Quantum Ball and Cup - Thought Experiment

[Lynch101]

TL;DR Summary

Thought experiment to try and get a clearer understanding of what the different interpretations say.

I'm sure most will be familiar with the well-known ball and cup trick. The dynamics of the game itself are unimportant, we just need to have the image of 5 cups with a single ball being revealed when the relevant cup is lifted.

The Set-up
Imagine a machine which has a conveyor belt coming out of it. When we turn on the machine the conveyor belt starts moving and 5 [upside down] cups appear from the machine, in a row across the conveyor belt.

On the bottom of the cup, such that we can read it, is a percentage. This figure represents the probability of finding the ball under the cup, if the cup is lifted.

At the end of the conveyor belt is a metal bar running the width of the belt such that, when the cups meet the bar all will be overturned* and a single ball will be revealed under one of them.
*something akin to the mechanical arm in a bowling pin setting machine, which clears the remaining pins before resetting.

The conveyor belt itself doesn't need to represent anything, it's more for the sake of visualisation. The cups could come out of a machine and "float" towards the bar to be knocked over.I will outline my understanding of what the different interpretations say. Hopefully, if anyone is so inclined they might be able to help correct my representation or elaborate further and also add the interpretations with which I am less familiar.The Interpretations
All interpretations agree on the percentages written on the bases of the cups and that only a single ball will be revealed when all the cups are overturned.

Bohmian Mechanics - says the ball was always under a single cup and the measurement just revealed to us which cup that was. It says that the position of the ball under the given cup was due to it riding along a guiding or pilot wave.

I'm not clear on where the FTL influence occurs here in BM, or is that only pertinent to tests of Bell's inequality?MWI - says that when the cups meet the bar and are overturned, the universe branches off into 5 "worlds" and in each world the ball is revealed to be under a different cup.

Does it say anything about the position of the ball prior to branching?Collapse Theories - says that a ball or the ball (or part of the ball) is under each cup, but when the cups are overturned by the bar, they spontaneously collapse (FTL) to a single ball, under one of the cups.Minimal Statistical - says the ball cannot be said to be definitely under any cup. We can only know the probability of finding it under any given cup. If we run enough iterations of the experiment the overall results will correspond to the statistical distribution on the cups.

If the ball is definitely not under a single cup, how is it that we find it under a single cup when all the cups are overturned?

 

[Andrew Mason]

Each of the approaches are just different ways of expressing the inherent uncertainty of position and momentum. As particle mass increases, the uncertainty as to its speed becomes small enough that the uncertainty as to position becomes immaterial - like the ball and cup.

The idea that a particle is not in any particular location at a particular time or is effectively in all of them at any given time (superposition) is just a way of expressing that uncertainty: the absolute inability to determine the particle's exact position in space and time.

AM

 

[StevieTNZ]

If the ball is definitely not under a single cup, how is it that we find it under a single cup when all the cups are overturned?

That's where interpretations come in and when 'collapse of the wave function' occurs.

 

[PeterDonis]

All interpretations agree on the percentages written on the bases of the cups

Yes, obviously, since this is part of the specification of the experiment.

and that only a single ball will be revealed when all the cups are overturned.

Not if the "overturning" is a quantum measurement; then the MWI says that all possible results are realized, since the "overturning" process has to involve entanglement between the cups and the overturning mechanism.

If your scenario stipulates that the "overturning" process does not involve anything like this, and is just a classical process that randomly selects which cup the ball is under on each run, then your scenario says nothing about QM and QM intepretations are irrelevant.

MWI - says that when the cups meet the bar and are overturned, the universe branches off into 5 "worlds" and in each world the ball is revealed to be under a different cup.

Note that this is inconsistent with the above statement that only a single ball will be revealed when all cups are overturned. If you were to limit the above claim to saying that only a single ball was observed to be revealed, that would remove the inconsistency--but then of course you would have to define what "observed" meant, and that would bring in all the familiar issues of entanglement, measurement, etc.

 

[PeterDonis]

Bohmian Mechanics - says the ball was always under a single cup and the measurement just revealed to us which cup that was. It says that the position of the ball under the given cup was due to it riding along a guiding or pilot wave.

Yes.

I'm not clear on where the FTL influence occurs here in BM

In the pilot wave, which can cause the ball to swerve due to information propagated at speeds faster than light from events far away.

or is that only pertinent to tests of Bell's inequality?

