\hat{x}$$ and $$\hat{p}$$ though obviously with a minimum resolution far above the quantum limit? Shouldn't there be canonical unambiguous joint PDF that we can consider such a measurement as operating upon? If not, why not?

The ultimate goal, though, by the way, of this question, is to ask if there is an unambiguous sense to which we can define the "total information content" of a quantum system or perhaps more accurately, quantum state. In classical probability theory, if we have a bunch of random variables, then to talk of the total information carried by all of them in some entropy measure, we generically need a joint PDF because of correlations. But this seems to be problematized in QM if the joint PDF (which is needed to fully specify "all information known about the system") is not unique, as the entropy is likely then not well-defined either. And it is definitely clear correlations can exist above and beyond the inherent incompatibility when you consider cases like the car where that in theory an agent may have information approximating a correlated classical probability distribution at levels well above the Heisenberg limit.

Hence it would be fallacious to, say, try a naive approach of defining the total information by something like summing over the probabilities of the various observables obtained individually, because that would in a sense "multiply count" things, just as it would be wrong to sum over a bunch of correlated random variables on the same underlying probability space in classical probability theory to create the entropy.

• You can't define a probability distribution $f(x,p)$ for position and momentum simultaneously. You can define a Wigner quasiprobability distribution. The wikipedia article linked discusses the motivation and technicalities. You can also take a classical limit of this object to recover classical probabilities, which would be a good approximation to scenarios involving macroscopic objects like a car. Commented Jul 14, 2022 at 19:57
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– Buzz
Commented Jul 16, 2022 at 23:39