Any good lay explanation of the Schrödinger cat duality?

[David Lewis]

Nothing hinders you to measure either Δx or Δp_x of the particle as precisely as you want (and being technically able to do).

[Emphasis added]

What if the slit width is very small, and the detector screen has a fine resolution. The particle will then make a small dot at a precise location when it lands. Then you know both the precise position and momentum of that particle at the same time, correct?

 

[vanhees71]

No, because the smaller the slit the broader will be the dispersion in transverse direction. So the better you locate the particle at the slit the less well known is the corresponding momentum component. That's the "wave-intuitive" reason for the xp-uncertainty relation.

 

[Nick Levinson]

@zonde: I accounted for two states occurring at different times even if they would have been mutually exclusive if simultaneous. I included simultaneity (e.g., "at once") but otherwise I generally didn't discuss time in my posts in this thread as I assumed everyone would incorporate that without my reminding them. Thank you for articulating it.

@Simon Phoenix:

Part of this I'm not qualified to address.

On lay books generally:

I understand the marketable length limitation (we can forget about selling a 20-volume QM encyclopedia to laity) means that a book will leave much unanswered, including some apparent contradictions within that book. But then other books should clarify those and leave different points unclarified, and not because editors get together to coordinate coverage but because different authors who are scholars in their fields hear about shortcomings elsewhere and want to rectify some of the more glaring ones. I read lay books but they're lay books by scholars, so I expect, and usually find, a higher editorial quality level than in books by people who make their livings mainly by being writers.

I read years ago of a professor who taught relativity to a third-grade class for a week and administered a test; the passing essays were many and I think were published. It may be that weirdness can be a teaching point for people ready for it or a potentially weird subject can be taught without the weirdness perspective to people who don't know (well) the contrary (non-weird) content but have some prerequisites (I'm only guessing that that's the third-grade level), leaving an in-between readership who's not ready for it, many of whom graduated college.

To me, this is beginning to look like the content may have a problem that can clear up not when the math is understood but when the terminology is more accurate or is more clearly defined. How to explain the weird without the math (average people tend to be only a step or two above innumerate) is a writer's challenge; it depends on invoking imagination (I gave the example of the hypercube, supra) but relying on readers' imagination is chancy, one solution to that being invoking imagination in different ways.

On why vs. how: If a vase falls, the why may be someone's push. The reason for the Big Bang (I assume that's not yet solved (I haven't yet read http://www.hawking.org.uk/the-beginning-of-time.html) (astrophysicist Paul Sutter on http://www.space.com/31192-what-triggered-the-big-bang.html nearly a year ago said "models that attempt to describe what 'ignited' or 'seeded' the Big Bang . . . at this stage . . . [a]re pure speculation")) may still be a matter only for hypotheticization from whatever is known, random guessing, speculation from whatever is known and some guessing, theology, or declaring it unknown, but four of those are legitimate paths to finding the reason (I exclude theology because it tends to insist on almost nothing important being unknown and impeding challenges to faith once something is declared as known). I would counterargue that physics is not mainly about the how but that physics mainly states the how but grows in large part from the why.

 

[Simon Phoenix]

To me, this is beginning to look like the content may have a problem that can clear up not when the math is understood but when the terminology is more accurate or is more clearly defined.

Well, in my view it's a bit more than clearing up the terminology. Let me attempt to describe a simple optical experiment without using too much jargon, and see how far I get. I hope to be able to persuade you that the difficulties I run into are more than just a matter of terminology. I will probably fail in this, but it's an interesting experiment that (I think) highlights the 'weirdness' inherent in QM.

So we're going to consider (rather idealized) lasers, detectors, beamsplitters and mirrors. Let's suppose we fire a laser at our detector. If we hooked the detector up to some kind of speaker we'd hear a kind of continuous noise as the laser hit the detector. As we turn down the laser intensity we reach a point where we start hearing breaks in the sound - there's a kind of graininess about it - and as we turn the wick down further we hear individual clicks as the detector fires.

Aha, we think, what we have here are blobs of light - a bit more experimentation convinces us that whatever these blobs of light are they cannot be broken down into still smaller blobs. So we have this notion that an intense laser beam is actually made up of zillions of these little blobs of light - which is why we hear the continuous noise at high intensities, but individual clicks at low intensities.

