B Does the EPR experiment imply QM is incomplete?

[bhobba]

The real problem with Many-Worlds at the moment is to mathematically prove that 3D semi-classical worlds like the one we experience actually arise. This has not yet been done, so the interpretation cannot be as of yet said to match experiment.

Some progress has been made in decoherent histories which some like Gell-Mann think is simply MW without the MW:
http://web.physics.ucsb.edu/~quniverse/papers/cop-ext2.pdf

But issues still remain. Some, including a number of very knowledgeable mentors on this forum, think until they are resolved QM is incomplete (ie Einstein was correct - he was anyway because we still have no complete quantum theory of gravity - but most take it to be something a bit different). Others like me think it's basically crossing your t's and dotting your i's. Really it's semantics - either way we are not there yet. Its interesting actually - everyone thought Einstein lost to his good friend Bohr in the Einstein Bohr debates - but here we are today and would say Einstein's intuition did not lead him astray - even here. What the future brings will indeed be interesting - however I may not be around to see what finally emerges. Another interesting thing is just before Einsteins death Bohr came to visit and Einstein would not see him - why - maybe he was too weary of the QM debates - or being so close to death didn't want Bohr to remember hijm like that - who knows - however there is little doubt during their prime they were very good friends - each admiring the other greatly. Bohr never ceased to believe GR was the greatest achievement of the human intellect ever.

Thanks
Bill

 

[vanhees71]

Well, I think nowadays it's pretty obvious that Einstein understood QT way better than Bohr, who was more a philosopher than a physicist after his young years, where he discovered "old quantum mechanics", i.e., did physics in the sense that it described something observable. He also qualitatively got the physical explanation for the chemistry of the periodic table right (to be a bit generous). As we understand QT today, his role in "interpretation" was more towards confusing the subject with quite imprecisely defined philosophical notions like "wave-particle duality" and, even worse, "complementarity". He is topped in obscurity only by Heisenberg ;-))). Of course, Einstein was wrong in thinking that the issue can be "repaired" by the hidden-variable argument, at least when you insist on locality of interactions (as realized in relativistic QFT) and separability, which has been clearly shown by Bell's work on his famous inequality, which brought the philosophical gibberish of EPR and Bohr's response to a clear physical statement which was testable in the lab (as was done by Aspect et al in the 1980ies and later on up to today at ever higher accuracy and with ever more fancy setups to exclude any, if not all!, "loopholes").

Nevertheless, I think that the minimal interpretation is the only thing we need to use the formalism to describe what's seen in Nature. In this sense QT is "complete" as a physical theory, but this completeness is only "for all practical purposes" (FAPP), and as Bell stated, that's not very satisfactory. Indeed, it is very well known that QT as we know it today is for sure incomplete, since there's no quantum description of gravity, and maybe one day some genius finds a solution to this puzzle, which overcomes maybe also the problems with ontology of QT.

For non-relativistic QM, I recently got convinced, that the Bohmian Mechanics, as interpreted and presented by Dürr et al, is an example for such a solution. Unfortunately it seems to be even "less complete" than QT, which also includes a very well working relativistic version in terms of local microcausal relativistic QFT, upon which the all too successful Standard Model of elementary-particle physics is based. So for QT as a whole we still don't have anything better than the FAPP interpretation called "minimal statistical interpretation".

 

[bhobba]

Well, I think nowadays it's pretty obvious that Einstein understood QT way better than Bohr, who was more a philosopher than a physicist after his young years, where he discovered "old quantum mechanics", i.e., did physics in the sense that it described something observable. He also qualitatively got the physical explanation for the chemistry of the periodic table right (to be a bit generous). As we understand QT today, his role in "interpretation" was more towards confusing the subject with quite imprecisely defined philosophical notions like "wave-particle duality" and, even worse, "complementarity". He is topped in obscurity only by Heisenberg ;-))).

First interesting thing about Heisenberg. Bohr's brother, Harald, who was actually a bit more famous in Denmark because he was a bit better soccer player, was a very good mathematician and friend of Hardy. Heisenberg was visiting and Hardy thought he would play a trick on Heisenberg. He said now you are becoming more involved with advanced physics we should develop your math a bit more. He gave him a problem that was then reasonably famous, and just recently solved. To Hardy's astonishment Heisenberg solved it - Hardy thought he had picked the wrong area - he should do math and come to work with him and Littlewood (too young to work with Ramanujan which would have been interesting). Of course he didn't take him up on the offer, but if he did he would have probably ended up working also with Dirac. So yes Heisenberg sometime spoke philosophical 'gibberish', but extremely talented in math and physics he certainly was. Personally I think pretty much everyone in the early days of QM got it wrong except Dirac, and we now know Einstein actually got it right as well - although it took him a while to reach his final view that QM was correct - but incomplete. To give Bohr his due I believe his debates with Einstein helped him in forming his final view even though Bohr was, to be kind, 'subtle' and with his mumble not a good communicator - but neither was Dirac for a different reason.

And yes Bell understood it as well, even though he was at least at the start into BM. To independently basically come up with a cut down version of what was known as a very difficult theorem to prove (Gleason's Theorem - from which Kochen-Speker followed as a simple corollary. Bell independently proved Kochen-Speker), taking one of the best mathematicians at the time, Gleason, to do it. I simply do not know how to categorize someone like that. Same with Feynman and Wilson - they were Putman Fellows - Feynman put on the team at the last minute, and Wilson twice even though he was accepted into Harvard early - 16 I think and was a Putman Fellow at 18 and 19 - but don't hold me to it. Interesting story - when Wilson got his first one he was given a celebration by the math department. He noticed one thing immediately - his father was a professor of Chemistry at Harvard so knew a number other professors. The math professors, while all very good mathematicians, were quite mad

Thanks
Bill

 

[vanhees71]

Sure, there's no question that Heisenberg was an ingenious theoretical physicist. After all he was the first to discover "modern QM" in 1925 in terms of matrix mechanics. The only thing I criticize is part of his "soft skills". In the beginning he fought Schrödinger's wave mechanics to defend his matrix mechanis as the only true quantum mechanics. Of course, this is nonsense given the fact that Schrödinger himself proved very early that both formalisms are equivalent representations.

