What has changed since the Copenhagen interpretation?

[PeterDonis]

a few weeks ago Smolin, Aaronson and some others commented on Woit's blog w.r.t. a question of Woit about interpretations

Please give a link to the specific blog post you are quoting from. Quotes should be sourced.

 

[kith]

I'm just reluctant to think read carefully or think too hard about a paper that claims Copenhagen is wrong and assumes that measurements are reversible, since it seems obviously wrong.

I don't think that you need to think about the paper in order to sort out the difference between your position and FR. It seems that the ordinary gedankenexperiment of Wigner's friend or even simpler experiments are enough. Let me spell out things in detail to clarify my own thoughts.

(1) In the context of Copenhagen, measurements are associated with macroscopic irreversibility. In my view, all of the terms "measurement", "macroscopic" and "irreversible" contain a notion of subjectivity because they can't be strictly derived from the physical theory alone - it doesn't tell us when certain approximations are good enough for the practical purposes of the observer. This isn't specific to QM. It is true also in classical mechanics, so it also applies to the classical apparatus in Copenhagen.

(2) In almost all cases, the subjectivity of (1) is inter-subjevtivity, i.e. if different observers look at the irreversible mark which has been left on a photo plate during an experiment, they all observe the same thing. So we can get away without worrying about subjevtivity and the only cases which force us to distinguish between subjectivity and inter-subjectivity are speculative Wigner's friend type experiments.

It's been a while since I have discussed things with you, so I am not sure about your position. Do you agree that irreversibility is subjective in the way I outlined above?

If you disagree, I don't see how to escape the conclusion of @DarMM's post #33, i.e. that collapse has an objectively real element. Although the wavefunction may be purely epistemic we get a physical change in the (classical) measurement apparatus. This would imply that the freedom in choosing the Heisenberg cut and the deferred measurement principle are tools for calculations but physically wrong.

If you agree, measurements may be reversed under very special circumstances and FR may have a point. Of course, it is very doubtful whether this is a point in practise. Their gedanken experiment seems to presuppose a step which corresponds to the quantum resurrection of dead cats which @stevendaryl outlined in the thread on FR. If this is true, Wigner's friend type experiments probably need to involve timescales which are wildly different in order to account for something like the Poincaré recurrance time.

I used to think that Wigner's friend type experiments are impossible and I still do but in the light of AI and quantum computers, I see a (slightly) growing need to justify this position. If the irreversibility of measurements just means that a mindbogglingly large time is needed to reverse it, an AI whose mind isn't boggled so easily because it works on a different timescale might make such an experiment feasible.

 

[stevendaryl]

Sorry, I didn't answer directly. I'm just reluctant to think read carefully or think too hard about a paper that claims Copenhagen is wrong and assumes that measurements are reversible, since it seems obviously wrong.

It seems obviously wrong that measurements are reversible, or it seems obviously wrong to call an interpretation with reversible measurements Copenhagen?

Measurements are certainly irreversible in a one-world ontology with objective collapse, but I'm not sure about a many-worlds ontology.

 

[Auto-Didact]

Please give a link to the specific blog post you are quoting from. Quotes should be sourced.

Sure, here.

The blogpost/comments even refers to some posts by @A. Neumaier on physicsoverflow. Arnold argued at length for his point of view on the matter here on PF in this thread which I then commented on after reading the thread, all of Woit's et al. comments and most of their references.

 

[A. Neumaier]

Sure, here.

Thanks for the pointer to this blog post by Peter Woit. I find his exposition of the question excellent. Still need to read through the many replies...

The blogpost/comments even refers to some posts by @A. Neumaier on physicsoverflow. Arnold argued at length for his point of view on the matter here on PF in this thread which I then commented on after reading the thread, all of Woit's et al. comments and most of their references.

I had missed your reply in the other thread, and continue the discussion of your contribution there in the other thread.

 

[Auto-Didact]

(1) In the context of Copenhagen, measurements are associated with macroscopic irreversibility. In my view, all of the terms "measurement", "macroscopic" and "irreversible" contain a notion of subjectivity because they can't be strictly derived from the physical theory alone - it doesn't tell us when certain approximations are good enough for the practical purposes of the observer. This isn't specific to QM. It is true also in classical mechanics, so it also applies to the classical apparatus in Copenhagen.