No. See above.

 

[PeterDonis]

Collapse Theories - says that a ball or the ball (or part of the ball) is under each cup

Depends on the collapse theory. Some don't make any claim at all about where the ball is when it is not being measured.

 

[PeterDonis]

Minimal Statistical - says the ball cannot be said to be definitely under any cup.

More precisely, it says that our mathematical descriptions of this system do not describe individual balls or cups; they only describe the statistical ensemble of large numbers of them.

If the ball is definitely not under a single cup, how is it that we find it under a single cup when all the cups are overturned?

The interpretation, even in your wording, does not say the ball is definitely not under a single cup. It only says that the ball cannot be said to be definitely under any cup. The latter is a weaker statement than the former.

 

[gentzen]

MWI - says that when the cups meet the bar and are overturned, the universe branches off into 5 "worlds" and in each world the ball is revealed to be under a different cup.

No, this is not what MWI says. The branching happens much earlier than the observation. You see, one of the points of MWI is that the observer should have no special role. If the act of observation would cause the universe to branch, then this would be exactly such a special role.

Does it say anything about the position of the ball prior to branching?

The actual branching in MWI is not a sharp event. Before the act of observation, the ball already has a position in the different branches, only the observer doesn't know yet which branch he belongs to.

Collapse Theories - says that a ball or the ball (or part of the ball) is under each cup, but when the cups are overturned by the bar, they spontaneously collapse (FTL) to a single ball, under one of the cups.

No, this is not what collapse theories say. Take a look at wikipedia, or SEP, or their references (if you object that an encyclopedia is not a valid reference), or some textbook mentioning that subject, like for example Dürr/Lazarovici, Understanding Quantum Mechanics (2020).

 

[DrChinese]

No, this is not what MWI says. The branching happens much earlier than the observation. You see, one of the points of MWI is that the observer should have no special role. If the act of observation would cause the universe to branch, then this would be exactly such a special role.

This view is mathematically inconsistent with the predictions of QM, specifically Bell. It is the relationship of the observers that determines the statistical predictions within QM (for entanglement cases). The statistical outcomes cannot be independent of these. I'm not saying that branching cannot occur prior, but IF branching occurred prior to an observation: then it is completely irrelevant to the predicted statistical correlations.

 

[PeterDonis]

The branching happens much earlier than the observation.

If by "observation" you mean "observation by a human using their eyes", then yes, branching, i.e., irreversible decoherence, will occur much earlier than that happens, at least if "much earlier" is allowed to refer to short enough time scales. After all, human observations occur on time scales of milliseconds to hundreds of milliseconds, depending on whether you count triggering of optic nerve impulses or conscious processing in the brain as "observation". These time scales are much longer than typical decoherence times for macroscopic systems, but they're still short in ordinary terms, so you have to be ok with, say, $10^{-10}$ seconds being "much earlier" than $10^{-2}$ seconds.

 

[gentzen]

This view is mathematically inconsistent with the predictions of QM, specifically Bell. It is the relationship of the observers that determines the statistical predictions within QM (for entanglement cases). The statistical outcomes cannot be independent of these.

The statistical outcomes depend on the type of measurement (like horizontal spin measurements, vertical spin measurement, ...), but not on the observers. In the described quantum ball and cup thought experiment, the type of measurement was fixed. The bar lifting the cups just revealed the results, just like looking inside the box for Schrödinger's cat.

I'm not saying that branching cannot occur prior, but IF branching occurred prior to an observation: then it is completely irrelevant to the predicted statistical correlations.

Not sure what you are getting at here, i.e. how to interpret your words. What was important to me was to make it clear that you don't get MWI by just taking Copenhagen, and then declaring that the world would split in the moment where you have to update the state. That is too simple, MWI is more subtle than that.

 

[DrChinese]

The branching happens much earlier than the observation.

The statistical outcomes depend on the type of measurement (like horizontal spin measurements, vertical spin measurement, ...), but not on the observers. ...

The important splitting must occur sometime close in time to when an observer's measurement occurs - which I refer to generically as the observer, because it includes the observer's choice of the basis of the measurement. As PeterDonis points out: if by "much" you mean the additional time it takes (after the physical interaction with the measurement device) for "observation by a human using their eyes" then that's fine. But I don't think that's what you intended, and it's certainly not what I intended. It doesn't matter (AFAIK) when or whether a human is involved.