Indeed of all "founding fathers" of QT, Dirac is the most clear of all, giving the bare formalism without too much interpretational gibberish, but also Schrödinger's paper are marvels in science-writing style, although he didn't have the correct probabilistic interpretation yet, and he was always critical against it until the end of his life. Dirac's textbook is still among the best ever written, but also his papers are not less clear and can be read easily by undergraduate students beginning to learn QT. That I can't say about Heisenberg's most famous Helgoland paper. I never got to understand it completely from scratch, i.e., not using the knowledge about QT I got from other sources. One also shouldn't underestimate the importance of Born, Jordan, and also Pauli, for the success of Heisenberg's matrix mechanics. The two papers by Born and Jordan as well as Born, Jordan, and Heisenberg ("Dreimännerarbeit") are quite well readable (containing even a part on field quantization by Jordan already then, but this one got forgotten, because people found this "too much", i.e., they though one should treat electromagnetism still via classical Maxwell theory, and it had to be reinvented by Dirac some time later to unerstand spontaneous emission). Pauli's contribution was not only the official one, namely the solution of the hydrogen energy-eigenvalue problem within matrix mechanics, using the O(4) symmetry and the associated Runge-Lenz vector from classical mechanics, but also as a "regulator" of Heisenberg's ingenious but not always very clearly stated ideas.

 

[DarMM]

Having read all their papers recently my personal impression was that Bohr and Heisenberg didn't spell out what they meant by phrases like "the electron doesn't exist between measurements". When you know what they mean a good portion of their writings make more sense, but I think they should have just written a summary account of their similar Copenhagen interpretations where they spell everything out.

By "the electron doesn't exist between measurements" they meant "electron" is a conceptual object we use to describe the type of marking "something" else leaves on our equipment. With that something being impossible to describe. It's a bit like how my signature isn't the same thing as me, but if you only have cheque books and your language only speaks of ink and paper, it's the only impression of me you'll have, but the signature doesn't exist between "cheque signing" events/measurements.
The electron is just this ineffable thing's signature on systems we can describe.

You can draw this out from their later papers, but unadorned and often unexplained it sounds like they don't think an independent reality exists.

Also Bohr assumes you've read Kant, many terms are being given their Kantian meaning rather than the everyday reading. Which just makes him even more difficult to understand.

 

[A. Neumaier]

just know its something every interpretation has and is what on this forum (and in all textbooks I am aware of) we call the Born Rule.

Except for my thermal interpretation! The latter says that as a property of measurements, Born's rule is only an approximation, valid only in very simple, idealized systems.
See the thread https://www.physicsforums.com/threads/many-measurements-are-not-covered-by-borns-rule.894095/ where this is argued from well-known cases in the literature.

 

[stevendaryl]

I actually do think that EPR shows that we don't completely understand QM. Whether that means it's incomplete or not is a matter of interpretation.

I don't think that we completely understand what a "measurement" is in QM. There are two competing ways of understanding measurements, and they are at odds. And both come into play in the EPR experiment.

One conception of measurement is that it's passive, at least in principle: When we measure some property of a system, we're just discovering a fact about that system. That's the classical notion of measurement.

A second conception of measurement is that it's active: The result of measurement is not a pre-existing quantity that you just become aware of, but is brought into existence through the participation and interaction of both the system being measured and the measuring device. The act of measuring disturbs the system being measured.

Both conceptions come into play in EPR. When it comes to measuring the polarization of a single photon, we can actually prove (at least in a one-world ontology) that the measurement disturbs the state of the photon being measured. The way that we measure a photon's polarization is by using a polarizing filter: If the photon passes through the filter, then its polarization is aligned with the orientation of the filter. If the photon is absorbed, then its polarization is perpendicular to the filter's orientation. To see that the filter is actually changing the state of the photon (as opposed to just filtering out photons of the wrong polarization), you can do the following:

• Send photons through one filter. The photons that come through will have a specific polarization.
• Have those that pass through go through a second filter, oriented at angle $\theta$ relative to the first.
• Have those that pass through the second filter go through a third.
• Etc.

With enough intermediate filters (actually, just three is enough, with a relative angle of 45^o between the orientations of successive filters), you can get photons coming through oriented 90^o away from the orientation they had after the first filter. It's obvious in this case that the filters have twisted the photons' polarizations, rather than simply passing those that had the wrong polarization. So the act of measurement of a photon's polarization actually does something to the photon. The fact that the photon coming out of the filter is polarization in a particular direction doesn't say anything definite about what it's polarization was before it went through.

In EPR, we have two correlated photons measured by Alice and Bob. Alice measures the polarization of one photon, and whatever answer she gets, she immediately knows the polarization of Bob's photon. Unless you allow for faster-than-light influences, Alice's measurement can't disturb Bob's photon. So if you disbelieve in FTL influences, you have to think of Alice's measurement as simply updating her information about Bob's photon, rather than changing Bob's photon.

But how can both be the case? When Alice measure's her own photon's polarization, her photon is interacting with her filter, and the state of her photon is affected by the filter. The fact that her photon comes through the filter polarized along a particular direction does not imply that the photon had the polarization prior to its passing through. But Alice can use the polarization of her photon after passing through the filter to deduce Bob's photon's polarization. How can that be possible? The disturbance model of measurement is justified in the case of Alice measuring her own photon, while the passive update model of measurement is justified in the case of Alice deducing something about Bob's photon.