(2) In almost all cases, the subjectivity of (1) is inter-subjevtivity, i.e. if different observers look at the irreversible mark which has been left on a photo plate during an experiment, they all observe the same thing. So we can get away without worrying about subjevtivity and the only cases which force us to distinguish between subjectivity and inter-subjectivity are speculative Wigner's friend type experiments.

The argument made in (1) is incorrect for subtle non-trivial reasons:
I) You are taking the physical theory to essentially be the mathematical scheme of QM, i.e. unitary evolution or any of its equivalent mathematical formulations*.

II) The physical theory of QM however does not solely consist of unitary evolution, but also of the fact that it is validated by experiment; were it not validated by experiment it wouldn't be a physical theory but a mere physical hypothesis.

III) This validation of hypothesis by experiment requires an operationalization - literally, any operationalization whatsoever - of measurement. This operationalization isn't a part of the theorists physical hypothesis, it is part of the experimentalists empirical analysis; the mathematics of this empirical analysis need not be consistent with the work of the hypothesis of the theorist.

IV) The only possible way to view QM as a physical theory is therefore to include a) the theorists hypothesis and b) the experimentalists analysis. This conjoined mathematically self-inconsistent object is the actual physical theory, not just a) as many often try to argue - almost by a sleight of hand - by equivocating the work of the theorist (hypothesis) with the word 'physical theory' (hypothesis validated by experiment).

QM is in essence the only physical theory** which really suffers from this inconsistency between the theoreticians mathematics and the experimentalists analysis, but certainly not the only scientific theory: just step out of exact science and almost all scientific theories have this problem.

*: this argument is completely unchanged if you include the mathematical scheme of decoherence and other physical theories such as statistical mechanics as well

**: theories literally completely derived (not merely shown to be mathematically derivable) from QM like Koopman-von Neumann classical mechanics notwithstanding

I used to think that Wigner's friend type experiments are impossible and I still do but in the light of AI and quantum computers, I see a (slightly) growing need to justify this position. If the irreversibility of measurements just means that a mindbogglingly large time is needed to reverse it, an AI whose mind isn't boggled so easily because it works on a different timescale might make such an experiment feasible.

Agreed w.r.t. the need to settle these matters due to the rise of AI

 

[kith]

The argument made in (1) is incorrect for subtle non-trivial reasons [...]

You don't seem to disagree with my main point that irreversibility is somehow subjective. The purpose of my post is to understand @atyy's view and in this, I'm prefering brevity over philosophical nuance.

 

[DarMM]

Option 3 is immediately self-inconsistent, demonstrable through proof by contradiction:
I. If collapse exists, then $\psi$ exists (i.e. collapse implies ontology of $\psi$)

Option 3 is specifically the epistemic case where $\psi$ is (a generalisation of) a probability distribution. Collapse then is just (a generalisation of) Bayesian conditioning, so it doesn't require $\psi$ to be real.

Spekkens toy model has a nice example of this. QBism is a full blown interpretation that takes this view and accepts the restriction on subjective collapse interpretations (they must be single user) implied by the Frauchiger-Renner theorem.

 

[Auto-Didact]

You don't seem to disagree with my main point that irreversibility is somehow subjective. The purpose of my post is to understand @atyy's view and in this, I'm prefering brevity over philosophical nuance.

Okay.

I do disagree with irreversibility being subjective, but that disagreement is based in the philosophically nuanced argument I just made; namely that any physical theory actually capable of incorporating a) and b) in a mathematically self-consistent manner will necessarily be based on mathematics which can introduce irreversibility as a physically occurring phenomenon into the theory.

In other words, the consistent physical theory resulting from making a) and b) mathematically self-consistent, must actually be time irreversible - demonstrating that unitarity is only an approximation, not a fundamental principle - with collapse of $\psi$ occurring as a predicted objective dynamical phenomenon. This dynamical phenomenon is not predicted by a) i.e. standard QM (unitary evolution of the Schrodinger equation) because a) is only a limiting case of the correct equation.