So now we're going to fire our laser beam at a 50:50 beamsplitter - this just splits the beam so that half goes one way and half goes the other. There are 2 output arms to this splitter - and in both arms we put one of these detectors. Sure enough for high intensities we see that we get a half and half split. As we reduce the intensity until we have these single blobs (and we'll assume that we end up with a single blob per given 'timeslot') we see that either the detector in arm 1 fires or the detector in arm 2 fires, but never both in a given timeslot. Furthermore, as far as we can determine with our entire array of tests for randomness, which detector fires is entirely random.

So far so good - nothing too difficult or weird here. We have an experiment that we can interpret in terms of little blobs that don't get split at a beamsplitter but go one way or t'other, entirely at random. And that makes sense - if we suppose that something went on both output arms - we'd have to explain how it is we never see both detectors click, and furthermore we'd have an issue with locality (if we assumed, for example, that the splitter gave us half a blob and that was sufficient to fire a detector, then why don't we ever see both detectors fire?)

Now let's change things a bit. Instead of detectors, we're going to put mirrors in the output arms and direct things on to another beamsplitter - splitter 2. So what do we expect to see? Well, we're happy (at the moment) with our notion that we have these little blobs that go one way or the other. So let's suppose we have a blob in arm 1 that's heading towards beamsplitter 2. This is just the same as our original experiment, surely? It's just a blob heading towards a 50:50 splitter and so we expect that it's going to have the same properties and just emerge into the output arms of splitter 2 at random.

The problem is - this isn't what happens. What we see when we run the experiment is that only one of the detectors after splitter 2 fires. All the blobs go out of one output arm of splitter 2. But how can this possibly happen? Well it can only happen if there's something from the other input arm that's influencing things - in other words there has to have been something in both output arms after the first beamsplitter.

Here's where it gets tempting to say the blob goes (in some sense) in both output arms after the first beamsplitter - and we're back to cats again being both alive and dead. It might be better to say that it is the possibility of a blob goes in both output arms - but I'm struggling here to describe things. Do we have blobs or don't we? Our first experiment (with no mirrors) seemed to be pretty cut and dried - the obvious explanation is in terms of these little blobs of light - but if we think like that we can't make much sense of experiment 2 (with the mirrors). On the other hand if we try to imagine some sort of wave property - which would neatly explain experiment 2, then we're a bit stumped to explain the properties of experiment 1.

We need, of course, the quantum description to properly 'explain' things - and we invoke superposition and properties of measurement - and the answer falls out (it's actually quite easy maths in this case). But now we have the problem to describe in words, what it is we really do have that's 'going through' both slits - clearly something happens in between the laser and the detectors - energy is getting from A to B. But does it just leave A and magically appear at B? What on earth is 'going on' in between?

It's these kinds of issues and trying to find the right language to describe things in terms of 'what's really going on' that leads to much of the difficulty. In practice, QM doesn't tell you much about what's 'really' happening in between things - it just tells us what probabilities of results we expect at the end. So at one level we can view the entire machinery of QM as just some fancy maths to get us from A to probabilities at B, without worrying too much about having a concrete picture of between A and B.

See - I told you I'd get into trouble.

I haven't found a way of thinking about this that I wouldn't describe, ultimately, as 'weird'. I'm happy with the QM formalism - very used to it - and most of the time I just talk about Hilbert spaces and vectors and operators and measurements and projections, and because I've talked like this for years I no longer think it's stark raving bonkers

 

[Stephen Tashi]

In practice, QM doesn't tell you much about what's 'really' happening in between things - it just tells us what probabilities of results we expect at the end.

Apparently a beam of light behaves differently that a beam of cats. Can we tell anything useful by trying to look at simpler systems than a cat ?

Intuitively, the alive-vs-dead state of a cat has to do with macroscopic structures either functioning "properly" or not. The proper functioning of a microscopic can be maintained while its microscopic pieces suffer various changes in position and momentum. What is the simplest toy model we can make of thing that has a "working" vs "non-working" state ? Can we do this with a system of a few particles ?

 

[Quandry]

On the subject of QM, here's what my serious QM text book has to say about the uncertainty principle (UP). Rather than talk about the infamous cat he says:

"You might wonder how the UP is enforced in the lab - why can't you determine the position and momentum of a particle? Niels Bohr was at pains to track down the mechanism by which the measurement of position destroys the previous value of momentum. The crux of the matter is that in order to determine the position of a particle, you have to poke it with something - shine light on it, say. But these photons impart to the particle a momentum you cannot control. His famous debates with Einstein include many delightful examples, showing in detail how experimental constraints enforce the UP."

