A Difference Between Collapse and Projection

[vanhees71]

It turns out you can in some sense do a sharp measurement of position. https://arxiv.org/abs/0706.3526 (section 2.3.2)

I'm not able to understand this so quickly, but is this realizable with a real-world experiment? Is there a device with which you can measure the position of a single particle with zero standard deviation, i.e., infinite position resolution?

If this is possible than the Heisenberg uncertainty principle tells us that it cannot be used as a preparation procedure for a particle's exact position!

 

[atyy]

If I understand it right II.4 is the generalization to "weak measurements". For them the same argument applies as for the projection measurements, they are only more comprehensive, i.e., describe more real-world experiments. Whether or not you really prepare the state given in II.4 depends on the specific experimental setup and cannot be considered a general postulate.

Sure, you can be more general than that, but whatever it is, you need a postulate here that is different from unitary evolution of the quantum state.

 

[Lord Jestocost]

The difference is that the collapse refers to a physical/dynamical process, claiming that it is a process that is not described by the quantum-theoretical theory of dynamics (unitary time-evolution). In this sense it's closely related to the assumption of a quantum-classical cut, claiming that quantum theoretical time evolution is not valid for "classical systems".

Of course, the quantum mechanical time evolution is valid for all "physical systems". That's the reason why finally everyting boils down - in mathematical language - simply to the purely quantum-mechanical von Neumann measurement chain. And what happens at the end of the purely quantum-mechanical von Neumann measurement chain? In case the “observer” is regarded as a “pure physical system”, mathematics is unambiguous: Nothing happens; the “purely physical observer” is simply part of the purely quantum-mechanical von Neumann measurement chain!

 

[Demystifier]

As I said, I don't understand, how one can state II.4 in this generality for the discussed reason that in the real world most measurements do not follow it. It depends on the specific realization of the measurement, and usually it needs special care to design a measurement realizing such a generalized projection-like prepatation procedure.

As I said, by Neumarks's theorem, all measurements are projective measurements in a larger Hilbert space.

 

[atyy]

I'm not able to understand this so quickly, but is this realizable with a real-world experiment? Is there a device with which you can measure the position of a single particle with zero standard deviation, i.e., infinite position resolution?

If this is possible than the Heisenberg uncertainty principle tells us that it cannot be used as a preparation procedure for a particle's exact position!

I'm not sure about a real-world experiment, but the proposed formalism gives a sharp position measurement in the sense that if the wave function is $\psi(x)$, the distribution of outcomes is $|\psi(x)|^2$. However, it does not allow preparation of a particle with a definite position (since the state is not in the Hilbert space). So sharp position measurements are possible in some sense, but they do not prepare position eigenstates.

 

[vanhees71]

As I said, by Neumarks's theorem, all measurements are projective measurements in a larger Hilbert space.

Well, so what?

 

[vanhees71]

I'm not sure about a real-world experiment, but the proposed formalism gives a sharp position measurement in the sense that if the wave function is $\psi(x)$, the distribution of outcomes is $|\psi(x)|^2$. However, it does not allow preparation of a particle with a definite position (since the state is not in the Hilbert space). So sharp position measurements are possible in some sense, but they do not prepare position eigenstates.

Sure, that's nothing new compared to the standard (minimal) interpretation. A particle cannot be precisely localized but the Heisenberg uncertainty relation always holds. You can of course always measure the position as accurately as you want, and if you do this accurately enough (i.e., with sufficient resolution) you get $|\psi(x)|^2$ as the position-probability distribution. What has this to do with the projection or collapse postulate?

 

[Demystifier]

Well, so what?

So one can say that the projection rule is universal, applicable to all types of measurements (not merely to some special idealized measurements), provided that one considers the whole closed system including the measuring apparatus.

 

[vanhees71]

Yes, but what has this to do with the projection or collapse postulate? It does not say that if I meausure a particles spin component to be "up" that I have necessarily prepared a particle with spin up, and that's what's claimed by the projection or collapse postulate.

