Difference Between Collapse and Projection

[Demystifier]

[Moderator's note: Spun off from a thread in the QM forum to separate out the interpretation discussion.]

"Collapse", as @vanhees71 was using the term in the post I was responding to in what you quoted from me, is an interpretation-dependent concept. "Projection" is just the basic mathematical operation described in Rule 7.

Yes, but I was hoping that you could give a precise specification of the difference. (In my opinion, @vanhees71 never specified the difference precisely.)

 

[vanhees71]

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".

In my opinion both has no foundation whatsoever. There's no quantum-classical cut ever observed. To the contrary recent measurements of quantum behavior of larger and larger objects have been achieved. So the quantum dynamics turns out to be valid for arbitrarily large systems, and thus it's as well valid for the description of the reaction of the measured system with the measurement apparatus. Thus an instantaneous-collapse assumption also contradicts relativistic local QFTs which postulate local interactions, i.e., the microcausality principle, so that also within QFTs there's no causal effects between space-like events.

In contradistinction to this the ensemble interpretation (statistical minimal interpretation) just takes the probabilities from Born's rule, which is just taken as another postulate independent of all the other postulates (see also Weinberg's, Lectures on Quantum Mechanics, where it is made convincingly plausible that Born's rule is not derivable from the other postulates). There's no collapse necessary to just interpret the quantum states as probabilities/probability distributions using Born's rule.

 

[Demystifier]

In contradistinction to this the ensemble interpretation (statistical minimal interpretation) just takes the probabilities from Born's rule, which is just taken as another postulate independent of all the other postulates (see also Weinberg's, Lectures on Quantum Mechanics, where it is made convincingly plausible that Born's rule is not derivable from the other postulates). There's no collapse necessary to just interpret the quantum states as probabilities/probability distributions using Born's rule.

My problem is the following. Let a possible result of measurement be defined by a projector $\pi$. Then, in the minimal statistical interpretation, there are two closely related rules:

(i) the Born rule: the probability of this measurement result is
$$p=\langle \psi|\pi|\psi\rangle$$
(ii) the projection rule: after the measurement, the knowledge about the system is given not by $|\psi\rangle$ but by
$$|\psi'\rangle =\frac{\pi|\psi\rangle}{\sqrt{p}}$$
Question: Can (ii) be derived from (i), or is (ii) an independent postulate?

 

[vanhees71]

My problem is the following. Let a possible result of measurement be defined by a projector $\pi$. Then, in the minimal statistical interpretation, there are two closely related rules:

(i) the Born rule: the probability of this measurement result is
$$p=\langle \psi|\pi|\psi\rangle$$
(ii) the projection rule: after the measurement, the knowledge about the system is given not by $|\psi\rangle$ but by
$$|\psi'\rangle =\frac{\pi|\psi\rangle}{\sqrt{p}}$$
Question: Can (ii) be derived from (i), or is (ii) an independent postulate?

(ii) is not part of the minimal interpretation. This can be the case for very special "measurements" ("von Neumann filter measurements). It's rather a way of preparation, possible only via local interactions. E.g., for a partial beam of a Stern-Gerlach experiment you can block one of the partial beams to prepare (in principle as accurately as you wish) a pure spin-up state.

(ii) is not a necessary postulate for the formulation of QT and is simply not true for almost all measurements in the real world.

 

[vanhees71]

Where is the Rule 7 in the Ballentine's book? I cannot find it written down explicitly. (The Rule 7 is the same as the rule (ii) in my #49 above.)

Exactly! It's not part of the minimal interpretation. I remember that at the time we discussed about the Insights article, people insisted on Rule 7, and it is okish in the way it's formulated because it's emphasized that it's a special (idealized) case of a state preparation. I'd not list it as a fundamental postulate but as a definition of a von Neuman filter measurement (and rather a preparation procedure than a measurement procedure).

 

[Lord Jestocost]

"According to the projection postulate, when a measurement of $F$ is made, the state vector undergoes a transition $| \psi_0 \rangle \mapsto | f_n \rangle$ to one of the eigenstates of $\hat F$. This occurs for a specified value of $n$ with the probability
$$ \pi_n=|\langle f_n| \psi_0 \rangle| ^2$$
and the result of the measurement in that case is the eigenvalue $f_n$. The associated transition $| \psi_0 \rangle \mapsto | f_n \rangle$ is called the state reduction or ‘collapse of the wave function’ arising from the measurement of $F$."