So, it's not so weird as a cat that's alive and dead at the same time, after all!

It seems to me that this is a very simple issue. To determine position you need a single measurement. To determine momentum you need two. There is no way to determine what has happened between the first and second measurement.

 

[PeroK]

It seems to me that this is a very simple issue. To determine position you need a single measurement. To determine momentum you need two. There is no way to determine what has happened between the first and second measurement.

If it were that simple, Bohr and Einstein would not have found so much to debate.

In any case, you can measure momentum of a bullet, say, by firing it into a block and measuring the displacement of the block. You would never need to measure the bullet directly.

 

[Nick Levinson]

I've read the SciAm article (@Nugatory) and it makes sense, especially when considering an observer as anything that can affect or be affected by the observable. The observer as human is metaphorical and for laity, but it's more metaphorical than I thought and it's too constraining. The abstraction helps me, since I tend to take wordings more literally. Perhaps an author for laity should combine both.

I'm getting to other sources (@Nugatory and @Simon Bridge).

On whether something unobserved should count as existing (@Simon Bridge, comment 7, end): I distinguish the unknown from the metaphysical (I'm not saying you don't). The scientific basis for the metaphysical is approximately zero. Once a scientific point is accepted, conflicting metaphysics moves over to another place, e.g., when the world was created has theologically moved from 4,004 B.P. or B.C.E. (I forgot which) to 13.8 billion B.P. (around the Big Bang). But not knowing is different. If there's ground to believe that generally in given circumstances neutrinos may come into existence and then leave existence but in a particular case having those circumstances neutrinos weren't detected, I'm not prepared to say they didn't exist. I can find that there is no omnipresent deity that is physically distinct from whatever else is physically present but I cannot find that there were no neutrinos in the given case. Even with instrumentation of the kind available today and assuming proper training, I just wouldn't know. By specifying it as unknown, I leave more room for investigation. There's less motivation to investigate what's supposedly known even if incorrectly. Declaring something unknown has the conversational hazard of getting misapplied Socratic method thrown back at me (as in, "c'mon, you know the answer . . .") but better that than being both certain and wrong.

 

[Nugatory]

I've read the SciAm article....
On whether something unobserved should count as existing .....If there's ground to believe that generally in given circumstances neutrinos may come into existence and then leave existence but in a particular case having those circumstances neutrinos weren't detected, I'm not prepared to say they didn't exist.

You may not have made all the connections yet.

If you've made it through earlier posts in this thread and the Scientific American article, you see how the unmeasured superposition does not have a definite value: when a particle is in the state "superposition of spin-up and spin-down" the only thing that we can say about its spin is that it will take on various values with various probabilities when and if we do make a measurement. That's superposition.

But now let's look at what it means to say "a neutrino has come into existence". In quantum field theory, particles are not little solid objects like tiny bullets. They are excitations of a quantum field. The state of that quantum field is itself a a superposition. For example the state of the neutrino field at a particular time and place (which is to say, at a particular point in spacetime) will be a superposition of the states "there are zero neutrinos here now", "there is one neutrino here now", "there are two neutrinos here now", and so forth. As with the superpositions discussed earlier this in thread, this means that until we put a detector there to interact with the field and collapse the superposition there is no number of neutrinos present - not zero, not one, not two, nor any other number either. Thus, asserting that the undetected neutrino is definitely there just because we would have detected one if we had a detector there is analogous to the assertion that the superposed spin has definite value even when we don't measure it - and we've already seen that doesn't work.

(I mentioned above that the use of the word "observation" in quantum mechanics is an unfortunate historical accident - the word doesn;t mean what it means in ordinary English usage. The same is true of the word "particle", and fewer people would form the wrong mental model of quantum objects if we had called them "quantized excitations of a quantum field" instead of "particles").

 

[Quandry]

If it were that simple, Bohr and Einstein would not have found so much to debate.

In any case, you can measure momentum of a bullet, say, by firing it into a block and measuring the displacement of the block. You would never need to measure the bullet directly.