 

[Demystifier]

Yes, but what has this to do with the projection or collapse postulate? It does not say that if I meausure a particles spin component to be "up" that I have necessarily prepared a particle with spin up, and that's what's claimed by the projection or collapse postulate.

Von Neumann projection/collapse postulate does not claim that. Instead, it claims that the state of the whole closed system (measured system + apparatus + environment + observer) gets projected into one term of the entangled superposition.

 

[vanhees71]

Ok, but that contradicts the formalism it claims to interpret.

 

[Demystifier]

Ok, but that contradicts the formalism it claims to interpret.

Von Neumann was completely aware of that, so to avoid the contradiction he postulated two types of evolutions. One is valid when conscious observations are present, the other when conscious observations are not present.

In your interpretation there are no two types of evolutions, but it doesn't make your interpretation simpler because in your interpretation there are two types of something else. Your interpretation has two standards of relevance. Some quantities (the values of observables) are relevant only when they are measured, while other quantities (e.g. the state) are relevant at all times. But you do not follow this relevance rule strictly. For instance, the values of conserved observables are in your interpretation relevant even when they are not observed. But it's impossible to prove your interpretation inconsistent because you never state your standards of relevance explicitly.

 

[kith]

Here is the full version of Nielsen & Chuang's postulate 3:

As far as I can see, this is the most general expression one can give for what happens to the quantum state when a measurement is performed (if one includes only the system of interest in the quantum description). If one tries to replace the Kraus operators M_m by something more general one runs into problems with probabilities not summing to one or the like. So this should apply to all kinds of measurements.

Whether the M_m are easy to determine for a real measurement device is of course a question worth asking. But it is independent of the question whether the full postulate 3 needs to be included in the set of postulates or whether the state-after-measurement rule (2.93) (which is a generalization of the projection postulate) can be omitted.

 

[PeterDonis]

By SG apparatus I mean the magnet plus the detector.

Then SG in your scenario is not an "SG apparatus" by your definition, since it has no corresponding detector.

 

[PeterDonis]

I assume that I can first measure the spin of atom in the z-direction, and later measure the spin of the same atom in the x-direction. Are you saying that it's impossible?

It depends on what you mean by "measure the spin".

If you mean put a filter after the z-direction SG magnet, which only allows the "up" beam to pass, and then put that beam through an x-direction SG magnet and put a detector after it, then yes, the filter can be considered a filter measurement (or, equivalently, as the preparation of a z-spin up state).

If you mean just put two SG magnets in series (or three--your SG, SG1, SG2 scenario), with no filtering, then the first magnet is not a measurement, because there is no filter and no detector; if there were a detector after the first magnet, it would absorb the atom and make it unavailable for the other magnets.

 

[vanhees71]

Von Neumann was completely aware of that, so to avoid the contradiction he postulated two types of evolutions. One is valid when conscious observations are present, the other when conscious observations are not present.

In your interpretation there are no two types of evolutions, but it doesn't make your interpretation simpler because in your interpretation there are two types of something else. Your interpretation has two standards of relevance. Some quantities (the values of observables) are relevant only when they are measured, while other quantities (e.g. the state) are relevant at all times. But you do not follow this relevance rule strictly. For instance, the values of conserved observables are in your interpretation relevant even when they are not observed. But it's impossible to prove your interpretation inconsistent because you never state your standards of relevance explicitly.

Yes, and I don't agree with this solipsistic interpretation. Concerning physics, never listen to mathematicians (and the other way: concerning mathematics better listen to the mathematicians ;-)).

I don't know, what you mean by "relevant". There is a formalism conisting of an algebra of observables with a unitary (ray) representation of this algebra on a Hilbert space, describing the system and its dynamics, leading to probabilistic predictions for the outcome of measurements and that's the physics. I don't care for philosophical speculations beyond the physics. What happens to the measured system and the measurement device when one measures an observable and to which extent I have to describe it depends on the specific situation and cannot be postulated as a general law defining the general formalism.

 

[PeterDonis]

I don't agree with this solipsistic interpretation.

Saying that it takes a conscious observer to trigger the second type of evolution (the one that converts an entangled superposition into just one of its terms) does not require saying that you are the only conscious observer that exists.

