Quantum state of system before measurement

[soviet1100]

Hello!

If we consider a single-particle system, I understand that the measurement of an observable on this system will collapse the wave function of the system onto an eigenstate of the (observable) operator.

Therefore, we know the state of the system immediately after the measurement. But as regards the state of the system before the measurement, what definitive statements can we make (in the Copenhagen/minimalist interpretation)? Are such questions physically meaningless? Or does the system exist in an unknown state in the Hilbert space? Or does the system not exist at all until the actual measurement is made?

Grateful for any answers, thanks!

 

[jfizzix]

We can't really know what the state of a single particle is before measurement unless we specifically prepare it to be in a given state.

However, if you have a lot of particles to measure in the same quantum state, you can use the statistics of different measurements to get a picture of what the average quantum state is. This process is known as quantum state tomography.

 

[jfizzix]

Also, to say that a quantum system has a definite quantum state before measurement is something not everyone agrees upon (interpretations abound), but it is a defensible position to take.

 

[soviet1100]

many thanks, jfizzix. that cleared my doubts on the issue. If or not a system has a definite quantum state before measurement is an interpretational issue then.

 

[jfizzix]

Glad I could help!

As a side point, whether or not a single quantum system has a definite quantum state is related to one's position on an underlying objective reality (i.e., that there is (perhaps unobtainable) information determining the physical state of any system independent of measurement).

There's an interesting theorem by Pusey, Barrett, and Rudolph (PBR) that discusses this relationship, though in truth, I don't fully understand its implications beyond showing that certain experimental results will prove that it is a defensible position.

 

[soviet1100]

Thanks again. I will read more into the interpretations of QM. University courses on QM generally ignore the interpretational aspects of QM and just teach you 'how to do QM'.

 

[atyy]

Also relevant http://arxiv.org/abs/quant-ph/0507214.

 

[nuclear_chris]

The statement you can make about a wave function before measurement is that it is a superposition or sum of basis wave functions, such that when a measurement is made, that wave function collapses into one of the basis.
Another words it is a wave packet made of many waves each with its own position or momentum value. A measurement will reduce the wave packet to a single wave and a single position or momentum.
Think of it as a straight-arrow vector that can be projected onto one of the axis (the projection being the measurement).

 

[bhobba]

Thanks again. I will read more into the interpretations of QM. University courses on QM generally ignore the interpretational aspects of QM and just teach you 'how to do QM'.

Read Ballentine:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

It integrates interpretation and understanding the formalism.

For example you will learn the true basis of Schroedingers equation is simply that probabilities are frame independent and the standard classical Galilean Transformations.

If you want to go deeper then check out
https://www.amazon.com/dp/3642071422/?tag=pfamazon01-20

Thanks
Bill

 

[soviet1100]

thank you for the reading recommendations, bhobba. I have a copy of Ballentine's excellent quantum mechanics text, although I haven't read it yet. I just completed a first course on quantum mechanics using R.Shankar's Principles of Quantum Mechanics. Shankar chooses not to explore interpretational issues either, although the text itself is a pedagogic masterpiece.

From what I've heard of Ballentine, he espouses the statistical/ensemble interpretation of quantum mechanics. Does the interpretation have any effect whatsoever on the formalism of quantum mechanics, or the results, or the interpretation of the results? Does Ballentine also make clear the distinction between formalism and interpretation in his book, in that he points out clearly those ideas and points that are open to interpretation and those that are indisputable and common to any interpretation?

 

[atyy]

From what I've heard of Ballentine, he espouses the statistical/ensemble interpretation of quantum mechanics. Does the interpretation have any effect whatsoever on the formalism of quantum mechanics, or the results, or the interpretation of the results? Does Ballentine also make clear the distinction between formalism and interpretation in his book, in that he points out clearly those ideas and points that are open to interpretation and those that are indisputable and common to any interpretation?

There are ensemble interpretations which are correct, such as bhobba's, but there are serious problems with Ballentine's ensemble interpretation. Ballentine's chapter 9 and section 12.2 in his 1998 book are misleading or wrong in their basic conception of quantum mechanics. Strikingly, he claims that his ensemble interpretation and the orthodox Copenhagen interpretation make different predictions, and he wrongly claims in two places that there is evidence against the orthodox Copenhagen interpretation. The first place he does this is the spin recombination experiment of Fig 9.2, which he analyzes wrongly because ignores that in Copenhagen a measurement involves the interaction of the apparatus with the system such that it gives a definite outcome, and that one can always shift the classical/quantum cut to include the measurement apparatus.