No, but you would have to measure the position of the block twice and you would never know whether the momentum was as a result of mass or velocity without measuring the bullet.
But all that is neither here nor there. We still don't know where the bullet came from, what its trajectory was or what other forces acted on it during its travels and you can't know that based on your example.

 

[PeroK]

No, but you would have to measure the position of the block twice and you would never know whether the momentum was as a result of mass or velocity without measuring the bullet.
But all that is neither here nor there. We still don't know where the bullet came from, what its trajectory was or what other forces acted on it during its travels and you can't know that based on your example.

You CAN measure the position and momentum of a bullet simultaneously to a very high level of accuracy. The uncertainty principle does apply to a macro object like a bullet, but the uncertainty is negligible. There is no practical issue measuring the momentum and position of a bullet simultaneously. A high-speed camera would do it. The camera does not affect the bullet to any measurable degree.

But, the uncertainty for a quantum object like an electron is not negligible. You cannot simply use a high-speed camera to follow an electron.

The question that Niels Bohr analysed (which is clearly not as simple as you would like to believe) is what happens practically when you carry out measurements on an electron. And how and why the uncertainty principle is enforced by experimental constraints.

 

[Quandry]

You CAN measure the position and momentum of a bullet simultaneously to a very high level of accuracy. The uncertainty principle does apply to a macro object like a bullet, but the uncertainty is negligible. There is no practical issue measuring the momentum and position of a bullet simultaneously. A high-speed camera would do it. The camera does not affect the bullet to any measurable degree.

But, the uncertainty for a quantum object like an electron is not negligible. You cannot simply use a high-speed camera to follow an electron.

The question that Niels Bohr analysed (which is clearly not as simple as you would like to believe) is what happens practically when you carry out measurements on an electron. And how and why the uncertainty principle is enforced by experimental constraints.

My purpose in continuing with your macro example was to illustrate that, in any terms your example was incorrect.
To continue with your example, no high speed camera can capture the position of the bullet at every instance of time and therefore you cannot have certainty of momentum between two frames.
But uncertainty is not about measurement.
"Historically, the uncertainty principle has been confused with a somewhat similar effect in physics called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system"
Heisenbergs uncertainty principle is independent of measurement, it says that there is a fundamental limit to the precision with which complimentary variables, such as position and momentum Can Be Known.

 

[PeroK]

My purpose in continuing with your macro example was to illustrate that, in any terms your example was incorrect.
To continue with your example, no high speed camera can capture the position of the bullet at every instance of time and therefore you cannot have certainty of momentum between two frames.
But uncertainty is not about measurement.
"Historically, the uncertainty principle has been confused with a somewhat similar effect in physics called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems, that is, without changing something in a system"
Heisenbergs uncertainty principle is independent of measurement, it says that there is a fundamental limit to the precision with which complimentary variables, such as position and momentum Can Be Known.

You are misunderstanding the point. If the HUP is independent of measurement, then what stops someone simply carrying out an experiment to a degree of accuracy less than the HUP predicts?

You could argue that other uncertainties in experimental measurements must always be greater than the theoretical HUP uncertainty. Okay, but why? What if you simply get better and better at experimentation: eventually you should get in under the HUP.

Also, the HUP predicts uncertainty only in certain so-called "incompatible" observables. For example, even for an electron, there is no HUP for the position in the x-direction and the momentum in the y-direction.

So, in theory, you COULD measure these quantities with unlimited accuracy.

So, if you measure the x-position and y-momentum to a certain accuracy, why can't you measure the x-position and x-momentum to the same degree of accuracy? Something must stop you. And, it's not non-commuting Hermitian operators - as they are not physically present in the lab.

A theory alone cannot stop you doing anything - there must be a physical manifestation of the theory. That's what Bohr was looking for.

For example, it's not the theory of relativity that stops you going faster than light. There must be a physical manifestation of the theory.

 

[PeroK]

But uncertainty is not about measurement.
.

If uncertainty is not about measurement, then what it is about?

 

[vanhees71]

The uncertainty is about preparation. You cannot prepare a particle to have accurate momentum and position at the same time due to the uncertainty relation
$\Delta x_j \Delta p_k \geq \delta_{jk}/2.$
You can measure momentum and position as accurately as you want. In fact, to verify the uncertainty relation you must measure both quantities with a much higher accuracy than the given standard deviations in the uncertainty relation. Of course, you cannot measure position and momentum at a single particle but you can measure either of these quantities on an ensemble of identically prepared particles with arbitrary accuracy, and in this sense the uncertainty relation is understood in quantum theory.