 

[vanhees71]

It's utter nonsense to be clear. It implies that only because I or some other physicist (or is a layman enough who doesn't understand what he sees or an amoeba, as Bell put it once?) a measurement is finished and the act of "conscious observation" would lead to some dynamics on the system.

Nowadays the measurement data are stored in some computer file and read by a "conscious observer" months after the data are taken. The measured system is long gone. For me von Neumann's interpretation is esoterics and has nothing to do with science.

 

[vanhees71]

Here is the full version of Nielsen & Chuang's postulate 3:

View attachment 276362
View attachment 276363

As far as I can see, this is the most general expression one can give for what happens to the quantum state when a measurement is performed (if one includes only the system of interest in the quantum description). If one tries to replace the Kraus operators M_m by something more general one runs into problems with probabilities not summing to one or the like. So this should apply to all kinds of measurements.

Whether the M_m are easy to determine for a real measurement device is of course a question worth asking. But it is independent of the question whether the full postulate 3 needs to be included in the set of postulates or whether the state-after-measurement rule (2.93) (which is a generalization of the projection postulate) can be omitted.

That's all fine with me when you just stop after Eq. (2.92). The claim what the state after the measurement is, is dependent on the situation (the experiment) you describe. In general I don't think that you have a pure state after the measurement, because the measured system is entangled with the measurement device ("immediately after the measurement") and thus the system's state given by the partial trace (tracing out the system), and this leads to a mixed state for the system rather than a pure state after the measurement.

In my opinion all that's predicted and really measurable are the probabilities (2.92). If the measurement is only an intermediate measurement and you measure something else later on that same system, then you have to describe it specifically for the given setup. You cannot in all generality state that (2.93) is "the state of the system immediately after the measurement".

 

[vanhees71]

It depends on what you mean by "measure the spin".

If you mean put a filter after the z-direction SG magnet, which only allows the "up" beam to pass, and then put that beam through an x-direction SG magnet and put a detector after it, then yes, the filter can be considered a filter measurement (or, equivalently, as the preparation of a z-spin up state).

If you mean just put two SG magnets in series (or three--your SG, SG1, SG2 scenario), with no filtering, then the first magnet is not a measurement, because there is no filter and no detector; if there were a detector after the first magnet, it would absorb the atom and make it unavailable for the other magnets.

If you just consider the magnet and not detecting the particles you have a preparation procedure for spin-component eigenstates (component in direction of the homogeneous part of the magnetic field), described by a unitary time evolution. The preparation is due to the entanglement (nearly 100% if you have a good design of the magnet and the incoming beam) between the spin component and the position (or momentum) of the particle.

I think in the usual sense that's indeed not yet a measurement, because with the preparation you have only established the correlation between the spin component and position of the particle, i.e., a correlation, telling you that if you register a particle in one partial beam you have with certainty "spin up" and if you register a particle in the other partial beam the particle has with certainty "spin down".

For an individual particle it is not predictable in which partial beam (and thus with which spin component) it leaves the magnet but it ends up with some probability in the one or the other beam depending on the initial state. In the original setup with a beam coming out of an oven you'll have probabilities of 1/2 for either outcome.

Of course, if you have just three magnets as described in the above arrangement then you just have to describe the interaction of the particle with the three magnets in the Hamiltonian to predict the outcome of the measurement after they went through all three magnets.

If you do a measurement after the 1st magnet or block one partial beam after the 1st magnet you have to describe this too, because then of course you change the state of the particle through the interaction with the measurement device or filter. Usually this is done in a FAPP description a la projection postulate, but in principle it's just an interaction of the particle with some material/fields making up the measurement device of filter and is described by the usual quantum dynamics (of the composite system consisting of the particle and the devices used to detect it), making the particle to an open quantum system. When "tracing out" the information about the state of the devices/environment of course the time evolution of the particle alone is no longer unitary.