 

[Mentz114]

I actually do think that EPR shows that we don't completely understand QM. Whether that means it's incomplete or not is a matter of interpretation.[].

About entangled photons QT says the the probability of a coincidence between A's and B's results is $\cos(\alpha-\beta)^2$ and no-one misunderstands that.

You are concerned about the things that QT does not tell us. The fact that it says so little could mean it is incomplete, but who cares anyway except philosophers

 

[DarMM]

Nice way of looking at EPR stevendaryl, I never thought of it like that.

Just a small thing, since the EPR correlations can be simulated by a local classical model, would maybe a Bell situation be better? I think your main point carries over without change.

There you would have the frustration between the active and passive view, but without the easy "out" that QM is incomplete due to the existence of a local classical model to simulate the correlations.

 

[Lish Lash]

When Alice measure's her own photon's polarization, her photon is interacting with her filter, and the state of her photon is affected by the filter. The fact that her photon comes through the filter polarized along a particular direction does not imply that the photon had the polarization prior to its passing through. But Alice can use the polarization of her photon after passing through the filter to deduce Bob's photon's polarization. How can that be possible?

Here's how this experiment is interpreted in Bohmian Mechanics: The photons detected by Alice and Bob are particles that each possesses just one (hidden) property: their actual position in 3D space at the point in time they are measured. All other quantum properties, such as polarization, are properties of the pilot wave with which these particles are entangled. When Alice measures the polarization of her photon, the wave function of her measuring device becomes entangled with the wave function of the photon. It is the pilot wave of the entangled photon/measuring device system that produces the measured outcome. So long as Bob's photon remains entangled with Alice's photon, its polarization will be correlated with the outcome of Alice's measurement.

 

[zonde]

Here's how this experiment is interpreted in Bohmian Mechanics: The photons detected by Alice and Bob are particles that each possesses just one (hidden) property: their actual position in 3D space at the point in time they are measured. All other quantum properties, such as polarization, are properties of the pilot wave with which these particles are entangled. When Alice measures the polarization of her photon, the wave function of her measuring device becomes entangled with the wave function of the photon. It is the pilot wave of the entangled photon/measuring device system that produces the measured outcome. So long as Bob's photon remains entangled with Alice's photon, its polarization will be correlated with the outcome of Alice's measurement.

You are explaining entanglement with a help of entanglement. It's circular explanation, don't you see?

 

[stevendaryl]

About entangled photons QT says the the probability of a coincidence between A's and B's results is $\cos(\alpha-\beta)^2$ and no-one misunderstands that.

You are concerned about the things that QT does not tell us. The fact that it says so little could mean it is incomplete, but who cares anyway except philosophers

I actually don't think that it "says so little". I think that what it does (quantum mechanics in the usual rule of thumb) is actually inconsistent, but it's a "soft" inconsistency. I've said before the reason I think that:

• You have the Born rule, which says that when a measuring device interacts with a system to measure some property of that system, then you get a result that is an eigenvalue of the corresponding operator with probabilities given by the squares of the relevant amplitudes.
• Getting a particular result means that the measuring device is in some definite configuration.
• However, if you treat the measuring device as a quantum system, then the interaction of the measuring device with the system it's measuring doesn't produce a single result, but produces a superposition of all possible results.

So on the one hand, you have a prediction from quantum mechanics (Schrodinger's equation) that the measurement device will be described by one state, a superposition of various possible macroscopic results. This prediction is deterministic. On the other hand, you have a prediction from quantum mechanics (Born's rule) that the measurement device will be described by one or another state, where each has a definite macroscopic result. Those are two different, and contradictory, predictions, both made by quantum mechanics.

I call that a "soft" contradiction, because nobody actually computes the state of a macroscopic device by treating it as a quantum-mechanical system and applying Schrodinger's equation. It's too complex to do that. They treat the device classically, or semi-classically, and only consider the Born prediction. But I think that it's actually inconsistent, logically. Maybe only philosophers care about inconsistencies, if there is a work-around.

 

[stevendaryl]

Here's how this experiment is interpreted in Bohmian Mechanics: The photons detected by Alice and Bob are particles that each possesses just one (hidden) property: their actual position in 3D space at the point in time they are measured. All other quantum properties, such as polarization, are properties of the pilot wave with which these particles are entangled. When Alice measures the polarization of her photon, the wave function of her measuring device becomes entangled with the wave function of the photon. It is the pilot wave of the entangled photon/measuring device system that produces the measured outcome. So long as Bob's photon remains entangled with Alice's photon, its polarization will be correlated with the outcome of Alice's measurement.

Yes, Bohmian mechanics doesn't actually treat measurements according to the Born rule, but instead, only uses the Born rule to give probabilities of various locations in configuration space (the position space for all particles involved). So the two aspects of measurement don't directly come into play.

 

[Mentz114]

[..]
So on the one hand, you have a prediction from quantum mechanics (Schrodinger's equation) that the measurement device will be described by one state, a superposition of various possible macroscopic results.
[..]

We've been here. I don't think macroscopic superpositions are possible. The SE does not assert that. People interpreting the SE say that.

 

[stevendaryl]

We've been here. I don't think macroscopic superpositions are possible. The SE does not assert that.

I don't agree. Quantum mechanics certainly has no criterion for when a system is too large to be described by the Schrodinger equation. And if it is really the case that sufficiently large systems are no longer described by the Schrodinger equation, then that would definitely mean that quantum mechanics is incomplete. (It would actually mean that it is false, but only an approximation good for small systems.)

 

[Mentz114]

I don't agree. Quantum mechanics certainly has no criterion for when a system is too large to be described by the Schrodinger equation. And if it is really the case that sufficiently large systems are no longer described by the Schrodinger equation, then that would definitely mean that quantum mechanics is incomplete. (It would actually mean that it is false, but only an approximation good for small systems.)