 

[Auto-Didact]

Option 3 is specifically the epistemic case where $\psi$ is (a generalisation of) a probability distribution. Collapse then is just (a generalisation of) Bayesian conditioning, so it doesn't require $\psi$ to be real.

Spekkens toy model has a nice example of this. QBism is a full blown interpretation that takes this view and accepts the restriction on subjective collapse interpretations (they must be single user) implied by the Frauchiger-Renner theorem.

I know that, I used to be a QBist; w.r.t. probability theory I still am a staunch Bayesian. The problem in QBism is there that the word 'existence' doesn't necessarily mean existence out in the world, but also existence inside the mind; this makes the argument far more slippery than the QBist realizes. All of these problems are inherited directly from the field foundations of probability theory.

 

[atyy]

I don't think that you need to think about the paper in order to sort out the difference between your position and FR. It seems that the ordinary gedankenexperiment of Wigner's friend or even simpler experiments are enough. Let me spell out things in detail to clarify my own thoughts.

(1) In the context of Copenhagen, measurements are associated with macroscopic irreversibility. In my view, all of the terms "measurement", "macroscopic" and "irreversible" contain a notion of subjectivity because they can't be strictly derived from the physical theory alone - it doesn't tell us when certain approximations are good enough for the practical purposes of the observer. This isn't specific to QM. It is true also in classical mechanics, so it also applies to the classical apparatus in Copenhagen.

(2) In almost all cases, the subjectivity of (1) is inter-subjevtivity, i.e. if different observers look at the irreversible mark which has been left on a photo plate during an experiment, they all observe the same thing. So we can get away without worrying about subjevtivity and the only cases which force us to distinguish between subjectivity and inter-subjectivity are speculative Wigner's friend type experiments.

It's been a while since I have discussed things with you, so I am not sure about your position. Do you agree that irreversibility is subjective in the way I outlined above?

I think so. Without thinking, I would like to believe in both Copenhagen (irreversible) and Bohmian Mechanics (reversible). Since Bohmian Mechanics would be more fundamental, the irreversibility is a subjective approximation.

If you disagree, I don't see how to escape the conclusion of @DarMM's post #33, i.e. that collapse has an objectively real element. Although the wavefunction may be purely epistemic we get a physical change in the (classical) measurement apparatus. This would imply that the freedom in choosing the Heisenberg cut and the deferred measurement principle are tools for calculations but physically wrong.

If you agree, measurements may be reversed under very special circumstances and FR may have a point. Of course, it is very doubtful whether this is a point in practise. Their gedanken experiment seems to presuppose a step which corresponds to the quantum resurrection of dead cats which @stevendaryl outlined in the thread on FR. If this is true, Wigner's friend type experiments probably need to involve timescales which are wildly different in order to account for something like the Poincaré recurrance time.

I used to think that Wigner's friend type experiments are impossible and I still do but in the light of AI and quantum computers, I see a (slightly) growing need to justify this position. If the irreversibility of measurements just means that a mindbogglingly large time is needed to reverse it, an AI whose mind isn't boggled so easily because it works on a different timescale might make such an experiment feasible.

Sure, if Bohmian Mechanics (the more fundamental theory) allows reversibility, and still produces definite outcomes at all times, I would expect it to be possible for QM to be wrong in cases where something seems irreversible to the QM observer, but it is in fact reversible in the underlying Bohmian Mechanics. I think they shouldn't sell it as Copenhagen or QM being wrong, especially if they break fundamental bits of QM in deriving their conclusions. Nor should they be surprised that Bohmian Mechanics does not reproduce QM exactly. Of course, if they have produced a specific case where QM fails but Bohmian Mechanics works, then that is an advance, since my arguments are just vague sketches.

 

[atyy]

In post #27, I quoted Haag's quote of Bohr as a source that says that measurements must be irreversible.