 

[Nick Levinson]

I read the Weirdness book despite it being 20 years old; the author is well-credentialled. It appears that the empirical evidence and the math should be taken as good and not as following the narrative but as the authority for the narrative, so that major defects belong to the major narratives as explanations of the evidence and math. And three major narratives differ and yet dominate. It was helpful to read the narratives as separate, not unified, which may have been my error in understanding other literature.

The many-worlds narrative is an infinitely-many-worlds narrative, which means measurements of the spatial dimensions of our universe (a universe of matter and energy, not of emptiness beyond) must be infinitely long, which scientists in cosmology fields reject by their finite measurements.

The hidden-variables approach makes sense. Every time we identify a new "smallest" atomic particle it's said to have no known internal structure until the day it's agreeably theorized to have a structure, thus a variable, thus a variable that's hidden till revealed, leaving some other particle to enclose its own hidden variable/s. If some seemingly-indivisible particles spin, some motivator must cause or continue the spinning, and if the motivator is not always external to the particle then at least sometimes it's internal, implying the particle has a hidden structure, thus a hidden variable.

But I don't want to trash the other approaches. They may contribute to a theoretical synthesis.

I had to return the book so I can't quote for its lacunae, but I think the assertion that what is unobserved therefore doesn't exist (Copenhagen) appears still as a faith-based statement, of a kind that lurks in various forms in various sciences until the existence is proven (e.g., biology's concept of "junk DNA" that now has acknowledged existing purpose (https://www.scientificamerican.com/article/hidden-treasures-in-junk-dna/ & http://healthland.time.com/2012/09/06/junk-dna-not-so-useless-after-all/ both as accessed Oct. 16, 2016)).

One problem with the unobserved being nonexistent: Suppose I, Nick, in a space suit with oxygen, am rocketed into outer space, ejected, and, becoming a complete system, momentarily unobserved. Through that moment, I think, feel, and talk to myself. From the standpoint of people on Earth, do I not exist? Am I thereafter, when re-observed, born an adult (at any arbitrary age not less than when being observed was lost plus the duration of nonobservation)? I don't think so. If not, then at least history is part of observationalism, since observation at one point in time would deny the ability to say that the observed could not have existed when unobserved.

If war and a natural calamity combine to wipe out all of Earth's telescopes except hand-held optical models and to lower our food supply so no more generations of people are very intelligent, the partial loss of humans' ability to observe and the passage of time sufficient for natural destruction of the formerly observed would, I think, not degrade the larger universe through progressive nonobservation by former particles of remaining particles until the universe's contents are much reduced to far less than we catalogue today.

Perhaps the math assigns the same value (e.g., 0) to a thing's nonacknowledgement and to its nonexistence, but, if so, that could be a convention, and perhaps not a solid donor to a pending narrative.

Maybe I'll look for Bohr's paper.

Einstein's objectivist view that we just may not understand what underlies the empirical findings and the math may be dated but right, I think. We may have to keep analyzing until one theory explains all that's known, and that need not be classical mechanics but likely will be quantum. I'll assume we even today have experimental results that are solid, replicated with variants, and yet not wholly explained ("weird"), maybe not the experiments that David Lindley wrote of but maybe newer results. Subjectivity is not without a role but maybe we're pushing objectivity too far away.

It won't be the first time a partial explanation was replaced.

 

[Avinash Arrah]

Hi Everyone,

One question that boggle my mind , now that they have been able to photograph light behaving as wave and particle at the same time , does the theory of multi universe still stands good?

 

[vanhees71]

Hi Everyone,

One question that boggle my mind , now that they have been able to photograph light behaving as wave and particle at the same time , does the theory of multi universe still stands good?

Can you give a reference in a scientific journal, where it is detailed, what's meant by "photograph light behaving as wave and particle at the same time" (which in my opinion is never the case since there is no wave-particle dualism since 1925 in our contemporary quantum theory anymore). I'm also not aware that any theory of "multi universe" has a particularly good standing anywhere in the science community.

 

[Avinash Arrah]

thanks Vanhees for the reply.