I am not saying that the SE cannot describe a macroscopic object. I'm objecting to the assumption of superpositions of solutions.
They are mathematically valid but so what. Not every solution to a set of constraints necessarily has a real counterpart.

 

[zonde]

So on the one hand, you have a prediction from quantum mechanics (Schrodinger's equation) that the measurement device will be described by one state, a superposition of various possible macroscopic results. This prediction is deterministic. On the other hand, you have a prediction from quantum mechanics (Born's rule) that the measurement device will be described by one or another state, where each has a definite macroscopic result. Those are two different, and contradictory, predictions, both made by quantum mechanics.

These two predictions are contradictory only if you consider QM complete.
If QM is incomplete there is no contradiction, because the state that is associated with different macroscopic configurations describes only one property of these configurations. That property is the same while other properties are different.

 

[PeterDonis]

I am not saying that the SE cannot describe a macroscopic object. I'm objecting to the assumption of superpositions of solutions.

These two statements are inconsistent. The SE gives you "superpositions of solutions" just by time evolution. You can't arbitrarily exclude "superpositions of solutions" and still use the SE at all.

 

[Mentz114]

These two statements are inconsistent. The SE gives you "superpositions of solutions" just by time evolution. You can't arbitrarily exclude "superpositions of solutions" and still use the SE at all.

I could be wrong but I thought

$\hat{H}=\sum c_n|E_n\rangle\langle E_n|$, so that $\hat{H}E_n\rangle =c_n |E_n \rangle$

The evolution is driven by $e^{\hat{H}t}$ but $\hat{H}$ is idempotent so I think my point below stands.

i.e. an eigenvalue cannot evolve to a superposition ? And a superposition can only evolve to a superposition.

 

[PeterDonis]

an eigenvalue cannot evolve to a superposition ?

An eigenstate of the Hamiltonian will stay an eigenstate of the Hamiltonian, yes. But an eigenstate of the Hamiltonian has no interactions whatever--nothing ever happens to it. So no real object is ever in an eigenstate of the Hamiltonian. Any state that is a reasonable candidate to describe a real object will change under time evolution; and any state that, at some instant of time, happens to look like a reasonable classical state of a classical object, will not stay that way; it will evolve into a "Schrodinger's Cat" type state that does not describe anything like a classical state of a classical object.

 

[Mentz114]

So on the one hand, you have a prediction from quantum mechanics (Schrodinger's equation) that the measurement device will be described by one state, a superposition of various possible macroscopic results.

On the other hand, you have a prediction from quantum mechanics (Born's rule) that the measurement device will be described by one or another state, where each has a definite macroscopic result.

Please can you express these alternatives as equations. I cannot make sense of the words.

[edit]

OK, the first one is a superposition and the second is not. Sorry for the dyslexia ( or something).

 

[Mentz114]

These two statements are inconsistent. The SE gives you "superpositions of solutions" just by time evolution. You can't arbitrarily exclude "superpositions of solutions" and still use the SE at all.

An eigenstate of the Hamiltonian will stay an eigenstate of the Hamiltonian, yes. But an eigenstate of the Hamiltonian has no interactions whatever--nothing ever happens to it. So no real object is ever in an eigenstate of the Hamiltonian. Any state that is a reasonable candidate to describe a real object will change under time evolution; and any state that, at some instant of time, happens to look like a reasonable classical state of a classical object, will not stay that way; it will evolve into a "Schrodinger's Cat" type state that does not describe anything like a classical state of a classical object.

I think it is implicit in the statement of @stevendaryl's 'soft paradox' that there is an operator that acts on the macroscopic wave function (in the appropriate basis) . The superposition must be between one or more of its eigenvalues. So it must have started as a preparation of a superposition. If these preparations are not allowed, then obviously we can't evolve into a superposition.

It all comes down to preparation again.

 

[PeterDonis]

So it must have started as a preparation of a superposition. If these preparations are not allowed, then obviously we can't evolve into a superposition.

That's not correct as you state it. An eigenstate of any operator other than the Hamiltonian will not stay an eigenstate of that operator under time evolution, unless that operator commutes with the Hamiltonian. It will evolve into a superposition of eigenstates.

Some useful operators, such as total momentum and total angular momentum, commute with the Hamiltonian. But I don't think any operator that might represent a realistic observable for a macroscopic system being measured--something like center of mass position, for example--will commute with the Hamiltonian.

 

[Mentz114]

That's not correct as you state it. An eigenstate of any operator other than the Hamiltonian will not stay an eigenstate of that operator under time evolution, unless that operator commutes with the Hamiltonian. It will evolve into a superposition of eigenstates.

Some useful operators, such as total momentum and total angular momentum, commute with the Hamiltonian. But I don't think any operator that might represent a realistic observable for a macroscopic system being measured--something like center of mass position, for example--will commute with the Hamiltonian.

Thank you. That is interesting. Two questions about the macroscopic apparatus come to mind,
If we are measuring a quantum property, would the apparatus have the same eigenvalues as the microscopic property ?
If the observable is an angle, would the operator commute with $\hat{H}$ ?

 

[PeterDonis]

If we are measuring a quantum property, would the apparatus have the same eigenvalues as the microscopic property ?

Eigenvalues of what operator?

When we model quantum measurements as applying an operator to the system, that operator is an operator on the Hilbert space describing the measured system, not the measured system + measuring device. The eigenvalues of the operator are therefore eigenvalues applying to the measured system only. If you think about it, this is already an admission that such a model is incomplete.

When we try to construct a more complete model, where we include the measuring device and the interaction between it and the measured system, then we no longer model measurement as applying an operator to the system; we model it as just time evolving the system using the Schrodinger Equation with the Hamiltonian including the interaction between the measuring device and the measured system. This time evolution then puts the whole system (measured system + measuring device) into an entangled state. This state is not an eigenstate of any operator (or at least, not of any operator that anyone is writing down and using in the analysis), so there aren't any useful eigenvalues that apply to it.

