Is the Born rule truly independent from other postulates in quantum mechanics?

[SCHROEDERFPM]

TL;DR Summary

I posted this as a comment on another thread but thought it might need its own.

My thought is, If a person opened the box, observed the cat, and then had their memory completely erased or died shortly afterwards would the cat still have been "observed"?

Also is the cat a good reference?
The cat is an observer and knows wether it is dead or alive.
If it knows it's not dead it's definitely not, and if it's dead it can't observe.
Does this not make the cat DEFINITELY one or the other?

Refer to ↑

 

[PeroK]

My thought is, If a person opened the box, observed the cat, and then had their memory completely erased or died shortly afterwards would the cat still have been "observed"?

Yes. It doesn't even need anyone to observe the cat.

Also is the cat a good reference?
The cat is an observer and knows wether it is dead or alive.

That's a good point. Replace the cat with a human being and they can tell you what happened.

If it knows it's not dead it's definitely not, and if it's dead it can't observe.
Does this not make the cat DEFINITELY one or the other?

If by "definitely one or the other", you mean a scenario involving a classical probability that involves an unknown state (rather than a quantum superposition of states), then yes.

The real point of this thought experiment is not to cast doubt on a classical scenario, but to explain where the purely QM reasoning goes wrong. We know all these things you are saying. But, QM appears to imply something different. Is QM wrong? And, if not, how do we explain the classical behaviour of a cat in this experiment.

 

[SCHROEDERFPM]

You're right, I shouldn't say anything is definitive.
Only Siths deal in absolutes, right?

I mean regarding a classical scenario yes, but yes, replace the cat with a human in a solid crate that won't shake and is sound proofed... (Hear me out)... The observer cannot observe, the human inside however is either aware they are alive or dead. In which case they wouldn't be in a super position they then would be one or the other at any given time... ?

 

[SCHROEDERFPM]

Starting to sound like a murderer but back to the guy dying.

I understood, I thought, that quantum experiments conducted have varying results when the variable was wether or not the experiment was observed.

If this is the case then if the observer was then rid of their memory shortly or immediately after. Say if they witnessed the light barrier test only it was radiation that fried their memory in the process. Would the results still have been "observed".

 

[PeroK]

You're right, I shouldn't say anything is definitive.
Only Siths deal in absolutes, right?

I mean regarding a classical scenario yes, but yes, replace the cat with a human in a solid crate that won't shake and is sound proofed... (Hear me out)... The observer cannot observe, the human inside however is either aware they are alive or dead. In which case they wouldn't be in a super position they then would be one or the other at any given time... ?

Schrodinger began to have doubts about QM, even though he contributed to the initial development of the theory. The point of the experiment was to show that QM couldn't possibly be a complete theory. Not so much that it was fundamentally wrong, but that it had missing elements. And, in this case, QM alone could not predict the behaviour of classical objects properly.

It is often misinterpreted as "QM says the cat is in a superposition of live and dead, so it must be true". This is what is claimed by people who do not understand the point of the experiment. It always amazes me when people learn about QM and assume that macroscopic objects must behave like elementary particles - only that we have somehow never noticed it!

The modern answer is that decoherence transforms quantum superpositions for large macroscopic objects (which have trillions of degrees of freedom) into definite either/or states. No one should seriously believe that just before the box is opened the cat is in a quantum superposition of alive and dead.

 

[PeroK]

Starting to sound like a murderer but back to the guy dying.

I understood, I thought, that quantum experiments conducted have varying results when the variable was wether or not the experiment was observed.

If this is the case then if the observer was then rid of their memory shortly or immediately after. Say if they witnessed the light barrier test only it was radiation that fried their memory in the process. Would the results still have been "observed".

It's not to do with observation (conscious or otherwise), it's whether irreversible interaction with a complex environment takes place.

 

[anuttarasammyak]

If this CAT BOX is a qubit in quantum computer, your Unexpected Observation with unknown result state is counted as an error and it harms the calculation process thereafter.

 

[SCHROEDERFPM]

The modern answer is that decoherence transforms quantum superpositions for large macroscopic objects (which have trillions of degrees of freedom) into definite either/or states. No one should seriously believe that just before the box is opened the cat is in a quantum superposition of alive and dead.