S L Adler, D C Brody, T A Brun and L P Hughston in "Martingale models for quantum state reduction" (Journal of Physics A: Mathematical and General, Volume 34, Number 42)
https://arxiv.org/abs/quant-ph/0107153v1

 

[Demystifier]

(ii) is not part of the minimal interpretation. This can be the case for very special "measurements" ("von Neumann filter measurements). It's rather a way of preparation, possible only via local interactions. E.g., for a partial beam of a Stern-Gerlach experiment you can block one of the partial beams to prepare (in principle as accurately as you wish) a pure spin-up state.

a) I agree that (ii) is not generally valid, but I don't agree that it's only true for filter measurements. It's also true for projective measurements.

b) An example is Stern-Gerlach (SG) without blocking, with two subsequent SG apparatuses, called SG1 and SG2. For instance, if the particle is detected at SG1 at the upper position, then the statistic for SG2 should be computed without the lower beam. That's an application of rule (ii) without blocking/filtering. How can the minimal statistical interpretation explain that?

c) The Neumark's theorem shows that any non-projective measurement can be represented by a projective measurement in a larger Hilbert space. Physically, the larger Hilbert space corresponds to the space in which the quantum state of the measuring apparatus is also taken into account. In this sense, the rule (ii) is valid for all measurements.

 

[vanhees71]

If you don't block one of the beams, it's not a preparation in an eigenstate of $\hat{s}_z$.

If you just put two more SG magnets behind each of the two partial beams there's not been made any projection anyway. You just write down the complete Hamiltonian for all three SG apparati do describe the situation. There's of course no problem with this setup within the minimal interpretation.

I don't understand c). It's true for measurements in the sense of the POVM formalism of course.

The projection postulate as stated in #54 is just wrong for much simpler cases, e.g., measuring a single photon with a photon detector. After the measurement the photon is simply absorbed, i.e., destroyed. It's not projected to the "eigenstate" corresponding to the outcome of the measurement but at best you can say that there's no more photon and thus you have "projected" the state to the vacuum state.

 

[PeterDonis]

I was hoping that you could give a precise specification of the difference.

There is no single precise specification of the difference because there is no single meaning of the term "collapse"; it's interpretation-dependent.

 

[PeterDonis]

the collapse refers to a physical/dynamical process

In some interpretations, yes. Not in all. In some interpretations it just refers to updating the observer's knowledge about the system.

 

[PeterDonis]

An example is Stern-Gerlach (SG) without blocking, with two subsequent SG apparatuses, called SG1 and SG2.

A Stern-Gerlach magnet by itself is not a measurement; it's just a unitary operation that entangles the particle's spin with its linear momentum. To actually make a measurement, you have to put a detector screen downstream of the magnet (or in this case, two such screens, one downstream of SG1 and one downstream of SG2). So there is no projective measurement taking place at SG itself.

 

[kith]

(ii) is not a necessary postulate for the formulation of QT and is simply not true for almost all measurements in the real world.

It makes sense to disentangle the two aspects here: the question whether (ii) is directly applicable to typical real-world measurements and the question whether something like (ii) is needed as a fundamental postulate.

I think everyone agrees that the answer to the first question is no (as long as one doesn't enlarge the Hilbert space by redefining the quantum system). The second question is what the disagreement seems to be about.

Nielsen & Chuang have a nice generalization of (ii) as part of their postulates. They distinguish between projective measurements and general measurements and only talk about general measurements in their postulates.

In their postulate 3, they include both the Born rule and a state-after-measurement rule which generalizes the projection postulate. Let me quote the relevant part: "Quantum measurements are described by a collection ${M_m}$ of measurement operators. [...] the state of the system after the measurement is
$$\frac{M_m |\psi \rangle}{\sqrt {\langle \psi | M_m^{\dagger}M_m | \psi \rangle}}.$$ The measurement operators satisfy the completeless relation [...]"

What do you think about the general state-after-measurement rule? Can it be omitted from the postulates without substitution?