In such a more complete model, "measurement results" are really encoded in the labels that are put on the terms in the entangled state. For example, say we put a qubit through a spin measurement. The resulting state will look something like (leaving out normalization factors): $\vert + \rangle \vert M_+ \rangle + \vert - \rangle \vert M_- \rangle$. Here the measurement result "measured spin up" is encoded in the kets in the first term, and the measurement result "measured spin down" is encoded in the kets in the second term. It is not encoded in eigenvalues of any operator.

If the observable is an angle, would the operator commute with $\hat{H}$ ?

I don't think so. Measuring an angle is not the same as measuring angular momentum.

 

[stevendaryl]

Please can you express these alternatives as equations. I cannot make sense of the words.

Let's make it simple, and suppose that we have some measurement device that measures the spin of a particle along the z-axis. For the particle, $|u\rangle$ is the state that is spin-up in the z-direction, and $|d\rangle is the state that is spin-down. Let's suppose that$|0\rangle$is the initial "ready" state of the device, and let$|U\rangle$mean "measured spin-up" and$|D\rangle## mean "measured spin down". Those are sometimes called "pointer" states.

So the assumption that the device actually works as a measuring device is that:

1. $|u\rangle |0\rangle \Rightarrow |u\rangle |U\rangle$
2. $|d\rangle |0\rangle \Rightarrow |d\rangle |D\rangle$

(where $\Rightarrow$ means "evolves into, taking into account the Schrodinger equation")

By linearity of the Schrodinger equation, it follows that:

$(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow \alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$

So if the particle starts off in a superposition of states, then the measuring device (and the rest of the universe, eventually, but we're not modeling that) ends up in a superposition of different "pointer" states.

So that's the prediction of the Schrodinger equation. The Born rule says, instead, that:

1. $(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow |u\rangle |U\rangle$ with probability $|\alpha|^2$
2. $(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow |d\rangle |D\rangle$ with probability $|\beta|^2$

The Born gives a probabilistic transition rule leading to a definite pointer state, while the Schrodinger equation gives a deterministic transition rule leading to a superposition of pointer states. Those are different, and contradictory, predictions.

 

[stevendaryl]

I think it is implicit in the statement of @stevendaryl's 'soft paradox' that there is an operator that acts on the macroscopic wave function (in the appropriate basis) . The superposition must be between one or more of its eigenvalues. So it must have started as a preparation of a superposition. If these preparations are not allowed, then obviously we can't evolve into a superposition.

Think about an actual measurement. You pass an electron through a Stern-Gerlach device. The electron is either diverted to the right, if it's spin-up, or it's diverted to the left, if it's spin-down. If the electron is diverted left, it makes a visible dark spot on the left side of a photographic plate. If it's diverted to the right, it makes a visible dark spot on the right side of a photographic plate.

So connecting this to the mathematics above, $|U\rangle$ is the Stern-Gerlach system plus photographic plate, in which there is a dark spot on the right. $|D\rangle$ is the state where the dark spot is on the left.

 

[zonde]

By linearity of the Schrodinger equation, it follows that:

$(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow \alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$

So if the particle starts off in a superposition of states, then the measuring device (and the rest of the universe, eventually, but we're not modeling that) ends up in a superposition of different "pointer" states.

So that's the prediction of the Schrodinger equation. The Born rule says, instead, that:

1. $(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow |u\rangle |U\rangle$ with probability $|\alpha|^2$
2. $(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow |d\rangle |D\rangle$ with probability $|\beta|^2$

The Born gives a probabilistic transition rule leading to a definite pointer state, while the Schrodinger equation gives a deterministic transition rule leading to a superposition of pointer states. Those are different, and contradictory, predictions.

They are not contradictory if they describe different aspects of reality.
One describes phase difference between different outcomes (HVs).
Other prediction describes actual outcome (when HV is revealed).

In case of microscopic systems you can't perform both measurements at the same time. In case of macroscopic systems we don't know how to measure phase relationship (perform interference measurement) between different outcomes, but hypothetically if we would know how to perform that interference measurement we can say that we would not be able to learn what was actual outcome in particular case.

 

[Mentz114]

Let's make it simple, and suppose that we have some measurement device that measures the spin of a particle along the z-axis. For the particle, $|u\rangle$ is the state that is spin-up in the z-direction, and $|d\rangle is the state that is spin-down. Let's suppose that$|0\rangle$is the initial "ready" state of the device, and let$|U\rangle$mean "measured spin-up" and$|D\rangle## mean "measured spin down". Those are sometimes called "pointer" states.

So the assumption that the device actually works as a measuring device is that:

1. $|u\rangle |0\rangle \Rightarrow |u\rangle |U\rangle$
2. $|d\rangle |0\rangle \Rightarrow |d\rangle |D\rangle$

(where $\Rightarrow$ means "evolves into, taking into account the Schrodinger equation")

By linearity of the Schrodinger equation, it follows that:

$(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow \alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$

So if the particle starts off in a superposition of states, then the measuring device (and the rest of the universe, eventually, but we're not modeling that) ends up in a superposition of different "pointer" states.

So that's the prediction of the Schrodinger equation. The Born rule says, instead, that:

1. $(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow |u\rangle |U\rangle$ with probability $|\alpha|^2$
2. $(\alpha |u\rangle + \beta |d\rangle)|0\rangle \Rightarrow |d\rangle |D\rangle$ with probability $|\beta|^2$

The Born gives a probabilistic transition rule leading to a definite pointer state, while the Schrodinger equation gives a deterministic transition rule leading to a superposition of pointer states. Those are different, and contradictory, predictions.