So rather than a "conscious observer", do particles interacting with each "observe" each other, defining them to either 1 or 0?

 

[PeroK]

So rather than a "conscious observer", do particles interacting with each "observe" each other, defining them to either 1 or 0?

No. You can't go to the opposite extreme and say that elementary particles behave like cats! The key is the complexity of the system and what that does to the mathematical calculations of QM. This is decoherence. There is a popular science book dedicated to explaining this called Where Does the Weirdness Go?

https://www.amazon.co.uk/Where-Does-Weirdness-Go-Mechanics/dp/0465067867

The bottom line is that when two elementary particles interact, they produce a superposition of the possible outcomes. But, when one of them interacts with the complex environment or a macroscopic measuring device, irreversible changes take place and we have a definite outcome. Think about a device to detect a photon or electron. The elemetary particle has a tiny amount of energy. In order to register something that we can see, the device must transform and amplify the signal in some way. It's this process of amplification that makes the act of measurement irreversible.

These ideas were first worked out by Niels Bohr in the 1920s, which led eventually to the concept of decoherence.

 

[kith]

As @PeroK has noted, the crucial question is whether (irreversible) decoherence occurs in the system (see my post in the other thread for an elaboration on terminology).

Since decoherence is something measurable, we aren't limited to talk about vague and philosophical notions but can simply take a look at some actual research.

From the introduction:

In this Letter we investigate the decoherence of molecular matter waves. We change the internal temperature
of the molecules in a controlled way before they enter a near-field interferometer, and observe the corresponding reduction of the interference contrast. The idea behind this effort is to demonstrate a most fundamental decoherence mechanism that we encounter in the macroscopic world: All large objects, but also molecules of sufficient complexity, are able to store energy and to interact with their environment via thermal emission of photons. It is generally believed that warm macroscopic bodies emit far too many photons to allow the observation of de Broglie interferences, whereas individual atoms or molecules can be sufficiently well isolated to exhibit their quantum nature. However, there must be a transition region between these two limiting cases. Interestingly, as we show in this study, C70 fullerene molecules have just the right amount of complexity to exhibit perfect quantum interference in our experiments [13] at temperatures below 1000 K, and to gradually lose all their quantum behaviour when the internal temperature is increased up to 3000 K

 

[kered rettop]

That depends what you mean by "erased". There is such a thing as quantum erasure, where the system, including the observer, is restored to the same quantum state as it was before the observation. AFAIK, this has only ever been done with very minimalist observations: a single photon carrying one (qu)bit of information, never actually being resolved by a detector. This does indeed reverse the observation.

But if you just mean a conventional wipe, then no, the information that the observer received, or indeed, that they could have received if they hadn't died first, is still in the environment, hopelessly scrambled, but there.

As to whether self-observation leads to definite information, that depends whether you believe that the wave function collapses. If it collapses, then of course, the state of the cat becomes definite. However interpretations such as MWI dispense with collapse. The cat-plus-everything around it remains in a superposition; a living cat world and a dead cat world. In either world, the cat experiences itself as definitely alive or definitely dead (so to speak!) But presumably its experiences are limited to the world in which it finds itself. (Any theory which allows sensory access to multiple worlds must be wrong since we don't, in fact, experience more than one at a time.)

 

[kered rettop]

Schrodinger began to have doubts about QM, even though he contributed to the initial development of the theory. The point of the experiment was to show that QM couldn't possibly be a complete theory. Not so much that it was fundamentally wrong, but that it had missing elements. And, in this case, QM alone could not predict the behaviour of classical objects properly.

It is often misinterpreted as "QM says the cat is in a superposition of live and dead, so it must be true". This is what is claimed by people who do not understand the point of the experiment. It always amazes me when people learn about QM and assume that macroscopic objects must behave like elementary particles - only that we have somehow never noticed it!

The modern answer is that decoherence transforms quantum superpositions for large macroscopic objects (which have trillions of degrees of freedom) into definite either/or states. No one should seriously believe that just before the box is opened the cat is in a quantum superposition of alive and dead.

That's not quite true. The modern answer is that decoherence transforms quantum superpositions for large macroscopic objects into superpositions of "definite either/or" states. In other words, the superposition remains even though you can resolve it into states A and B, the much-desired "definite states". But the complete state is still a superposition and therefore indefinite.