Here is another example from Peres's textbook:

"It must now be shown that no inconsistency arises if the measuring instrument, or the “observer” (in the above example, the center of mass position r), is considered as a genuine quantum object for which no classical description is used, but this quantized instrument is, in turn, observed by some other instrument, having both quantum and classical descriptions. For example, it should not matter whether Schrödinger’s cat is considered as the observer, or is an intrinsic part of the atom-cat dynamical system which is observed by someone else. The reason for suspecting a possible inconsistency is the following: If the observer were not a cat or some other, possibly inanimate but utterly complicated measuring apparatus, the unitary evolution leading to Eq. (12.1) would be reversible. Simple, highly idealized models of measuring apparatuses can easily be concocted, which have that property. After the measurement is achieved (i. e., the instrument is correlated with the measured system), it still is possible to undo the whole process: A superobserver, capable of fully analyzing the dynamical behavior of the measuring apparatus (e. g., capable of writing explicitly the Hamiltonian of Schrödinger’s cat) could cause the original observer (or apparatus—this makes no difference) to decorrelate itself from the measured system and to “unlearn” the result of the measurement. For example, the cat would be resurrected. And then, the superobserver, by measuring again the same system, could obtain a different result for his measurement.

If such a scenario were indeed possible, the notion of measurement would become meaningless, as no measurement would ever be conclusive. Consistency thus requires the measuring process to be irreversible."

 

[DarMM]

I think they shouldn't sell it as Copenhagen or QM being wrong, especially if they break fundamental bits of QM in deriving their conclusions.

Having read the paper I don't see where they are saying Copenhagen is wrong or what fundamental parts of QM they are breaking.

The quotes from Peres and Haag are fine, but there's nothing there that disagrees with the paper. They break assumption Q, just as Many Worlds would break S.

The main result is really about subjective collapse being inconsistent if QM applies everywhere and you combine different agents conclusions. That's it, nothing about Copenhagen being wrong, just certain forms of it.

 

[DarMM]

I know that, I used to be a QBist; w.r.t. probability theory I still am a staunch Bayesian. The problem in QBism is there that the word 'existence' doesn't necessarily mean existence out in the world, but also existence inside the mind; this makes the argument far more slippery than the QBist realizes. All of these problems are inherited directly from the field foundations of probability theory.

Yeah, I find the interpretation of probability theory quite difficult. Perhaps a better example would be the retrocausal theories which have subject collapse without $\psi$ being ontic.

Or some of the $\psi$-epistemic constructions, like those of Aaronson et al: https://arxiv.org/abs/1303.2834

 

[atyy]

Having read the paper I don't see where they are saying Copenhagen is wrong or what fundamental parts of QM they are breaking.

The quotes from Peres and Haag are fine, but there's nothing there that disagrees with the paper. They break assumption Q, just as Many Worlds would break S.

The main result is really about subjective collapse being inconsistent if QM applies everywhere and you combine different agents conclusions. That's it, nothing about Copenhagen being wrong, just certain forms of it.

I think Haag's and Peres's statements are pretty fundamental parts of QM. If Frauchiger and Renner break it, I'm not sure what they are talking about can be called QM.

Anyway, would you agree with Demystifier that collapse in BM is objective (post #17)?

 

[DarMM]

I think Haag's and Peres's statements are pretty fundamental parts of QM. If Frauchiger and Renner break it, I'm not sure what they are talking about can be called QM.

I don't know if measurements being in principal irreversible is a fundamental part of QM. Haag and Peres think so, others think not, why are Haag and Peres's particular views on measurement fundamental to QM? In many interpretations it is reversible in principal, are those interpretations simply wrong then?

Also Frauchiger-Renner rather than using or discussing reversibility of measurements, more make the assumption that Wigner is correct to assign a superposed state to his friend rather than a mixed state, is that correct in your view or do you think Wigner should be assigning a mixed state?

(This is necessary before talking about objective collapse in Bohmian Mechanics)

 

[atyy]

I don't know if measurements being in principal irreversible is a fundamental part of QM. Haag and Peres think so, others think not, why are Haag and Peres's particular views on measurement fundamental to QM? In many interpretations it is reversible in principal, are those interpretations simply wrong then?

Frauchiger and Renner are trying to show that QM is not a universally valid theory. From the point of view of Copenhagen, QM has long been argued not a universally valid theory because the observer has a special status and the first thing that one does is to make the classical-quantum cut - this is the point of view of those who believe there is a measurement problem. Attempts to solve the measurement problem that have reversible measurements are BM and MWI. From the BM point of view, QM is not a universally valid theory (because of the measurement problem), and even BM is not a universally valid theory (because of what BM calls "quantum equilibrium"). Since BM is not an interpretation that believes that Copenhagen QM is a universally valid theory, it seems that it is not in FR's version of QM. I think MWI is controversial enough even among proponents that it doesn't deserve the label of "intellectually coherent" at the moment. [I guess I am saying that Copenhagen (with a classical-quantum cut, irreversible measurements, and collapse) should be the only "textbook" interpretation.]