I don't see any $e^{\hat{H}t}$ in there. Unless you have explicitly different evolutions I can't see any contradiction. The state of the macroscopic apparatus needs to be there also because the point in question is final state. You need to give the assumptions concerning preparation of the microscopic state and the apparatus.

 

[stevendaryl]

I don't see any $e^{\hat{H}t}$ in there.

Yeah, well that's the stumbling block for any discussion of the application of quantum mechanics to large systems is that it's difficult to do it rigorously.

Unless you have explicitly different evolutions I can't see any contradiction.

That's why I call it a "soft contradiction". The computations are intractable to actually derive the state from first principles.

However, we know that an electron hitting a photographic plate does cause a dark spot on the plate. To believe that this is not described by quantum mechanics seems to be equivalent to believing that quantum mechanics is incorrect or incomplete.

I spent a certain amount of time studying AI and one of the systems I looked at was the CYC project (I don't know whether that's been abandoned, or not). CYC had a notion of "microtheories" which described a particular small domain. For a physics example, statics. It knew how to reason within a microtheory. Then there were heuristics on top of the microtheories to decide which microtheory to use in what circumstances.

It was suspected that the microtheories were actually inconsistent, but it was a "soft" inconsistency, because you never applied more than one microtheory at a time.

Quantum mechanics is similarly composed of two microtheories for describing systems: For small systems, you use the Schrodinger equation. For large systems, you use the Born rule.

 

[Mentz114]

Eigenvalues of what operator?

[cut for brevity]

Thanks for trying to unconfuse me. I have to keep asking stuff that should be included in the description of the problem and I have asked stevendaryl for clarification.
There may be a contradiction in the requirement that the apprartus state should be highly correlated with the incoming system unless the apparatus has some similarities with the incoming state - viz. common eigenvalues which implies an operator (?). The states being evolved must include the apparatus and the outcome depends only on the Hamiltonian and the initial state. This does not always result in a superposition but I don't know if that is relevant because the 'contradiction' is not unambiguously defined.

 

[Mentz114]

They are not contradictory if they describe different aspects of reality.
One describes phase difference between different outcomes (HVs).
Other prediction describes actual outcome (when HV is revealed).

In case of microscopic systems you can't perform both measurements at the same time. In case of macroscopic systems we don't know how to measure phase relationship (perform interference measurement) between different outcomes, but hypothetically if we would know how to perform that interference measurement we can say that we would not be able to learn what was actual outcome in particular case.

Yes, that is true. In 'collapse' terms the the first case is uncollapsed, but the second is after measurement. That is pretty important.

 

[stevendaryl]

They are not contradictory if they describe different aspects of reality.

They predict different future probabilities, so they are contradictory predictions. To see this, suppose there is a final state $|final\rangle$. Let $\psi_{U}$ be the probability of making a transition from $|u\rangle |U\rangle$ to the state $|final\rangle$. Let $\psi_{D}$ be the probability of making a transition from $|d\rangle |D\rangle$ to the state $|final\rangle$.

Then, the probability of later observing the system in the state $|final\rangle$ will be

$|\psi_{U}|^2 |\alpha|^2 + |\psi_{D}|^2 |\beta|^2$

under the assumption that the composite system nondeterministically transitioned to $|u\rangle |U\rangle$, with probability $|\alpha|^2$ or to $|d\rangle |D\rangle$, with probability $|\beta|^2$.

In contrast, if the composite system is in the superposition $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$, then the probability of ending up in state $|final\rangle$ will be given by:

$|\psi_{U}|^2 |\alpha|^2 + |\psi_{D}|^2 |\beta|^2 + \alpha^* \psi_{U}^* \beta \psi_{D} +\beta^* \psi_{D}^* \alpha \psi_{U}$

Those are different, and inconsistent, predictions.

 

[Mentz114]

They predict different future probabilities, so they are contradictory predictions. To see this, suppose there is a final state $|final\rangle$. Let $\psi_{U}$ be the probability of making a transition from $|u\rangle |U\rangle$ to the state $|final\rangle$. Let $\psi_{D}$ be the probability of making a transition from $|d\rangle |D\rangle$ to the state $|final\rangle$.

Then, the probability of later observing the system in the state $|final\rangle$ will be

$|\psi_{U}|^2 |\alpha|^2 + |\psi_{D}|^2 |\beta|^2$

under the assumption that the composite system nondeterministically transitioned to $|u\rangle |U\rangle$, with probability $|\alpha|^2$ or to $|d\rangle |D\rangle$, with probability $|\beta|^2$.

In contrast, if the composite system is in the superposition $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$, then the probability of ending up in state $|final\rangle$ will be given by:

$|\psi_{U}|^2 |\alpha|^2 + |\psi_{D}|^2 |\beta|^2 + \alpha^* \psi_{U}^* \beta \psi_{D} +\beta^* \psi_{D}^* \alpha \psi_{U}$

Those are different, and inconsistent, predictions.

That is clearer. Coherence is the difference as Zonde pointed out. Is this not the problem that the decoherence theory addresses ?

In the case where $\psi_{U}$ and $\psi_D$ are orthogonal do the cross-terms disappear ?

 

[zonde]

They predict different future probabilities, so they are contradictory predictions. To see this, suppose there is a final state $|final\rangle$. Let $\psi_{U}$ be the probability of making a transition from $|u\rangle |U\rangle$ to the state $|final\rangle$. Let $\psi_{D}$ be the probability of making a transition from $|d\rangle |D\rangle$ to the state $|final\rangle$.

Then, the probability of later observing the system in the state $|final\rangle$ will be

$|\psi_{U}|^2 |\alpha|^2 + |\psi_{D}|^2 |\beta|^2$

under the assumption that the composite system nondeterministically transitioned to $|u\rangle |U\rangle$, with probability $|\alpha|^2$ or to $|d\rangle |D\rangle$, with probability $|\beta|^2$.