So I'm afraid I do believe that according to standard quantum mechanics, the cat is in a superposition of alive and dead (until linearity is suspended). To be more technical, there is an entanglement comprising a dead-cat system superposed with a living-cat system.

Whether it collapses at the point of opening the box is another matter. Schroedinger only mentioned opening the box as a concession to the prevailing idea that observation results in definite outcomes. The results of decoherence theory actually support the idea of superposed data-states, or, as they are commonly called, "worlds": the worlds of MWI.

Of course I find the idea of "all those other worlds" as disconcerting as the next man, but I have to admit, nature seems to pay scant attention to what I think she should be like.

 

[PeroK]

That's not quite true. The modern answer is that decoherence transforms quantum superpositions for large macroscopic objects into superpositions of "definite either/or" states. In other words, the superposition remains even though you can resolve it into states A and B, the much-desired "definite states". But the complete state is still a superposition and therefore indefinite.

The question is not whether there are superpositions, the question is whether the total state of the cat approximates a mixed state - in particular, in terms of the property of "alive" or "dead". The first problem with SC is that it confuses a simple either/or QM property, with a complex macroscopic property. There is no well-defined live state for a cat. In fact, it's not easy to say why a poison would kill a cat and what that even means in purely QM terms. Even cell biology is inexplicable in terms of QM.

So I'm afraid I do believe that according to standard quantum mechanics, the cat is in a superposition of alive and dead (until linearity is suspended). To be more technical, there is an entanglement comprising a dead-cat system superposed with a living-cat system.

My understanding of decoherence is that the (almost unlimited) number of states that represent a superposition of live and dead have effectively zero probability amplitude. In that sense, although they exist, they are never outcomes of a measurement. In other words, there is a vanishingly small probability of observing a state that represents a cat that is, literally, half alive and half dead.

Of course I find the idea of "all those other worlds" as disconcerting as the next man, but I have to admit, nature seems to pay scant attention to what I think she should be like.

I don't find MWI disconcerting at all. The issue with MWI is the probabilities and Born's rule. MWI is too simple, IMO.

 

[PeterDonis]

I'm afraid I do believe that according to standard quantum mechanics, the cat is in a superposition of alive and dead (until linearity is suspended).

My understanding of decoherence is that the (almost unlimited) number of states that represent a superposition of live and dead have effectively zero probability amplitude.

I think the two of you are talking past each other. Perhaps this will help: decoherence means that the "alive" and "dead" terms in the wave function, while they are still both there, cannot interfere with each other; in other words, there is no way to reverse the interaction that created the "alive" and "dead" terms in the wave function, the way that, for example, you can take the two terms of the wave function of a qubit that has just had its spin measured, and recombine them by reversing the spin measurement (putting the two output arms of one spin measurement device into the input arms of a second spin measurement device to recombine them so only one output arm of the second device will have a signal).

In other words, both of you are making true statements with the appropriate intepretations of the terms. The cat is in a superposition of "alive" and "dead" (an entangled superposition, in which the "alive" and "dead" states of the cat are entangled with the "not decayed" and "decayed" states of the radioactive atom), but there are no terms in the wave function for states that are linear combinations of "alive" and "dead" (corresponding to, for example, a spin-X up qubit's state written in the spin-Z basis).

 

[kered rettop]

We may not be able to write an explicit dead/alive operator, but it is reasonable to assume that a surrogate can be used. With a minor modification to the apparatus, a simple positional measurement will determine whether the cat is trying to get out of the box or is lying there curled up dead...

I think you have misunderstood MWI a little. I was quite surprised to discover how simple the derivation of the Born Rule actually is. I don't think Everett got very far with it, but it all comes together with decoherence.

The quick and dirty derivation considers the interaction of a particle in a superposition with the complicated environment. The latter is one vast superposition. The result is therefore an enormous superposition of microstates (entanglements), which fall into two categories: those which appear, both externally and to their inhabitants, as dead-cat worlds and those that appear as living-cat worlds.

The Born Rule then arises by consideration of the statistics. The total vector amplitude of all the dead-cat worlds must equal that of the original dead-cat component. However, these are vectors, with random phase angles derived from the environment. Therefore these micro-vectors add up as the square root of their number - I'm using a discrete model. Simultaneously the observer in any particular world does not know whether they are in a dead-cat world until they look. So, with a little justification for assigning equal probabilities to otherwise equivalent worlds, the probability of being in a dead-cat world is simply proportional to the number of such worlds. Which from the previous is the total vector amplitude squared. Probability is proportional to the original amplitude squared. That's the Born Rule.