Also Frauchiger-Renner rather than using or discussing reversibility of measurements, more make the assumption that Wigner is correct to assign a superposed state to his friend rather than a mixed state, is that correct in your view or do you think Wigner should be assigning a mixed state?

(This is necessary before talking about objective collapse in Bohmian Mechanics)

OK, I'll have to think about that and discuss another time then.

 

[ftr]

I think all these QM interpretations, especially the "collapse", are so useless. That is because none of them address the origin of the problem, why the wavefunction, where does it come from and not why it does this or that upon measurement. I have seen many "derivations" of Schrodinger equation but they all seem to be like a mathematical trick with no fundamental/logical principle involved.

 

[atyy]

So let's say in the Wigner's friend scenario, Wigner should be using a mixed state, not the pure state:
$$\frac{1}{\sqrt{2}}\left(|\uparrow, A_{\uparrow}\rangle + |\downarrow, A_{\downarrow}\rangle\right)$$
with $A_{\uparrow}, A_{\downarrow}$ device states?

Ok, I've now read the Wigner's friend scenario. If the pure state refers to the friend (A) and his spin, and if Wigner has not interacted with or measured his friend and the spin, then Wigner uses the pure state, since no measurement has occurred.

 

[DarMM]

then Wigner uses the pure state, since no measurement has occurred.

The friend has performed a measurement though right, just Wigner hasn't yet. Do you mean Wigner can use the superposed state because he hasn't performed a measurement, even though the friend has?

 

[atyy]

The friend has performed a measurement though right, just Wigner hasn't yet. Do you mean Wigner can use the superposed state because he hasn't performed a measurement, even though the friend has?

Yes, from the point of view of Wigner, the friend has not performed a measurement.

 

[DarMM]

Yes, from the point of view of Wigner, the friend has not performed a measurement.

So then you are working with subjective collapse and the Frauchiger-Renner paper shows you cannot combine reasoning from Wigner with that of his friend, i.e. a statement that the friend considers true (or another way of saying it: one they assign probability one to, i.e. certain ) A and another that Wigner considers true B, cannot be considered at once without contradictions.

 

[atyy]

So then you are working with subjective collapse and the Frauchiger-Renner paper shows you cannot combine reasoning from Wigner with that of his friend, i.e. a statement that the friend considers true (or another way of saying it: one they assign probability one to, i.e. certain ) A and another that Wigner considers true B, cannot be considered at once without contradictions.

Why do you need FR to do that? Wigner does not consider his friend a user of QM, since the friend is on the quantum side of the cut.

 

[DarMM]

Why do you need FR to do that? Wigner does not consider his friend a user of QM, since the friend is on the quantum side of the cut.

So the cut is an objective feature of reality?

 

[atyy]

So the cut is an objective feature of reality?

No, the cut is a subjective feature that Wigner imposes on his reasoning about measurement outcomes. Within QM, Wigner is agnostic about the reality of things on the quantum side of the cut. If he wants to reason about the reality of his friend, Wigner cannot use Copenhagen, but he must use a more comprehensive theory such as Bohmian Mechanics (or GRW etc).

 

[DarMM]

Then you are consistent with Frauchiger-Renner. QM taking the subjective collapse view, unlike General Relativity or Kolmolgorov probability theory, is not consistent with inter-agent logic.

You cannot consider other agents and their outcomes, even in principal.

Take the Wigner's friend case where the friend agreed to measure a spin. It seems odd to say that Wigner cannot even consider "If my friend got spin up", but if you think so then Frauchiger-Renner has nothing new to say for you. It's simply that some thought there was a version of Copenhagen where you can at least consider the outcomes of other agents in general.

 

[atyy]

Then you are consistent with Frauchiger-Renner. QM taking the subjective collapse view, unlike General Relativity or Kolmolgorov probability theory, is not consistent with inter-agent logic.