 

[vanhees71]

a) I agree that (ii) is not generally valid, but I don't agree that it's only true for filter measurements. It's also true for projective measurements.

b) An example is Stern-Gerlach (SG) without blocking, with two subsequent SG apparatuses, called SG1 and SG2. For instance, if the particle is detected at SG1 at the upper position, then the statistic for SG2 should be computed without the lower beam. That's an application of rule (ii) without blocking/filtering. How can the minimal statistical interpretation explain that?

c) The Neumark's theorem shows that any non-projective measurement can be represented by a projective measurement in a larger Hilbert space. Physically, the larger Hilbert space corresponds to the space in which the quantum state of the measuring apparatus is also taken into account. In this sense, the rule (ii) is valid for all measurements.

Ad a) For me filter measurements and projective measurements are synonymous. What's the difference in your definition?

Ad b) Then there's no measurement between the first and the second+third SG magnets. That has to be formulated within the Hamiltonian for the particle (silver atom).

Ad c) You mean for the description of general measurements in terms of the POVM formalism.

That filter and projective measurements are a very special case is much more simple to see. If you measure a photon with some detector, usually it gets absorbed, i.e., concerning the em. field the measurement leads to a transition from a one-photon state to the vacuum and no projection to the single-photon state measured by the detection event.

 

[vanhees71]

It makes sense to disentangle the two aspects here: the question whether (ii) is directly applicable to typical real-world measurements and the question whether something like (ii) is needed as a fundamental postulate.

I think everyone agrees that the answer to the first question is no (as long as one doesn't enlarge the Hilbert space by redefining the quantum system). The second question is what the disagreement seems to be about.

Nielsen & Chuang have a nice generalization of (ii) as part of their postulates. They distinguish between projective measurements and general measurements and only talk about general measurements in their postulates.

In their postulate 3, they include both the Born rule and a state-after-measurement rule which generalizes the projection postulate. Let me quote the relevant part: "Quantum measurements are described by a collection ${M_m}$ of measurement operators. [...] the state of the system after the measurement is
$$\frac{M_m |\psi \rangle}{\sqrt {\langle \psi | M_m^{\dagger}M | \psi \rangle}}.$$ The measurement operators satisfy the completeless relation [...]"

What do you think about the general state-after-measurement rule? Can it be omitted from the postulates without substitution?

Yes, it's not a generally valid rule. You have to analyse your measurement process depending on the used measurement device to know, how the observed system looks after interacting with the measurement device.

If, e.g., I measure a photon by detecting it, usually it's absorbed. What was observed was the em. field prepared in a single-photon Fock state before the measurement. After the measurement it's in the vacuum state. This is not described by the projection nor the here considered more general POVM formalism.

 

[vanhees71]

In some interpretations, yes. Not in all. In some interpretations it just refers to updating the observer's knowledge about the system.

Sure, but for that you still have to analyze each individual experiment. It cannot be a generally valid postulate.

 

[AlexCaledin]

- even in the Copenhagen, I can't help suspecting two collapses:

1. The universal collapse that has generated (or is generating) the whole "macroscopic" reality - thus making the textbook QM FAPP-applicable;

2. The abstract collapse of a particular model wavefunction used for probability calculation.

 

[PeterDonis]

The universal collapse

There is no such thing in Copenhagen. In fact, there's no such thing in any QM interpretation that I'm aware of. The only interpretations I'm aware of that assign any useful meaning at all to a wave function for the universe as a whole are no collapse interpretations like the MWI.

 

[kith]

If, e.g., I measure a photon by detecting it, usually it's absorbed. What was observed was the em. field prepared in a single-photon Fock state before the measurement. After the measurement it's in the vacuum state. This is not described by the projection nor the here considered more general POVM formalism.

Why not? Nielsen & Chuang seem to think it is described by their postulate 3: "For instance, if we use a silvered screen to measure the position of a photon we destroy the photon in the process. This certainly makes it impossible to repeat the measurement of the photon’s position! Many other quantum measurements are also not repeatable in the same sense as a projective measurement. For such measurements, the general measurement postulate, Postulate 3, must be employed." (p. 91)

(I'm not sure if this is relevant because I'm not well-acquainted with the POVM formalism but Nielsen & Chuang consider their postulate 3 to be more general than the POVM formalism because it includes a concrete set of Kraus operators ${M_m}$ instead of only the POVM elements (which are compatible with multiple sets of Kraus operators).)