In contrast, if the composite system is in the superposition $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$, then the probability of ending up in state $|final\rangle$ will be given by:

$|\psi_{U}|^2 |\alpha|^2 + |\psi_{D}|^2 |\beta|^2 + \alpha^* \psi_{U}^* \beta \psi_{D} +\beta^* \psi_{D}^* \alpha \psi_{U}$

Those are different, and inconsistent, predictions.

You are considering setup where $|d\rangle |D\rangle$ and $|u\rangle |U\rangle$ can end up in the same state $|final\rangle$. This is clearly interference measurement. But you ignore phase relationship between two initial states, so you are assuming that the two initial states have decohered.
On the other hand for the state $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$ you assume no decoherence.
Can you explain your reasoning vs coherence/decoherence?
To me it seems like you are using two different microtheories for the two cases. In one microtheory each individual measurement outcome is completely independent from the rest of the world and/or from any other measurement in ensemble and so there can be no physical basis to consider any relative phase relationship between outcomes.
In the other microtheory two measurement outcomes sort of exist in two parallel but interacting worlds at the same time so there are means how to consider relative phase relationship between two outcomes.

Well, I'm not sure that inconsistency will persist if you would consider both cases in the same microtheory.

 

[stevendaryl]

You are considering setup where $|d\rangle |D\rangle$ and $|u\rangle |U\rangle$ can end up in the same state $|final\rangle$. This is clearly interference measurement. But you ignore phase relationship between two initial states, so you are assuming that the two initial states have decohered.

Yes, that's the reason I call it a "soft" contradiction. You can't actually calculate phases accurately enough to calculate interference effects between macroscopic objects. But if we assumed unlimited computation power, we could in principle see the difference between the two evolution equations.

Decoherence is a matter of computational ability. Two systems have decohered if it is in practice impossible to accurately predict the phase relationship between them. I actually do not think that decoherence has any role in understanding the foundations of quantum mechanics, although it does explain why in practice we don't see interference effects for macroscopic objects.

Well, I'm not sure that inconsistency will persist if you would consider both cases in the same microtheory.

That's my point: the two microtheories are: (1) smooth evolution according to Schrodinger's equation, and (2) getting definite results of measurements according to the Born rule. It's the inconsistency between these two that I'm pointing out.

 

[stevendaryl]

That is clearer. Coherence is the difference as Zonde pointed out. Is this not the problem that the decoherence theory addresses ?

In the case where $\psi_{U}$ and $\psi_D$ are orthogonal do the cross-terms disappear ?

They don't disappear. They become negligible. That's why I call it a "soft" contradiction. The differences between the predictions are in practice impossible to observe. But they are different predictions.

Just for clarification, $\psi_U$ and $\psi_D$ are just complex numbers, not functions.

$\psi_U = \langle final |e^{-iHt}|u,U\rangle$
$\psi_D = \langle final | e^{-iHt}|d,D\rangle$

 

[stevendaryl]

Decoherence is a matter of computational ability. Two systems have decohered if it is in practice impossible to accurately predict the phase relationship between them.

Actually, there are two things going on decoherence: One, as I said, is just the practical impossibility of computing phase relationships between states of a large system. The second is that if the system becomes entangled with yet other systems, then interference effects are not possible between states of the subsystem, only between different states of the larger composite system. And rapidly, that larger system becomes the entire universe (or the near-by part of it).

 

[zonde]

You can't actually calculate phases accurately enough to calculate interference effects between macroscopic objects. But if we assumed unlimited computation power, we could in principle see the difference between the two evolution equations.

I'm not sure I understand.
Do not assume there is any decoherence. The state that you would use for predicting outcome of interference measurement between $|d\rangle |D\rangle$ and $|u\rangle |U\rangle$ then should be $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$. Exactly the same as in the second case because they are the same case. And your Born probabilities remain hidden variables that you can't observe because you performed interference measurement.

On the other hand if you assume decoherence in both cases then the state $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$ becomes unphysical idealization. Again there is no inconsistency.

I guess you have your viewpoint without HVs and you are saying it is inconsistent. And my suggestion that bringing HVs into the picture makes it consistent is not satisfactory for you.

 

[zonde]

I guess one of the questions about decoherence is why one basis is preferred over another. I would say that position basis is preferred because particles form bond states when they are near each other. That determines that macro world "lives" in position basis.

 

[stevendaryl]

I'm not sure I understand.
Do not assume there is any decoherence. The state that you would use for predicting outcome of interference measurement between $|d\rangle |D\rangle$ and $|u\rangle |U\rangle$ then should be $\alpha |u\rangle |U\rangle + \beta |d\rangle |D\rangle$. Exactly the same as in the second case because they are the same case. And your Born probabilities remain hidden variables that you can't observe because you performed interference measurement.

I don't know what you mean by "remain hidden".

If you use the postulate that a measurement always results in an eigenvalue of the thing that is measured, then there would be no interference between the two alternatives. If you use Schrodinger's equation to compute the probabilities, then there would be interference. So the two axioms are contradictory. They predict different probabilities for winding up in the state $|final\rangle$.

[edit]This assumes that measurement is a physical process by which a microscopic variable is amplified to make a macroscopic difference in the measuring device. If you define measurement to mean "a conscious observer becomes aware of the result" then nothing that devices do can be considered a measurement.

 

[zonde]

If you use the postulate that a measurement always results in an eigenvalue of the thing that is measured, then there would be no interference between the two alternatives.

Can you explain why do you claim that? Because I can imagine different reasons why do you think so, and I would like to avoid guessing.