I don't properly understand the preferred basis issue/problem nor its relationship to those vanishing off-diagonal terms that you allude to. So I won't muddy the waters any more :)

I think the two of you are talking past each other. Perhaps this will help: decoherence means that the "alive" and "dead" terms in the wave function, while they are still both there, cannot interfere with each other; in other words, there is no way to reverse the interaction that created the "alive" and "dead" terms in the wave function, the way that, for example, you can take the two terms of the wave function of a qubit that has just had its spin measured, and recombine them by reversing the spin measurement (putting the two output arms of one spin measurement device into the input arms of a second spin measurement device to recombine them so only one output arm of the second device will have a signal).

In other words, both of you are making true statements with the appropriate intepretations of the terms. The cat is in a superposition of "alive" and "dead" (an entangled superposition, in which the "alive" and "dead" states of the cat are entangled with the "not decayed" and "decayed" states of the radioactive atom), but there are no terms in the wave function for states that are linear combinations of "alive" and "dead" (corresponding to, for example, a spin-X up qubit's state written in the spin-Z basis).

Well, thanks for that. But my mental block is "why are there no such terms?" And can you spell out the connection between the absence of such terms and the irreversibility of a measurement? Thanks.

 

[PeterDonis]

my mental block is "why are there no such terms?"

There are no such terms after decoherence has taken place. Decoherence eliminates them. The full theory of how that happens is fairly complex and should be learned from the literature, but a quick heuristic description is that decoherence spreads the entanglement out among a huge number of untrackable degrees of freedom in the environment, and that spreading of entanglement drives the interference terms to zero.

 

[PeterDonis]

I was quite surprised to discover how simple the derivation of the Born Rule actually is.

Please give a reference for your preferred derivation of the Born Rule in the MWI. And please be aware that there is no reference in the literature that is generally accepted as being a valid derivation, not even by all MWI proponents, much less by MWI skeptics.

 

[PeterDonis]

Probability

The MWI is deterministic: there is no probability anywhere. That is one of the key obstacles to any derivation of the Born Rule in the context of the MWI. The issue is not showing that the relative amplitudes of the branches are the "right" ones--that is obvious and is in fact independent of any interpretation, since it is a simple calculation using the basic math of QM. (Collapse interpretations end up eliminating all but one of the branches, but up to that point they agree with no collapse interpretations on all the calculations, including the relative amplitudes of the branches.) The issue is showing how the relative amplitudes correspond to probabilities of different outcomes in an interpretation where there is no collapse, all possible outcomes happen, and everything is deterministic with no probabilities anywhere.

 

[kered rettop]

I don't understand the problem. Obviously MWI cannot create randomness where everything is deterministic but probability does not depend on randomness, it depends only on ignorance. This is so in the classical case, where the outcome from tossing a coin is actually deterministic, but we can apply probability theory because we can't actually predict it. Ignorance-based probability can therefore emerge in each branch/world even though random-event-based probability doesn't exist anywhere. What am I missing?

 

[kered rettop]

Please give a reference for your preferred derivation of the Born Rule in the MWI. And please be aware that there is no reference in the literature that is generally accepted as being a valid derivation, not even by all MWI proponents, much less by MWI skeptics.

Did you recognise the derivation that I outlined? It's just a Schmidt decomposition - of the entanglement between the system and the environment, followed by "counting microstates". I can't give a definitive reference, I'm afraid. However if you do recognise it, perhaps you would be so kind as to say where the flaw is? Unless, of course there is no serious flaw, in which case the lack of general acceptance would appear to be irrational. But if you have never seen such a thing or don't recognise it from my sloppy summary, I am quite happy to step through it. If it's not straying too far off-topic.

 

[PeterDonis]

probability does not depend on randomness, it depends only on ignorance

Probability depends on there being more than one possible outcome, only one of which actually happens. That isn't the case for the MWI. Yes, "ignorance" comes into play with probability because we don't know which of the possible outcomes will actually occur; that can be the case if the underlying dynamics is deterministic--i.e., the "possibility" in this case is epistemic. But it can't be the case if all outcomes actually occur; if that is the case, there is no such thing as "possibility" even in the epistemic sense.