--------------------------

If we are using the OP example: The important splitting is NOT when the ball is dropped. But that example is not a good example for discussing when MWI splitting occurs, because it obscures the actual quantum process. Normally: we have the creation of a quantum system in a superposition (say T=0), followed by a quantum measurement (say T=1). The human observer sees the result at T=2. I am stating that no splitting occurs at T=0 worth discussing, because it does not figure into the predicted quantum statistical results. It is the splitting that occurs when a measurement occurs that must be the important one. Otherwise, Bell is not respected (which would be more clear if the example were a Bell test).

If there is branching occurring at T=0.1, T=0.2, T=0.3 - certainly a possibility - then such branching cannot contribute to the statistical outcomes. Why? Because those would then be operating in effect as hidden variables. And we know those are not viable, because quantum mechanics is contextual. And by contextual, I mean observer dependent. That is the opposite of "one of the points of MWI is that the observer should have no special role". I don't know enough about MWI to agree or disagree with that characterization of MWI.

 

[gentzen]

As PeterDonis points out: if by "much" you mean the additional time it takes (after the physical interaction with the measurement device) for "observation by a human using their eyes" then that's fine.

In the OP example, I interpreted the ball and the 5 cups as macroscopic objects. Therefore, the quantum interaction giving rise to the probabilities must happen much earlier than when the cups meet the bar and are overturned.

In the sense that a human using his eyes has no chance to observe the ball while it is still covered by a cup, I do indeed mean "the additional time it takes for observation by a human using their eyes".

But I don't think that's what you intended, and it's certainly not what I intended.

It depends. It made sense to PeterDonis himself, and it was not wrong. On the other hand, Lynch101's characterization of MWI was wrong, and my intention was to provide an explanation in his own picture on how that characterization can be improved. Sean Caroll calls this "self-locating uncertainty".

Normally: we have the creation of a quantum system in a superposition (say T=0), followed by a quantum measurement (say T=1). The human observer sees the result at T=2. I am stating that no splitting occurs at T=0 worth discussing, because it does not figure into the predicted quantum statistical results. It is the splitting that occurs when a measurement occurs that must be the important one.

Agreed.

If there is branching occurring at T=0.1, T=0.2, T=0.3 - certainly a possibility - then such branching cannot contribute to the statistical outcomes. Why? Because those would then be operating in effect as hidden variables.

Indeed, the branches in MWI should not be interpreted to act as hidden variables.

And we know those are not viable, because quantum mechanics is contextual. And by contextual, I mean observer dependent.

an observer's measurement occurs - which I refer to generically as the observer

I am not a big fan of redefining words like that. If the only purpose of an observer is to establish permanent measurement records, then it may be fine to call any device providing such permanent records an observer. But I can't see the point in redefining observer to mean measurement.

That is the opposite of "one of the points of MWI is that the observer should have no special role". I don't know enough about MWI to agree or disagree with that characterization of MWI.

Of course, the meaning of my sentence changed when you changed the meaning of observer. Now each branch has its own permanent records, so you might still question that characterization even without changing the meaning of observer. But the intention of my sentence was to explain how Lynch's characterization can be improved, and I prefer being concrete and understandable over being unobjectionable.

 

[Lynch101]

In the OP example, I interpreted the ball and the 5 cups as macroscopic objects. Therefore, the quantum interaction giving rise to the probabilities must happen much earlier than when the cups meet the bar and are overturned.

In the sense that a human using his eyes has no chance to observe the ball while it is still covered by a cup, I do indeed mean "the additional time it takes for observation by a human using their eyes".

Just to clarify, the observation made by a human is not necessarily important in the analogy. The "revealing" of the ball is meant to be practically equivalent to the quantum system making a "mark" on the detector screen. This doesn't need to be seen by someone until much later.

We can remove the cups altogether and just have a ball spontaneously appearing when it passes some arbitrary marker. The cups (with the probabilities written on their base) are merely meant to represent the probabilistic predictions of QM and represent the idea that we cannot determine the exact position of the system prior to measurement. The cups can also just disappear when they pass the "measurement line" to represent the update of the wavefunction with the measurement of a definite position - if I'm using the terminology correctly here (basically just to represent the measurement).

I'm just trying to employ a more visual analogy to try to help me think about it in a slightly different way.

 

[DrChinese]

... Agreed.

Indeed, the branches in MWI should not be interpreted to act as hidden variables.