A simple and in some sense extreme example is a Brownian particle in a "heat bath". Over some steps of "coarse graining" you end up with a (in general non-Markovian) Langevin equation describing the particle's motion or even just an equation of motion with some friction for its average position, i.e., a classical equation of motion. Then nothing is left from the unitary time evolution of the entire huge system consisting of the particle and the heat bath, but the derivation shows that this classical behavior is completely consistent with unitary time evolution on the full quantum level when considering the entire closed system, which of course you never can calculate in the full detail, because with around $10^{24}$ particles it's hard to solve the fully resolved dynamics of all these "microscopic degrees of freedom".

 

[vanhees71]

I suspect that @vanhees71 refers to a closed system, because he believes that we can in principle include the observer and measurement apparatus in the quantum state, so that there is only unitary evolution. This is also my reading of what Ballentine means in his textbook, given his criticism of standard QM. I believe that postulating unitary evolution without state reduction is not correct unless one introduces additional postulates (eg. as attempted by many worlds, hidden variables, which also remain non-standard).

That's precisely the question. Which additional postulates do you mean? Why precisely do you think that the minimal statistical interpretation is incomplete? You always claim Ballentine's textbook is fundamentally wrong, but there's no convincing argument for that claim. It's just an opinion.

I think it's a philosophical standpoint rather than a valid critique of QT as a physical theory. You seem to consider QT as incomplete in the same sense as Einstein did, i.e., Einstein indeed seems to have accepted QT in its minimal (ensemble) interpretation but then considered it incomplete, because for Einstein a complete theory should be deterministic in the sense that, provided you have "complete knowledge" about the system's state, you should be able to precisely predict the outcome of all possible measurements on the individual system and not only about the statistics for these outcomes when measured on an ensemble of equally prepared systems.

Einstein's postulate what "complete knowledge" in the sense of a more comprehensive theory than QT means was that there are unkown "hidden variables" which we just ignore or don't have discovered yet to exist. In other words QT was incomplete in the same sense as the description of a classical mechanical in terms of a coarse-grained probabilistic description using only partial information about the system in statistical mechanics, because you simply cannot keep track of the full phase-space trajectories of $10^{24}$ particles, i.e., it's the step from the full Liouville equation (describing full knowledge about the phase-space evolution of the entire system) to some transport equation, throwing away a lot of information in terms of correlations by cutting the BBGKY hierarchy at a certain order. In the Boltzmann equation you only consider single-particle phase-space distribution functions and throwing away correlations already at the two-particle level, introducing the "molecular-chaos assumption" and at this point throwing away a lot of detailed information about the system, leading to increasing macroscopic entropies and dissipation, implying the H-theorem (with the thermodynamic time arrow identical with the causal time arrow implicit in the full mechanical dynamics from the beginning).

 

[atyy]

That's precisely the question. Which additional postulates do you mean? Why precisely do you think that the minimal statistical interpretation is incomplete? You always claim Ballentine's textbook is fundamentally wrong, but there's no convincing argument for that claim. It's just an opinion.

I think it's a philosophical standpoint rather than a valid critique of QT as a physical theory. You seem to consider QT as incomplete ...

The question is not whether the minimal statistical interpretation is complete. My claim is that Ballentine does not give the minimal statistical interpretation. He gives a wrong theory due to omission of state reduction. The minimal statistical interpretation contains state reduction.

 

[vanhees71]

But the minimal statistical interpretation doesn't contain a state-reduction postulate. That's the main point distinguishing it form some flavors of Copenhagen interpretations including a state-reduction postulate. So you are of the opinion that you cannot use QT without a state-reduction postulate. That's an opinion I have to respect, but I don't understand where you need the state reduction to apply the formalism to real-world experiments nor do I understand, why I should accept a postulate contradicting the other postulates and is not generally consistent with what's done in experiments. Projective measurements are very rare and very hard to realize!

 

[atyy]

Projective measurements are very rare and very hard to realize!

The elementary description of Bell experiments uses projective measurements, and these experiments are realized routinely nowadays. If the measurements are spacelike separated, there will be a frame in which you can consider the measurements simultaneous. But that means in another frame, the measurements are sequential, and the projection postulate ensures consistency of outcomes regardless of what frames are used to calculate.