He specifically rejects that it is possible to treat a pure state as the complete state of an individual physical system, and he rejects state reduction. This is possible if one introduces hidden variables or adopts Many-Worlds. However, neither is clearly stated in his book, which is therefore at least misleading. Also it is unclear how his ensemble interpretation differs from the very notions he rejects. For example, he says that the state labels a conceptual ensemble that is assigned to a single system. If the members of the conceptual ensemble are assigned additional labels that distinguish them, then it is a hidden variables interpretation. If the members are not assigned additional labels, then because the ensemble labels a single system, and the state labels the ensemble, one ends up with a pure state being the most complete possible specification of an individual system (because the pure state is an extremal point), which is the very thing he rejected. Also his Eq 9.28 is indistinguishable from the state reduction postulate of Copenhagen, and cannot be derived from unitary evolution alone, which again means that he postulates what he rejects. These conceptual errors lead to his second wrong analysis in section 12.2, where he ignores the formalism of continuous measurement which can be used to explain the classical tracks in cloud chambers.

One should also note that Ballentine's 1970 review on an earlier version of his ensemble interpretation, which also attacked the orthodox Copenhagen interpretation, uses the concept of a classical trajectory in quantum mechanics to wrongly claim that canonically conjugate position and momentum can be simultaneously and accurately measured. So it is not clear that Ballentine has ever produced a clear and correct exposition of his ensemble interpretation.

I would stick with standard textbooks such as Shankar. If you want some that mention interpretation try Landau & Lifshitz or Weinberg. The state reduction postulate in those books are a bit old fashioned, and specifically have trouble with continuous variables. One can find more modern versions of the state reduction postulate in Nielsen and Chuang, or Holevo.

 

[bhobba]

From what I've heard of Ballentine, he espouses the statistical/ensemble interpretation of quantum mechanics. Does the interpretation have any effect whatsoever on the formalism of quantum mechanics, or the results, or the interpretation of the results?

No.

Its simply a sharpening of the Born rule so that states apply to ensembles of similarly prepared systems. The formalism of QM doesn't actually spell out an interpretation of probability but most applied mathematicians have a, not necessarily explicitly stated, frequentest view - in my experience anyway from studying it at university. In Feller's Introduction To Probability Theory And Its Applications, for example, he gives that view explicitly. All the ensemble interpretation does is use that view and apply it to the ensemble so that when you observe a system in that state it conceptually selects an outcome from that ensemble. That will be in proportion to the probability of its occurrence from the frequentest interpretation so it conforms to the Born Rule. The difference with Copenhagen (most versions anyway) is the interpretation of probability is subjective which is more along the lines of a Bayesian view of probability. Some applied mathematicians involved in areas like decision theory or Bayesian inference use that view. You will find the two camps go at it a bit but the fact is they are both valid, but without a specific reason most seem to gravitate toward frequentest. My background in applied math was in statistical modelling and that seems more naturally frequentest because you deal with things like queues in a bank etc and it natural to think of the queue that way.

Does Ballentine also make clear the distinction between formalism and interpretation in his book, in that he points out clearly those ideas and points that are open to interpretation and those that are indisputable and common to any interpretation?

Unfortunately no. He examines a straw man argument against Copenhagen for example. Its value however is not in a balanced view of interpretations but in integrating the interpretive issues into the text so you see what they are and how ensemble resolves it. A balanced view is found in Schlosshauer's book.

The other big advantage of Ballentine is it does an axiomatic development from just two axioms. That way you see what the real assumptions in QM are which is of great value in nutting out exactly what interpretations are saying. For example I did a recent post on Zurecks Quantum Darwinism because his comments on the axiomatic assumptions he uses is incorrect - as can be easily seen from Ballentine's axioms and an important theorem called Gleason's theorem. In fact there is really only one key axiom (see post 137):
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Once you understand that you get a lot clearer view of different interpretations.

Regarding Atty's comment on Ballentine being wrong - Atty is one of my favourite posters and I nearly always agree with him, and he has taught me a lot, but on the issue of Ballentine being incorrect we part company. My ignorance ensemble interpretation is basically a slight variation on Ballentine's simply applying it just after decoherence - mine can't be correct and Ballentine wrong. The only issue with Ballentine is the straw man argument I mentioned and he doesn't believe decoherence has anything to say on interpretive issues - a point I most emphatically disagree with.