I am not a big fan of redefining words like that. If the only purpose of an observer is to establish permanent measurement records, then it may be fine to call any device providing such permanent records an observer. But I can't see the point in redefining observer to mean measurement. ... Of course, the meaning of my sentence changed when you changed the meaning of observer. ...

Great, I think we're on the same page*. As best I can tell of MWI:

1. At the time T a quantum measurement is made, there is a branching into 2 or more worlds, corresponding to the possible outcomes (and weighted in some fashion according to the quantum expectation). This process is deemed "deterministic" for some reason that escapes me; the apparent "indeterminism" (what we see as a random outcome) is then a result of us finding ourselves in a particular branch and unable to see the alternative branches.

2. Question: Let's say we have an HV polarizing beam splitter (PBS) with a 45 degree polarized photon passing through it at some time circa T1. The photon becomes polarized at H or V when passing through, is subsequently detected by one of 2 detectors place near the respective output ports H and V circa time T2. Can anyone tell me if the splitting occurs at T1, or is it at T2?

3. Assuming the answer to 2. above is that branching occurs at T1: the issue I have with this is that the PBS output ports H and V could alternately be brought back together to reconstruct the original 45 degree polarization. I.e. we intend to "erase" the previous measurement. But the information to accomplish such does not exist in my branch - because there was previously a determinate outcome. I would say this is a paradox.

4. The only other answer for the question then is to say the branching occurs upon detection. That in turn implies that each detector causes a branching of some kind. But since the H detection or V detections can occur at different T2 times (these can be made arbitrarily different by path length adjustment), do the worlds so created only come into being at those arbitrary times? This seems paradoxical to me as well.

Of course, every interpretation contains an apparent paradox (or inconsistency) of some kind anyway. ----------------------

*You had said "one of the points of MWI is that the observer should have no special role" which created some confusion for me, as I don't know of any significant interpretation that assigns a special role to the observer's actions after a quantum measurement. We usually assume that the human observer has no special role, except perhaps to (freely) make a choice of measurement basis prior to a quantum measurement. I call this an observation (an observer making a measurement) and I don't personally distinguish the human (or other recording device) from the measurement itself, once that assumption is accepted. But I get that you and others might.

 

[PeterDonis]

1. At the time T a quantum measurement is made, there is a branching into 2 or more worlds, corresponding to the possible outcomes (and weighted in some fashion according to the quantum expectation). This process is deemed "deterministic" for some reason that escapes me

The reason is simple: because it's just unitary evolution, and unitary evolution is deterministic. The unitary evolution is already there in the basic math of QM: a "measurement" involves a unitary interaction that entangles the system to be measured with the measuring apparatus. That process is deterministic, and its deterministic result is an entangled wave function with multiple "branches", each one corresponding to a possible measurement outcome, i.e., each "branch" is a product of the measured system's state vector that corresponds to the measured outcome, and the measuring device's state vector that corresponds to the observation and recording of the measured outcome.

Note that the entangled state I've just described is there in every interpretation, because, as I said before, it's just what the basic math of QM gives you by unitary evolution as a result of the measurement interaction. The MWI just differs from other interpretations about what happens next: other interpretations have some form of "collapse", where only one of the multiple outcomes described by the entangled wave function I described above actually happens--only one of the branches of the entangled wave function is actually "real". MWI says they all happen; they're all "real".

the apparent "indeterminism" (what we see as a random outcome) is then a result of us finding ourselves in a particular branch and unable to see the alternative branches.

Yes, because the branches can never interact with each other. Until decoherence theory was developed, this had to be taken as an unexplained postulate of the MWI; but decoherence theory explains why the branches can't interact: because they're decohered.

2. Question: Let's say we have an HV polarizing beam splitter (PBS) with a 45 degree polarized photon passing through it at some time circa T1. The photon becomes polarized at H or V when passing through, is subsequently detected by one of 2 detectors place near the respective output ports H and V circa time T2. Can anyone tell me if the splitting occurs at T1, or is it at T2?

I would say that, according to the MWI, the "branching" happens at T2, because that is when the irreversible measurement interaction occurs; i.e., that is where the MWI has to say that both possible outcomes happen, each in its own "branch" of the entangled wave function, whereas other interpretations would say that only one of the two possible outcomes happens. This can't be said at T1 because what happens at T1 is reversible; you could recombine the two output beams of the PBS and recover the initial state (as in, for example, an Mach-Zehnder interferometer), and all interpretations would agree that there is only one possible outcome in that case.