 

[Demystifier]

if there were a detector after the first magnet, it would absorb the atom and make it unavailable for the other magnets.

Is it possible to detect an atom without absorbing it?

 

[Demystifier]

Concerning physics, never listen to mathematicians

Von Neumann was much more than just a mathematician. He was the first who understood what physically happens with the state of the closed system during the measurement, using non-rigorous physical arguments. It's now called Von Neumann theory of measurement. He was also the first (or maybe the second, after Landau) who introduced the concept of density matrix in quantum physics, in a physical fashion.

Of course, as a mathematician he also contributed significantly to rigorous functional-analysis style formulation of quantum mechanics, but that's another story.

 

[Lord Jestocost]

Nowadays the measurement data are stored in some computer file and read by a "conscious observer" months after the data are taken. The measured system is long gone. For me von Neumann's interpretation is esoterics and has nothing to do with science.

Please, stay with the mathematics of the purely quantum-mechanical von Neumann measurement chain (and, please, avoid terms like "esoterics" as scientific arguments).

Regarding the Heisenberg cut (or “collapse” or “state reduction” or whatever one might chose to call it), it is a “purely epistemological move without any counterpart in ontology” (as N.P. Landsman characterizes Heisenberg’s and Bohr’s reasoning with respect to the Heisenberg cut in his paper “Between classical and quantum”). Or, to elaborate it a little bit more (Klaas Landsman in “Foundations of Quantum Theory / From Classical Concepts to Operator Algebras”):

“Summarizing the preceding discussion, 'our' measurement problem states that:
• Certain pure post-measurement states of an (ontologically quantum-mechanical!) apparatus coupled to a microscopic quantum object induce mixed states on the apparatus (and on the composite) once the apparatus is described classically.
This is a precise version of Schrödinger’s Cat problem (rather than von Neumann’s purely quantum-mechanical measurement problem), making it clear that at heart the problem does not lie with the (dis)appearance of interference terms (which is a red herring) but with the inability of quantum mechanics to predict single outcomes.” [Italics and bold in original, LJ]

It is extremely important to understand this point: In case you want to avoid this epistemological move (one has to grasp that it's - so to speak - a mental decision where to arbitrarily interrupt the purely quantum-mechanical von Neumann measurement chain; epistemologically without any ontological basis), you end up at the end of the von Neumann measurement chain. There is no esotericism, it's pure mathematics.

Statements like “Nowadays the measurement data are stored in some computer file and read by a "conscious observer" months after the data are taken.“ exactly mirror the fundamental misunderstanding of the messages send by quantum mechanics. Such statements are nothing but "classical common sense" imaginations, but the mathematics of the purely quantum-mechanical von Neumann measurement chain speaks something else.

 

[Demystifier]

You always claim Ballentine's textbook is fundamentally wrong, but there's no convincing argument for that claim.

In the book he explicitly makes a testable prediction that the quantum Zeno effect does not exist. Experiments prove him wrong.

 

[Demystifier]

In general I don't think that you have a pure state after the measurement, because the measured system is entangled with the measurement device ("immediately after the measurement") and thus the system's state given by the partial trace (tracing out the system), and this leads to a mixed state for the system rather than a pure state after the measurement.

If so, then the state of the closed system (the measured system + the measurement device) is in the macroscopic superposition. It's a superposition of different possible measurement outcomes. Yet only one outcome actually realizes, we never observe superpositions of different possible outcomes. Do you agree that, when we learn what the actual outcome is, we can update our knowledge by using the state (2.93) in post #43?

 

[vanhees71]

Von Neumann was much more than just a mathematician. He was the first who understood what physically happens with the state of the closed system during the measurement, using non-rigorous physical arguments. It's now called Von Neumann theory of measurement. He was also the first (or maybe the second, after Landau) who introduced the concept of density matrix in quantum physics, in a physical fashion.

Of course, as a mathematician he also contributed significantly to rigorous functional-analysis style formulation of quantum mechanics, but that's another story.

Yes, and the latter is his great achievement concerning quantum theory. His part on "interpretation" is not so brillant. That's known since Herman's work in the 30ies.