You cannot consider other agents and their outcomes, even in principal.

Take the Wigner's friend case where the friend agreed to measure a spin. It seems odd to say that Wigner cannot even consider "If my friend got spin up", but if you think so then Frauchiger-Renner has nothing new to say for you. It's simply that some thought there was a version of Copenhagen where you can at least consider the outcomes of other agents in general.

OK, I haven't studied FR closely enough yet, but at least at the qualitative level we seem to agree.

One can consider "other" agents, but they must all be on the same classical side of the classical-quantum cut, maybe for example the demonstration that collapse and unitary evolution from all different special relativistic frames of reference produce consistent results (even though the states in different frames are not related by unitary transformation)

Are there really Copenhagen versions in which an agent can be on both sides of the quantum-classical cut? I mean, it would be like saying that the cat is both "dead and alive" and "dead or alive", which seems ridiculous. Maybe QBists try to do that, since I think FR say QBism is inconsistent?

 

[DarMM]

Before I respond to that post I just want to ask, when you say an agent on both sides of the cut, in the Wigner's friend case, the friend is on the classical side of the cut for himself, but on the quantum side for Wigner.

So he is on both sides of the cut, but it's a different side for different agents. I assume you are asking about being on both sides for one agent.

 

[atyy]

Before I respond to that post I just want to ask, when you say an agent on both sides of the cut, in the Wigner's friend case, the friend is on the classical side of the cut for himself, but on the quantum side for Wigner.

So he is on both sides of the cut, but it's a different side for different agents. I assume you are asking about being on both sides for one agent.

Yes, I believe the usual version of Copenhagen is one in which Wigner only grants "agent" or "reality" status to things on the same side of Wigner's classical-quantum cut.

 

[atyy]

So let's say in the Wigner's friend scenario, Wigner should be using a mixed state, not the pure state:
$$\frac{1}{\sqrt{2}}\left(|\uparrow, A_{\uparrow}\rangle + |\downarrow, A_{\downarrow}\rangle\right)$$
with $A_{\uparrow}, A_{\downarrow}$ device states?

Let's try a different answer from the one I gave in post #54, where I said Wigner uses a pure state.

I think another possibility is that Wigner accepts that his friend has done a measurement. In this case, Wigner and his friend are on the same classical side of the classical-quantum cut, and A represents the friend's ancilla. If the friend performs a measurement, but does not tell Wigner the result, and if Wigner accepts his friend's measurement as a measurement, then Wigner should use a mixed state.

 

[stevendaryl]

I think Haag's and Peres's statements are pretty fundamental parts of QM. If Frauchiger and Renner break it, I'm not sure what they are talking about can be called QM.

Anyway, would you agree with Demystifier that collapse in BM is objective (post #17)?

It's a little ambiguous. One meaning of "wave function collapse" is that after a measurement, the wave function changes discontinuously to the collapsed wave function. That does not happen in BM. Another meaning is that upon collapse, one or more possible outcome is selected. In BM, all outcomes are predetermined, so in a sense, everything is pre-collapsed. I'm not exactly sure what @Demystifier would say about it, but it seems to me that BM is not consistent with the usual idea of wave function collapse.

The "pilot wave" wave function always evolves according to Schrodinger's equation. But that's the wave function for the entire universe, in the sense of Many-Worlds. If you instead focus on an effective wave function for just part of the universe, a single system, then the effective wave function might have something like collapse.

 

[stevendaryl]

Let's try a different answer from the one I gave in post #54, where I said Wigner uses a pure state.

I think another possibility is that Wigner accepts that his friend has done a measurement. In this case, Wigner and his friend are on the same classical side of the classical-quantum cut, and A represents the friend's ancilla. If the friend performs a measurement, but does not tell Wigner the result, and if Wigner accepts his friend's measurement as a measurement, then Wigner should use a mixed state.

That's the practical approach to resolving paradoxes in QM: Assume that there is no such thing as a pure state consisting of a superposition of macroscopically different states. There can only be mixed states.

However, I find that assumption to be a "soft contradiction". QM does not in any way limit the size or complexity of the systems that can be described by it.