4. The only other answer for the question then is to say the branching occurs upon detection. That in turn implies that each detector causes a branching of some kind. But since the H detection or V detections can occur at different T2 times (these can be made arbitrarily different by path length adjustment), do the worlds so created only come into being at those arbitrary times? This seems paradoxical to me as well.

I think the MWI would say that if there are two different T2's, say T2H and T2V, the "branching" would have to occur at the first one, because once the first interaction has decohered, there is no longer any possibility of recombining the two beams or having them interact in any way.

But I suppose an MWI advocate might also say that the "branching" is not a single event that can be assigned a definite time; it's just whatever happens to the wave function by unitary evolution, and unitary evolution has no problem in accommodating different path lengths to the H and V detectors and corresponding interactions at different times. In other words, on this view, the question "when does the branching occur?" is an artifact of our human intuition that says the "worlds" have to "branch" at some definite time; there actually is no such thing in the wave function itself, there's just unitary evolution.

 

[PeterDonis]

I don't know of any significant interpretation that assigns a special role to the observer's actions after a quantum measurement. We usually assume that the human observer has no special role, except perhaps to (freely) make a choice of measurement basis prior to a quantum measurement. I call this an observation (an observer making a measurement) and I don't personally distinguish the human (or other recording device) from the measurement itself, once that assumption is accepted.

I agree with this. I think decoherence theory makes it clear why we don't need to assign any special role to human observers: because lots of things besides human perceptual apparatus and brains can cause decoherence, and in pretty much any experimental situation, by the time the effects of a measurement outcome have become perceptible by humans, decoherence has already occurred somewhere earlier in the process.

 

[gentzen]

Let's say we have an HV polarizing beam splitter (PBS) with a 45 degree polarized photon passing through it at some time circa T1. The photon becomes polarized at H or V when passing through, is subsequently detected by one of 2 detectors place near the respective output ports H and V circa time T2. Can anyone tell me if the splitting occurs at T1, or is it at T2?

Today, the answer would be that the splitting occurs at T2, or more precisely at both T2H and T2V (in PeterDonis' terminology), because that is where the amplification of the quantum events and the decoherence happens. But Everett would have had no serious issues with branching at T1 either, because his branches can still interfere, at least in principle. However, if they do, then any permanent records distinguishing those interfering branches will be gone. (My intention here is to reinforce the point that the branches in MWI should not be interpreted to act as hidden variables.)

Great, I think we're on the same page*.

Thanks for confirming this.

*You had said "..." which created some confusion for me, as I don't know of any significant interpretation that assigns a special role to the observer's actions after a quantum measurement.

The observer plays a special role in epistemic interpretations. For example, Heisenberg defended the interpretation of the quantum state as epistemic, especially the update of the quantum state after a measurement.

 

[PeterDonis]

Everett would have had no serious issues with branching at T1 either, because his branches can still interfere, at least in principle. However, if they do, then any permanent records distinguishing those interfering branches will be gone.

Everett wrote his thesis before decoherence theory was developed, though, so he didn't consider the fact that being able to make branches interfere like this, and consequently destroying any permanent records distinguishing them, amounts to being able to undo decoherence. I don't think the MWI in general is committed to any such claim, although particular MWI proponents might be. I think it's consistent with the MWI to acknowledge that decoherence is irreversible, so branches can't interfere once decoherence has occurred (and in fact the occurrence of decoherence is what defines "branches" in the first place).

 

[DrChinese]

Everett wrote his thesis before decoherence theory was developed, though, so he didn't consider the fact that being able to make branches interfere like this, and consequently destroying any permanent records distinguishing them, amounts to being able to undo decoherence. I don't think the MWI in general is committed to any such claim, although particular MWI proponents might be. I think it's consistent with the MWI to acknowledge that decoherence is irreversible, so branches can't interfere once decoherence has occurred (and in fact the occurrence of decoherence is what defines "branches" in the first place).

1. Clearly, there are a) times when NO decoherence has occurred. Such "branches" can *theoretically) be recombined to create the original superposition. Further, such branches can interfere with each other. There are b) other times when there has been SOME (partial) decoherence, i.e. some branches have been irreversibly measured (and therefore decohered). And finally, there are c) times when ALL branches have been observed (full decoherence) and no subsequent recombination/reversal is possible. So MWI must follow all of these scenarios to match the statistical predictions of QM.