 

[vanhees71]

Not having followed this thread in detail, I think the problem between your mutual understanding is the usual one not to distinguish clearly between measurements of observables on a system and the preparation of the system in some state. Let's keep the story as simple as possible and assume a non-degenerate observable $A$ to be measured. This means that the eigenspaces for each eigenvalue of the corresponding representing self-adjoint operator $\hat{A}$ are all one-dimensional. Let's denote the corresponding normalized eigenvectors as $|u_a \rangle$, where $a$ denotes the eigenvalue. Also let's suppose these are true eigenvectors, normalizable to 1 and the spectrum of possible eigenvalues $a$ thus discrete.

Let's furter assume the system is prepared in a pure state, represented by the statistical operator $|\psi \rangle \langle \psi|$ with $|\psi \rangle$ a normalized vector. The only meaning, within the minimal interpretation, of this state is that the probability to find the value $a$ when the observable $A$ is measured is given according to Born's rule by
$$W_{\psi}(a)=|\langle u_a|\psi \rangle|^2.$$
This implies, for this most simple case, that the system is prepared with a determined value $a$ of the observable $A$ if and only if $|\psi \rangle \langle \psi|=|u_a \rangle \langle u_a|$, and then $W_{\psi}(a)=1$ as it must be.

Otherwise
$$|\psi \rangle=\sum_{a} \psi_a |u_a \rangle$$
is in a superposition of eigenvectors of $\hat{A}$ and thus
$$W_{\psi}(a)=|\psi_a|^2.$$
For such an ideal measurement, each outcome at an individual system is always one of the possible values $a$ of $A$, which are the eigenvalues of $\hat{A}$, but since the observable $A$ doesn't take a determined value there are only the given probabilities and nothing else due to the preparation of the system in the state $|\psi \rangle \langle \psi|$.

What's usually meant when one talks about the "measurement problem" is this very assumption that there's always a well-defined outcome when $A$ is measured, no matter whether the value of $A$ is determined or note due to the preparation of the system. This is, however, only a metaphysical problem. From a physical, i.e., operational point of view of preparations (defining the quantum state) and measurements, there's no such problem as long as all observations agree with this postulate of Born's rule.

 

[zonde]

What's usually meant when one talks about the "measurement problem" is this very assumption that there's always a well-defined outcome when $A$ is measured, no matter whether the value of $A$ is determined or note due to the preparation of the system. This is, however, only a metaphysical problem. From a physical, i.e., operational point of view of preparations (defining the quantum state) and measurements, there's no such problem as long as all observations agree with this postulate of Born's rule.

The question is not about "measurement problem" itself but rather about inconsistency that this "measurement problem" creates (or does not create) in the model.
In simple words the question is where do the relative phase factors go when measurement is performed. Schrodinger's equation says that relative phase factors live on, while Born rule leaves no place for them to live on. @stevendaryl says there is inconsistency because relative phase factors are fine according to Schrodinger's equation, but they die according to Born rule. I say that relative phase factors could be just fine even after we apply Born rule.

 

[stevendaryl]

What's usually meant when one talks about the "measurement problem" is this very assumption that there's always a well-defined outcome when $A$ is measured, no matter whether the value of $A$ is determined or note due to the preparation of the system. This is, however, only a metaphysical problem. From a physical, i.e., operational point of view of preparations (defining the quantum state) and measurements, there's no such problem as long as all observations agree with this postulate of Born's rule.

My point is that the assumption that there is a well-defined (single) outcome is contradicted by unitary evolution, if you consider the measurement device itself to be a quantum system.

 

[vanhees71]

The question is not about "measurement problem" itself but rather about inconsistency that this "measurement problem" creates (or does not create) in the model.
In simple words the question is where do the relative phase factors go when measurement is performed. Schrodinger's equation says that relative phase factors live on, while Born rule leaves no place for them to live on. @stevendaryl says there is inconsistency because relative phase factors are fine according to Schrodinger's equation, but they die according to Born rule. I say that relative phase factors could be just fine even after we apply Born rule.

This I don't understand. The relative phase factors are the very point that "matter waves" have been introduced and modern QT was discovered in the first place. The relative phases are crucial for, e.g., the result of the double-slit experiment, showing interference fringes in the probability distribution (later demonstrated to be correct by Davisson and Germer with electrons for the first time).

 

[vanhees71]

My point is that the assumption that there is a well-defined (single) outcome is contradicted by unitary evolution, if you consider the measurement device itself to be a quantum system.

This I never understood either. Measurement devices as macroscopic objects are never described by unitary time evolution but by "master equations" of "open quantum systems". That's the whole point of the decoherence program. There's no contradiction between unitary time evolution for closed systems and effective descriptions of macroscopic, i.e., heavily coarse-grained, observables.

 

[stevendaryl]

This I never understood either. Measurement devices as macroscopic objects are never described by unitary time evolution but by "master equations" of "open quantum systems".

Sure. But I'm talking about an idealized situation in which you have an isolated composite system that consists of a measurement device plus the system that it's measuring. You can describe the composite system quantum mechanically.

In the real world, of course, there are interactions between any macroscopic measuring device and the rest of the universe: It's interacting gravitationally and electromagnetically and so forth. But in principle, we can consider an isolated system that contains a measuring device. If the theory is inconsistent in that case, it shows that the theory is inconsistent, period.

 

[vanhees71]

Measurements are done with real-world macroscopic apparati. I let these problems happily to the philosophers to have some food of thought for writing papers with more footnotes than main text...

 

[stevendaryl]

Measurements are done with real-world macroscopic apparati. I let these problems happily to the philosophers to have some food of thought for writing papers with more footnotes than main text...

That's why I said that possibly only philosophers care whether our theories are consistent. But regardless of your attitude toward it, if a theory is inconsistent, then it can't be actually correct. So that gets back to the claim that quantum mechanics is incomplete, or at least, our understanding of it is incomplete.