Thanks
Bill

 

[atyy]

Regarding Atty's comment on Ballentine being wrong - Atty is one of my favourite posters and I nearly always agree with him, and he has taught me a lot, but on the issue of Ballentine being incorrect we part company. My ignorance ensemble interpretation is basically a slight variation on Ballentine's simply applying it just after decoherence - mine can't be correct and Ballentine wrong. The only issue with Ballentine is the straw man argument I mentioned and he doesn't believe decoherence has anything to say on interpretive issues - a point I most emphatically disagree with.

Even there I don't think we really disagree. If I understand you correctly, your ensemble interpretation is essentially Copenhagen (except you usually take a bigger Hilbert space to allow decoherence to give the pointer basis), and has essentially the same postulates and the same problems as Copenhagen (you usually state the measurement problem a bit differently as the problem of definite outcomes, again because you usually take decoherence into account, but the measurement problem is still there).

What I most object to is that Ballentine presents Copenhagen as inconsistent with data (not true) and as having troubling things like state reduction (more or less true), and the natural reading of his text is that the ensemble interpretation doesn't have these troubling things and thus at least partly solves the measurement problem (not true, especially since he doesn't accept decoherence, which can be argued to remove the choice of pointer basis being made by an external observer, if a system-apparatus division has been chosen). So if one reads Ballentine's ensemble interpretation as being defined in opposition to and solving the problems of Copenhagen, then Ballentine's ensemble interpretation doesn't work since it is essentially Copenhagen with essentially the same problems.

Of course if one removes Ballentine's objections to Copenhagen, then the contradictions in Ballentine's ensemble interpretation go away, and the interpretation can be taken as a valid quantum mechanical interpretation that is simply some version of Copenhagen with the same problems. So whether Ballentine's ensemble interpretation is right or wrong depends on whether his erroneous criticism of Copenhagen and a claim to solve some problems of Copenhagen is an essential part of his interpretation.

 

[lucas_]

In the original thread https://www.physicsforums.com/threads/zurecks-quantum-darwinism-paper.802260/page-2#post-5039988 (the original poster Bhobba said the thread is about Gleason theorem so let me post it here so it's not off topic there)

Yes it does.

In decoherence we usually read that a pure density matrix evolves toward a mixed one diagonal in a basis and that one of the vectors of this basis will be the output of the measurement.

Here Zurek reverses the way to see things. When a system decoheres while interacting with the environment it leaves many imprints in it. Eg take a grain of dust in a superposition of what you want. When it is illuminated it returns many photons in different directions. Many observers can receive them. they will agree that they come from a same source. An observable emerges from the decoherence Zurek calls it $\pi$ and writes:

Observers probing fraction of the universe can act AS IF the system has a state of its own (one of the eigenstates of $\pi$)

What would happen if a laser beam is targetted directly to the grain of dust (here it is not intercepting the photons), what would it see? You said "Observers probing fraction of the universe can act AS IF the system has a state of its own (one of the eigenstates of $\pi$)". Since the laser is not probing fragments from the system but directly perturbing the system, why does it still act like "system has a state of its own (one of the eigenstates of $\pi$)", shouldn't it see something else?

 

[naima]

The laser is not an observer. Zurek writes that the observers of a particle often observe it indirectly. Eg they receive somewhere else a photon coming from the laser but reflected by the particle. the laser is a part of the environmant that is the cause of decoherence. the relected photons are the imprints that observer can measure. their many datas will be coherent with one eigenvector of an emerging observable. (if i understand it!)

 

[lucas_]

The laser is not an observer. Zurek writes that the observers of a particle often observe it indirectly. Eg they receive somewhere else a photon coming from the laser but reflected by the particle. the laser is a part of the environmant that is the cause of decoherence. the relected photons are the imprints that observer can measure. their many datas will be coherent with one eigenvector of an emerging observable. (if i understand it!)

Ah, you saying anything that can interact/perturb the object can form part of the imprints? Is there no way for an observer to view the object directly without using the imprints.. that is what I meant by the laser to observe the grain of dust spectroscopic state.. what other instruments do you know that can probe the grain of dust and without influencing or become part of the reflected imprints?

 

[naima]

When you see "directly" something your brain analizes the imprints on your retina which sends a cascade of signals through your optic nerve. those signals are also imprints.

 

[soviet1100]

many thanks to all of you for the comments. Bhobba, atyy, thank you for the comments on Ballentine's text. I will keep in mind your points when I read the book.