2. Further, when any branch has split irreversibly: any observed "fact" (such as measuring spin=up on an entangled particle) must cause all related "facts" to be globally updated in that world so they match. In other words, a distant entangled particle must take on the appropriate value as to match what was measured. There may not be "spooky action at a distance", but the newly created world is globally consistent - the result could not be considered "local" by any reasonable standard.

3. And somehow, when new worlds are created due to an irreversible measurement on a system "here" (which we agree results from a decohering branching): all other "pending" (i.e. those that are still reversible) branches - related to many many quantum systems occurring elsewhere in the universe - are not affected. Their pending branching continues unaffected. New worlds are being created constantly that feature "pending branching in process", for lack of a better term.

I admit this is really no more complex than any quantum interpretation, as they all must have something like this somewhere. But it certainly seems as if whatever we gained from following MWI disappears quickly once we think through some of the more complex quantum scenarios (such as erasing a quantum outcome). I can't even how I would begin to describe delayed choice quantum teleportation (entanglement swapping) using MWI. My brain is exploding just trying to picture all the permutations.

 

[PeterDonis]

when any branch has split irreversibly: any observed "fact" (such as measuring spin=up on an entangled particle) must cause all related "facts" to be globally updated in that world so they match.

This automatically happens by unitary evolution. Nothing needs to be added to standard QM to make it happen. The overall wave function already contains all the information about which facts go with which other facts in which branches. The entanglement takes care of all that automatically.

the result could not be considered "local" by any reasonable standard.

Agreed. The entangled wave function itself is nonlocal in the obvious sense that a single wave function includes terms describing different spatial locations for the quantum subsystems it includes (for example. the H and V output beams of the PBS in your scenario). MWI proponents sometimes try to handwave this away by concocting other definitions of "local" according to which the MWI is local, but I don't think any of those alternative definitions are reasonable.

when new worlds are created due to an irreversible measurement on a system "here" (which we agree results from a decohering branching)

I generally do not like the phrase "new worlds are created" because unitary evolution doesn't create or destroy anything. Strictly speaking, the individual quantum systems included in the overall wave function do not have any definite state at all, since they are entangled; only the overall wave function is definite. What the MWI has to add is the claim that, even though no individual quantum system is in a definite state by itself, each "branch" of the entangled wave function still somehow corresponds to a definite "world" in which a particular definite result of a measurement is irreversibly recorded, which implies that particular individual quantum systems are in definite states that correspond to the recorded outcome. Everett used the term "relative state" to describe this claim.

all other "pending" (i.e. those that are still reversible) branches - related to many many quantum systems occurring elsewhere in the universe - are not affected.

The reason for this is simple: the overall wave function is a product state with respect to the two groups of subsystems (the ones that have "branched" and the ones that haven't). Unitary evolution of product states is decoupled: each individual subsystem (or group of subsystems) is not affected by whatever happens to the other.

 

[gentzen]

2. Further, when any branch has split irreversibly: any observed "fact" (such as measuring spin=up on an entangled particle) must cause all related "facts" to be globally updated in that world so they match. In other words, a distant entangled particle must take on the appropriate value as to match what was measured. There may not be "spooky action at a distance", but the newly created world is globally consistent - the result could not be considered "local" by any reasonable standard.

MWI is formulated with a global time variable. On the one hand, this allows to let unitary evolution automatically take care of all that complicated updating. On the other hand, this implies the use of a fixed inertial system for the description, together with the notion of simulataneity that comes with that fixed system. This doesn't feel very "local", but it is nevertheless "more local" than Bohmian mechanics.

Trying to eliminate the global time variable from MWI would be misguided from my POV, because we mean the existing interpretation with all its proponents and publications when we talk of MWI, not some hypothetical not yet existing construct.

 

[collinsmark]

MWI - says that when the cups meet the bar and are overturned, the universe branches off into 5 "worlds" and in each world the ball is revealed to be under a different cup.

At least 5 branches ("worlds"). Five branches will pedagogically suffice for the special case where the quantum experiment is prepared with the probabilities all being equal, with a value of exactly 0.2 (i.e., 20%) for each and every cup.

But if the experiment is prepared with different probabilities, with arbitrarily large precision, the number of branches can be vast.

 

[PeterDonis]

if the experiment is prepared with different probabilities, with arbitrarily large precision, the number of branches can be vast.