 

[Demystifier]

A Stern-Gerlach magnet by itself is not a measurement; it's just a unitary operation that entangles the particle's spin with its linear momentum. To actually make a measurement, you have to put a detector screen downstream of the magnet (or in this case, two such screens, one downstream of SG1 and one downstream of SG2). So there is no projective measurement taking place at SG itself.

By SG apparatus I mean the magnet plus the detector.

 

[Demystifier]

The projection postulate as stated in #54 is just wrong for much simpler cases, e.g., measuring a single photon with a photon detector. After the measurement the photon is simply absorbed, i.e., destroyed. It's not projected to the "eigenstate" corresponding to the outcome of the measurement but at best you can say that there's no more photon and thus you have "projected" the state to the vacuum state.

The photon detector is not perfectly efficient, there is a probability that the photon will not be detected. Hence the interaction with the detector prepares the full QED state (of EM field and the detector) into a superposition, of which one term has 1 photons and the other term has 0 photons. That's called premeasurement. But if the detection actually happens, then, by measurement, the state gets projected into the second term, that is, the state with 0 photons. So it's still a projection, but in a bigger Hilbert space.

 

[vanhees71]

Why not? Nielsen & Chuang seem to think it is described by their postulate 3: "For instance, if we use a silvered screen to measure the position of a photon we destroy the photon in the process. This certainly makes it impossible to repeat the measurement of the photon’s position! Many other quantum measurements are also not repeatable in the same sense as a projective measurement. For such measurements, the general measurement postulate, Postulate 3, must be employed." (p. 91)

(I'm not sure if this is relevant because I'm not well-acquainted with the POVM formalism but Nielsen & Chuang consider their postulate 3 to be more general than the POVM formalism because it includes a concrete set of Kraus operators ${M_m}$ instead of only the POVM elements (which are compatible with multiple sets of Kraus operators).)

Yes, exactly. I don't have this book. So how is Postulate 3 formulated?

 

[vanhees71]

The photon detector is not perfectly efficient, there is a probability that the photon will not be detected. Hence the interaction with the detector prepares the full QED state (of EM field and the detector) into a superposition, of which one term has 1 photons and the other term has 0 photons. That's called premeasurement. But if the detection actually happens, then, by measurement, the state gets projected into the second term, that is, the state with 0 photons. So it's still a projection, but in a bigger Hilbert space.

If you take into account inefficiency of the detector, if there is anything prepared, it's a mixed state. If the photon gets detected and is absorbed you have prepared (if you call that preparation at all) the vacuum state not the single-photon eigenstate corresponding to the measurement.

I also never understood, why I have to assume this projection or collapse postulate to begin with. All that I describe with QT is what's measured in the real world, i.e., some setup or "preparation", which can be by actively "preparing" something like a proton beam in the LHC or which is observed somehow, but this "preparation by observation" is nothing that you can formulate as a fundamental law of nature or postulate but it depends on each specific setup. Then you take the so prepared ensemble and do measurements of observables you want to measure and you get some statistics about the outcomes which can be compared to the predicted probabilities of QT. That's all. What do I need to know the state after that meausurement except to do further experiments. Then you have again a preparation procedure, which however never can be described by a generally valid postulate but must be described by the analysis of the specific situation.

 

[atyy]

Even in Copenhagen there is no need to use the state reduction unless one does a sequence of measurements. The point that it only applies to "filter" measurements is implicit, since Copenghagen is just an FAPP theory. If that is the complaint, one can just add the condition that the postulate is only used if one needs to calculate probabilities for sequential measurements. It doesn't change that as far as most people can tell, a postulate is needed, which is why it is given by Nielsen and Chuang.

In ensemble language, one can say that without the postulate, there is no definition of the quantum state of subensembles. In general, for a given quantum state, there is no unique decomposition into subsensembles, which is why measurement is needed to pick out subensembles. The Born rule allows us to use measurement outcomes to label subensembles, but that is not sufficient to associate a measurement outcome and a quantum state to the subensembles.