Did you recognise the derivation that I outlined? It's just a Schmidt decomposition

Sorry, but your unsupported statement is not sufficient here.

I can't give a definitive reference, I'm afraid.

Then you should not be making such claims as if they were established fact, at least not here. If you really believe your derivation is correct, publish it in a peer-reviewed paper and see what the rest of the physics community says. PF is not the place to try to make such claims if they are not already in the peer-reviewed literature.

 

[PeroK]

I don't understand the problem. Obviously MWI cannot create randomness where everything is deterministic but probability does not depend on randomness, it depends only on ignorance. This is so in the classical case, where the outcome from tossing a coin is actually deterministic, but we can apply probability theory because we can't actually predict it. Ignorance-based probability can therefore emerge in each branch/world even though random-event-based probability doesn't exist anywhere. What am I missing?

I've never believed this whole argument. There's a difference between ignorance and unknowability. Even without QM the ability to pin down the position and momentum of every elementary particle in the universe to infinite precision must be impossible. There will always be uncertainty, randomness and probabilities. Especially when you consider the instability of complex dynamic system under small variations in initial conditions.

 

[kered rettop]

It is not my derivation, it's Zurek's, that much I do know.

 

[Nugatory]

It is not my derivation, it's Zurek's, that much I do know.

Do you have a link or other citation?

 

[kered rettop]

Do you have a link or other citation?

Well, I've just found this post on Quora by Michael Price which a) summarises the matter and b) cites a handful of papers, no doubt including the original one by Zurek.

https://www.quora.com/How-does-the-...ding-to-the-Born-rule/answer/Michael-Price-29

 

[kith]

Did you recognise the derivation that I outlined? It's just a Schmidt decomposition - of the entanglement between the system and the environment, followed by "counting microstates". I can't give a definitive reference, I'm afraid. However if you do recognise it, perhaps you would be so kind as to say where the flaw is?

I'm reluctant to comment because you really should track down a reference for the precise argument as a basis for discussion. If you think it is from Zurek why don't you look through his papers?

So just a quick pointer to one critique of Born rule derivations which is very general: the postulates of QM including the Born rule are usually used to motivate the usage of basic structures in composite systems like the tensor product. Some authors explicitly mark the tensor product for composite systems as a postulate of its own but it seems to be rigorously derivable from the other postulates (including the Born rule). In any case, if it isn't independent from the Born rule there's a big danger of circularity when using it to derive the Born rule. At least, one has to be very careful which is precisely why we need the actual argument.

 

[kered rettop]

I'm reluctant to comment because you really should track down a reference for the precise argument as a basis for discussion. If you think it is from Zurek why don't you look through his papers?

So just a quick pointer to one critique of Born rule derivations which is very general: the postulates of QM including the Born rule are usually used to motivate the usage of basic structures in composite systems like the tensor product. Some authors explicitly mark the tensor product for composite systems as a postulate of its own but it seems to be rigorously derivable from the other postulates (including the Born rule). In any case, if it isn't independent from the Born rule there's a big danger of circularity when using it to derive the Born rule. At least, one has to be very careful which is precisely why we need the actual argument.

Well, I'm glad you have commented because you raise an important point. The argument that I outlined, (having described it as "quick and dirty") does refer to vector amplitudes. It's a world-counting argument so the way that we add states is crucial. If it depended on the Born Rule, then the argument would be disastrously circular.

So OK, on a quick check, the paper cited by Price:
https://www.quora.com/How-does-the-...ding-to-the-Born-rule/answer/Michael-Price-29

seems to cover everything , even if it's not the first paper by Zurek on the subject:
https://arxiv.org/pdf/0707.2832v1.pdf

Right at the beginning Zurek states that the first postulate of QM is
"(i) State of a quantum system is represented by a vector in its Hilbert space HS".
So the argument, as formulated, simply postulates the vector as a standard part of QM. Which is consistent with Postulates 1 & 2 of:
https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/

In addition, later on in the paper, under "Envariance", Zurek warns against using constructs like trace and reduced density matrices, which rely on the Born Rule "which we do not yet have".

Which is about as far as I can go with this discussion.