No, this is not correct. For an experiment with discrete outcomes, there is one branch per outcome, no matter how the probabilities are distributed between the outcomes. In fact, one of the issues with the MWI is how to assign meaning to the probabilities since they don't affect the branch count.

 

[Morbert]

No, this is not correct. For an experiment with discrete outcomes, there is one branch per outcome, no matter how the probabilities are distributed between the outcomes. In fact, one of the issues with the MWI is how to assign meaning to the probabilities since they don't affect the branch count.

The reduction of branches with unequal probabilities to coarse-grainings of many more branches with equal probability is one of the proposed solutions to the MWI probability problem. See Zurek's envariance paper and his discussion of no-collapse interpretations ( https://arxiv.org/pdf/quant-ph/0405161.pdf )

 

[DrChinese]

No, this is not correct. For an experiment with discrete outcomes, there is one branch per outcome, no matter how the probabilities are distributed between the outcomes. In fact, one of the issues with the MWI is how to assign meaning to the probabilities since they don't affect the branch count.

When there are outcomes that have an essentially continuous range of values (momentum, position, energy, etc.), there is one per possible value? That doesn't seem reasonable either (since large values would have a high probability even when the average outcome is lesser).

 

[PeterDonis]

When there are outcomes that have an essentially continuous range of values (momentum, position, energy, etc.), there is one per possible value?

A continuous spectrum does raise issues that aren't raised by a discrete spectrum. That's why I explicitly said discrete outcomes in my post, since that was the case the post I was responding to appeared to be discussing.

For a continuous spectrum, if such is taken literally, there would have to be a continuous infinity of branches at each measurement. From what I have read, while some MWI proponents are willing to accept that, others are hesitant about it.

Another option would be to use the unavoidable finite resolution of any real measurement to say that there are no actual continuous measurements, and any actual measurement has effectively a discrete spectrum, although the number of possible results might be very large (for example, a position measurement over a possible range of 1 meter with, say, 1 nanometer resolution would have a billion possible results). On this view, the number of branches on any real measurement is a finite integer, though possibly a very large one.

 

[Morbert]

As an aside, @Lynch101 a heads up: Never look up the three-box paradox

 

[Lynch101]

These are not strictly interpretational questions, but more a prelude to such.

In the ball and cup analogy, there is a probability associated with each cup and the cups move towards the "measurement device"" (bar). Is that aspect representative of the predictions of QM? My understanding is that, for any given position in the experimental set-up, there is an associated probability of measuring the system at that position, at a given time. Is this the case? Do the probabilities for the cups change [within individual runs of the experiment] as the cup advances, or do they remain the same?

Also, let's say for the first run of the experiment, are the associated probabilities the same for each position or is it more likely to be measured in some positions than others? E.g. say, is it more likely to be measured at cup A than cup E?

After the first measurement, do the probabilities for each cup change?

 

[Nugatory]

Is that aspect representative of the predictions of QM?

Not very much.
It’s seldom effective to use macroscopic objects in forming a mental model of a quantum system. One difficulty is that the ball is not a simple point particle that we can drop into Schrodinger’s equation and solve for $\psi(x,t)$, the way we would if this were a five-well problem with Hamiltonian $p^2/2m+V(x)$ and initial state being a superposition of equal amplitude in each well. Instead, the ball is something like $10^{23}$ individual particles interacting with one another as well as the potential walls formed by the cups.

Even if we could prepare the system in a pure state superposition with equal probability of the multi-particle ball being in each cup, the forward evolution of that system won’t be anything like the idealized point particle solution. It will very quickly decohere into a classical ball in one cup and we just don’t know which; this state provides no insight into the predictions of QM.

 

[Morbert]

These are not strictly interpretational questions, but more a prelude to such.

In the ball and cup analogy, there is a probability associated with each cup and the cups move towards the "measurement device"" (bar). Is that aspect representative of the predictions of QM? My understanding is that, for any given position in the experimental set-up, there is an associated probability of measuring the system at that position, at a given time. Is this the case? Do the probabilities for the cups change [within individual runs of the experiment] as the cup advances, or do they remain the same?

Also, let's say for the first run of the experiment, are the associated probabilities the same for each position or is it more likely to be measured in some positions than others? E.g. say, is it more likely to be measured at cup A than cup E?

After the first measurement, do the probabilities for each cup change?