 

[Auto-Didact]

It's a little ambiguous. One meaning of "wave function collapse" is that after a measurement, the wave function changes discontinuously to the collapsed wave function. That does not happen in BM. Another meaning is that upon collapse, one or more possible outcome is selected.

One or more? Surely only one; given that it is only one then those two meanings are equivalent.

In other words, the (wave function) collapse is a discontinuous collapse resulting in a random selection of one of the possible outcomes.

This discontinuity in the time evolution of $\psi$ is the key clue that $\psi$ needs to be modeled by some non-continuous equation; the linear PDE is a simplification which preserves unitarity.

 

[stevendaryl]

That's the practical approach to resolving paradoxes in QM: Assume that there is no such thing as a pure state consisting of a superposition of macroscopically different states. There can only be mixed states.

However, I find that assumption to be a "soft contradiction". QM does not in any way limit the size or complexity of the systems that can be described by it.

Let me state, more precisely, what I think the contradiction is. But first, let me formulate an alternative to the Born rule: Instead of saying "A measurement of a system always produces an eigenvalue of the operator that is measured, with a probability given by blah blah blah", you say: "At any given time, the probability that the universe is in some particular macroscopic configuration is given by blah blah blah". This is a specific case of the Born rule, in which the system is the entire universe, and the observable is the universe's macroscopic configuration. You don't need the general Born rule because by definition, a measurement of a microscopic quantity means setting up a measurement so that the microscopic quantity is amplified to produce a macroscopic effect. So if you know the probabilities for macroscopic configurations, then that tells you the probabilities for various measurement results (since they are macroscopically different).

This form of the Born rule can be mathematically described this way: If we let $\Pi_j$ be the projection of the state of the universe onto the macroscopic state number $j$ (you can't have continuum many macroscopically distinguishable configurations, so it's enough to consider a countable collection of projection operators), and you let $|\psi(0)\rangle$ be the initial state, at time $t=0$, then the probability of being in state $j$ at a later time $t > 0$ is given by:

$P(j) = \langle \psi| e^{iHt/\hbar} \Pi_j e^{-iHt/\hbar} |\psi\rangle$

Then the issue of collapse can be stated this way: What if, we ask what the probability is of being in state $j$ at time $t_1$ and then later being in state $k$ at time $t_2$? Here are two possible answers:

1. $P_{collapse}(j,k) = \langle \psi| e^{iH t_1/\hbar} \Pi_j e^{iH(t_2 - t_1)/\hbar} \Pi_k e^{-iH(t_2 - t_1)/\hbar} \Pi_j e^{-iHt_1/\hbar} |\psi\rangle$
2. $P_{no-collapse}(j,k) = \langle \psi| e^{iH t_2/\hbar} \Pi_k e^{-iH t_2/hbar} |\psi\rangle$

If the state $|\psi\rangle$ is itself a pure macroscopic state ($\Pi_i |\psi\rangle = |\psi\rangle$ for some macroscopic configuration $i$), then there will be negligible difference between these two numbers. That's because the macroscopic configuration $k$ includes the record of having previously gotten some particular result for some past measurement. So for each $k$, the probability $P_{collapse}(j,k)$ will be approximately zero for all except one value of $j$. For that value of $j$, there will be a negligible difference between $P_{collapse}$ and $P_{no-collapse}$.

So under the assumption that the world currently is in a macroscopically pure state, the collapse assumption is harmless. Assume it or not, it makes no difference.

Eventually, though, the state of the universe will drift away from being macroscopically pure, and the distinction between $P_{collapse}$ and $P_{no-collapse}$ will grow larger and larger.

 

[stevendaryl]

One or more? Surely only one; given that it is only one then those two meanings are equivalent.

Well, in an experiment such as measuring the spin in the x-direction of an electron that has been prepared to be spin-up in the z-direction, there is, a priori, more than one possible outcome, spin-up or spin-down. In BM, which is deterministic, only one of them is actually possible, and we only consider them both possible because we have incomplete knowledge of the current state.

In other words, the (wave function) collapse is a discontinuous collapse resulting in a random selection of one of the possible outcomes.

In that sense, BM has no collapse. Nothing discontinuous ever happens. Except in the classical sense of changing a probability distribution based on acquiring more information.