 

[Demystifier]

Ad a) For me filter measurements and projective measurements are synonymous. What's the difference in your definition?

Filtering is deterministic, while projection is random. I can choose to block the upper beam in SG, but without blocking I cannot choose which (upper or lower) detector will click.

Ad b) Then there's no measurement between the first and the second+third SG magnets. That has to be formulated within the Hamiltonian for the particle (silver atom).

I assume that there is. More precisely, 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?

 

[vanhees71]

Even in Copenhagen there is no need to use the state reduction unless one does a sequence of measurements. The point that it only applies to "filter" measurements is implicit, since Copenghagen is just an FAPP theory. If that is the complaint, one can just add the condition that the postulate is only used if one needs to calculate probabilities for sequential measurements. It doesn't change that as far as most people can tell, a postulate is needed, which is why it is given by Nielsen and Chuang.

In ensemble language, one can say that without the postulate, there is no definition of the quantum state of subensembles. In general, for a given quantum state, there is no unique decomposition into subsensembles, which is why measurement is needed to pick out subensembles. The Born rule allows us to use measurement outcomes to label subensembles, but that is not sufficient to associate a measurement outcome and a quantum state to the subensembles.

Of course, it's not sufficient, because it cannot be generally stated what happens at a measurement, because that depends on the specific way you measure. It depends on the "design" of the experiment and the technical feasibility of the task, how well the experimenter is able to realize a wanted preparation-measurement sequence (aka experiment).

 

[atyy]

Of course, it's not sufficient, because it cannot be generally stated what happens at a measurement, because that depends on the specific way you measure. It depends on the "design" of the experiment and the technical feasibility of the task, how well the experimenter is able to realize a wanted preparation-measurement sequence (aka experiment).

It's general enough in the version given by Nielsen and Chuang. I don't have that here but you can see similar postulates in the following

The modern tools of quantum mechanics
Matteo G. A. Paris
https://arxiv.org/abs/1110.6815
p9 Postulate II.4

"No Information Without Disturbance": Quantum Limitations of Measurement
Paul Busch
https://arxiv.org/abs/0706.3526
"It is understood that upon reading an outcome, symbolized in the diagram with a discrete label $k$, the apparatus is considered to be describable in terms of a pointer eigenstate $T_{A,k}$, and this determines uniquely the associated final state $T_{k}$ of the object, as will be shown below."

However, the better reason that the projection postulate is not general enough is that it doesn't work for continuous variables. For discrete variables, POVMs can be derived from the projection postulate by considering a model of measurements. Preskill explains that as the reason he gives the postulates with the simpler projection postulate.
http://theory.caltech.edu/~preskill/ph219/ph219_2020-21.html
http://theory.caltech.edu/~preskill/ph219/chap2_15.pdf (axioms of QM stated using projection postulate in Eq 2.8)
http://theory.caltech.edu/~preskill/ph219/index.html#lecture (section 3.1.2 derives POVMs from the projection postulate)

 

[vanhees71]

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.

For measurements of continuous observables of course you can never prepare an eigenstate, because the eigenstates are not normalizable vectors but distributions belonging to the dual of the domain of the corresponding self-adjoint operator. That's treated in any QM1 textbook with more or less rigor (see Ballentine, de la Madrid, or Pascual for a modern approach using the rigged-Hilbert-space formalism). It's also very intuitive: You can measure a continuous observable only with some finite resolution.

 

[atyy]

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.

I think you are looking at the wrong equation in https://arxiv.org/abs/1110.6815. He first states it using the projection postulate (p8, 2.5), but later gives the more general postulate (p9, II.4).

 

[atyy]

For measurements of continuous observables of course you can never prepare an eigenstate, because the eigenstates are not normalizable vectors but distributions belonging to the dual of the domain of the corresponding self-adjoint operator. That's treated in any QM1 textbook with more or less rigor (see Ballentine, de la Madrid, or Pascual for a modern approach using the rigged-Hilbert-space formalism). It's also very intuitive: You can measure a continuous observable only with some finite resolution.

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

 

[vanhees71]

I think you are looking at the wrong equation in https://arxiv.org/abs/1110.6815. He first states it using the projection postulate (p8, 2.5), but later gives the more general postulate (p9, II.4).

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.

 

[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.