There isn't any particular distinction between classical and quantum physics in this scenario. The probabilities will change as the cups are turned over, in the sense that each time a cup is turned over, new information is made available to the observer which can be used to update probabilities. E.g. The probability that the ball will be found under cup 3 is $\mathbf{Tr}\left[\rho\Pi_3\right]$ at the start of the experiment. If cup one is overturned and turns up empty, the probability changes to $\mathbf{Tr}\left[\not\Pi_1\rho\not\Pi_1\Pi_3\right]$

 

[Lynch101]

Not very much.
It’s seldom effective to use macroscopic objects in forming a mental model of a quantum system. One difficulty is that the ball is not a simple point particle that we can drop into Schrodinger’s equation and solve for $\psi(x,t)$, the way we would if this were a five-well problem with Hamiltonian $p^2/2m+V(x)$ and initial state being a superposition of equal amplitude in each well. Instead, the ball is something like $10^{23}$ individual particles interacting with one another as well as the potential walls formed by the cups.

Even if we could prepare the system in a pure state superposition with equal probability of the multi-particle ball being in each cup, the forward evolution of that system won’t be anything like the idealized point particle solution. It will very quickly decohere into a classical ball in one cup and we just don’t know which; this state provides no insight into the predictions of QM.

Thanks for that clarification Nugatory. I should probably have specified that the thought experiment wasn't meant to be used to infer anything at the macroscopic level. The ball in this case is meant to represent the idealised point particle upon measurement, not to be treated as a multi-particle system. The cups are just meant to represent the fact that there is a probability associated with each position and that we don't know the position of the particle until it is measured.

The question I'm asking re: QM predictions, might be thought of in terms of a "map of predictions". I'm wondering if QM makes probabilistic predictions only for where the particle will be measured on the screen or if it makes predictions for the space between the preparation device and the screen.

My understanding is that it does give us this "map of predictions" and not just predictions at the screen. It would also make sense because we can just imagine moving the screen forward, towards the preparation device. I just wanted to check that my understanding of this is correct or if there is some nuance that I'm not picking up on.

 

[Lynch101]

There isn't any particular distinction between classical and quantum physics in this scenario. The probabilities will change as the cups are turned over, in the sense that each time a cup is turned over, new information is made available to the observer which can be used to update probabilities. E.g. The probability that the ball will be found under cup 3 is $\mathbf{Tr}\left[\rho\Pi_3\right]$ at the start of the experiment. If cup one is overturned and turns up empty, the probability changes to $\mathbf{Tr}\left[\not\Pi_1\rho\not\Pi_1\Pi_3\right]$

Thanks Morbert. I'm not trying to get at the idea that individual cups are overturned during individual runs of the experiment, rather whether the probability is the same for each cup, for each individual run of the experiment.

For example, let's say it's the first run of the experiment and there are 5* cups. Is the probability for the position of the particle the same for each cup (20%), or in that individual run of the experiment (with no cups overturned) does QM predict different probabilities for different cups? E.g.
Cup 1 = 20%
Cup 2= 30%
Cup 3 = 10%
Cup 4 = 15%
Cup 5 = 25%

*The cups represent the possible positions the particle could be measured in and the number can be scaled to represent the number of possible positions predicted by QM.

 

[Morbert]

For example, let's say it's the first run of the experiment and there are 5* cups. Is the probability for the position of the particle the same for each cup (20%), or in that individual run of the experiment (with no cups overturned) does QM predict different probabilities for different cups? E.g.
Cup 1 = 20%
Cup 2= 30%
Cup 3 = 10%
Cup 4 = 15%
Cup 5 = 25%

That depends on the preparation. An experimenter could prepare the experiment such that the probability for each cup is 20% or they could prepare the experiment such that the probability is different for each cup.

 

[Lynch101]

That depends on the preparation. An experimenter could prepare the experiment such that the probability for each cup is 20% or they could prepare the experiment such that the probability is different for each cup.

Ah right. Thank you for the clarification.

Do the different interpretations propose different reasons for the differences in probability, for each position? I would be inclined to presume yes, but presumptuousness is probably ill advised, particularly with regard to QM

In relation to the idea of a "map of probabilities", I would distinguish this from say a "wall of probabilities", where a "wall of probabilities" would represent the predictions on the measurement device/screen only. A "map of probabilities" would include the space between the preparation device and the screen which register the particles. Am I right in thinking that QM predicts such a "map of probabilities"?