I was refering to following link
https://phys.org/news/2015-03-particle.html (light behaving as wave and particle)

And for the mutli universe :
http://science.howstuffworks.com/innovation/science-questions/quantum-suicide4.htm

I am really a newbie in quantum physics but i like to read a lot on that. What i may have ask may be totally irrelevant. But please be kind enough to go through it and give me your thought.

 

[vanhees71]

Well, the photoelectric effect does NOT demonstrate wave-particle duality in any sense. At the level described by Einstein, it shows the quantum nature of the electrons but not the quantum nature of the em. field. I highly recommend to read good no-nonsense textbooks on the subject. As a first starter, I recommend Susskinds "Theoretical Minimum" book.

Concerning the many-worlds interpretation, it's just up to you what you like to follow. It's irrelevant for quantum theory as a physical theory. All you need is the minimal statistical interpretation to relate the formalism of the theory to real-world experiments and observations, and that's all that counts in science. Interpretations are, to a large extent, just philosophical speculation and is rather hindering an understanding of QT instead of helping to understand it.

 

[Quandry]

If uncertainty is not about measurement, then what it is about?

Been away, so I guess there was some uncertainty about whether or not I would answer. How long did that uncertainty last? When did you conclude I would not answer? You had no way of knowing, and were probably wrong if you made an assumption.

Uncertainty is about ability to know. No matter how good your measurement system is you cannot know a particles absolute position and its velocity at the same time because velocity is a measure dependent on time change. You can't know two things at the same time if time is a factor in determining one of them.

 

[Nugatory]

You can't know two things at the same time if time is a factor in determining one of them.

That's not where the uncertainty principle comes from. For example, time is in no way a factor in determining the spin of a particle along either of two different axes, but the uncertainty principle still applies and prevents us from determining both at the same time.

 

[Quandry]

That's not where the uncertainty principle comes from. For example, time is in no way a factor in determining the spin of a particle along either of two different axes, but the uncertainty principle still applies and prevents us from determining both at the same time.

Heisenbergs Uncertainty Theory was based on measuring the conjugate variables, principally position and momentum. Originally he argued that determining the position by measurement affected the momentum. He later argued that if you knew the velocity, then measured the position, the uncertainty principal did not apply, i.e. there was no uncertainty in the past.
Therefore it seems fairly clear that the uncertainty principle 'comes from' consideration of a particles position and momentum. Which was the topic being discussed.
Certainly, it is only one example of a great many including your example.
As to whether or not time is a factor you state that the UP "prevents us from determining both at the same time" So time is a factor.
More specifically, if one of the variables is velocity, time is a critical factor because velocity cannot be determined at one point in time. It takes at least two.

 

[vanhees71]

The uncertainty is not about measurement but about preparation. It's important to understand that. It says that in any (pure or mixed) state the standard deviations of two observables $A$ and $B$ obey
$$\Delta A \Delta B \geq \frac{1}{2} |\langle [\hat{A},\hat{B}] \rangle|.$$
This implies that, if the observables are incompatible, i.e., the operators do not commute, you cannot prepare the system to be in a state, where both variables are determined, and the more determined you make one of them the less determined the other one gets.

This does not limit the accuracy with which you can measure the observables $A$ and $B$. According to the Born rule, i.e., the statistical meaning of quantum states, you measure an observable by preparing a lot of quantum systems in the same way (avoiding any correlations between these preparation procedures) and measure either $A$ or $B$. To verify the uncertainty relation for the given state, you must perform the measurements such that the accuracy of each measurement is much better than the standard deviations in order to verify these standard deviations by collecting "enough statistics", i.e., by repeating the preparation and measurement procedure often enough to be able to determine the statistical properties of the ensemble, defined by the state you prepare it in, with sufficient confidence (in particle physics for a discovery you need at least $5 \sigma$ significance level).

 

[Quandry]

The uncertainty is not about measurement but about preparation.

Measurement and preparation both imply that uncertainty would change if we could prepare and measure. However, it is no more dependent on our ability to prepare than it is on our ability to measure. The point is that it is not possible to determine with exactness two conjugate variables at a single point in time. No matter how short the time is between the two determinations, assumptions have to be made about what has happened in the intervening interval.
The Born rule (sometimes called the Born Law - but Law it is not) is a matter of probabilities. It is possible, with sufficient information, to determine to a high degree of accuracy the probability of A and B having specific states. But this has nothing to do with determining a single particles position and momentum. Born's rule implies that so long as the standard deviation is other than zero, we have uncertainty.