Is the collapse of the particle wave function?

[vanhees71]

That is not accurate at all. The question is a mathematical statement. Collapse is the statement that after a measurement, we can use a classical measurement outcome to label a subensemble. In Ballentine, this is equation 9.28 and the statement preceding that this is the state of the sub-ensemble that is labelled by the previous measurement outcome. Ballentine does not call that state reduction, but that is only a matter of terminology since other people do call it state reduction or collapse. So yes, Ballentine does have collapse. So the question is can that be derived from unitary evolution and the Born rule alone?

The question is a physical statement. What Ballentine describes with his Eq. (9.28) is the Statistical operator after a von Neumann "filter measurement", which I'd consider a state-preparation procedure. For those who don't have available this excellent textbook, here's the statement

Suppose an ensemble of a quantum mechanical system is prepared such that it is discribed by the statistical operator $\hat{\rho}$. Then a measurement of an observable $R$. He takes an observable with a continuous spectrum. Let $|r,\beta \rangle$ denote the generalized eigenvectors, normalized to a $\delta$ distribution and $\beta$ labeling a possible degeneracy, which I take as a discrete set (you can also have a continuous set, but that doesn't change much). Then he considers a state-preparation procedure which filters out all systems out of the ensemble for which $R$ takes a value in an interval $\Delta_a$. The remaining sub-ensemble is then described by a new statistical operator, which is constructed as follows: We first define the projection operator
$$\hat{M}_R(\Delta_a)=\int_{\Delta_a} \mathrm{d} r \sum_{\beta} |r,\beta \rangle \langle r,\beta|$$
and then define
$$\hat{\rho}'=\frac{1}{Z} \hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a), \quad Z=\mathrm{Tr} \left [\hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a)
\right]$$
as the statistical operator of the corresponding sub-ensemble.

This is just a definition what you call preparation through an idealized von Neumann filter measurement. An example, where you can describe this entirely with quantum theory is the Stern-Gerlach experiment. In this example there's then no collapse but just the ignorance of all particles with the spin component not in the desired state. The "collapse" in the Copenhagen doctrine is nothing but the choice of the subensemble by the experimenter through filtering out all undesired complementary subensembles. There's no spooky non-quantum action-at-a-distance magic called "collapse"!

 

[atyy]

The question is a physical statement. What Ballentine describes with his Eq. (9.28) is the Statistical operator after a von Neumann "filter measurement", which I'd consider a state-preparation procedure. For those who don't have available this excellent textbook, here's the statement

Suppose an ensemble of a quantum mechanical system is prepared such that it is discribed by the statistical operator $\hat{\rho}$. Then a measurement of an observable $R$. He takes an observable with a continuous spectrum. Let $|r,\beta \rangle$ denote the generalized eigenvectors, normalized to a $\delta$ distribution and $\beta$ labeling a possible degeneracy, which I take as a discrete set (you can also have a continuous set, but that doesn't change much). Then he considers a state-preparation procedure which filters out all systems out of the ensemble for which $R$ takes a value in an interval $\Delta_a$. The remaining sub-ensemble is then described by a new statistical operator, which is constructed as follows: We first define the projection operator
$$\hat{M}_R(\Delta_a)=\int_{\Delta_a} \mathrm{d} r \sum_{\beta} |r,\beta \rangle \langle r,\beta|$$
and then define
$$\hat{\rho}'=\frac{1}{Z} \hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a), \quad Z=\mathrm{Tr} \left [\hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a)
\right]$$
as the statistical operator of the corresponding sub-ensemble.

This is just a definition what you call preparation through an idealized von Neumann filter measurement. An example, where you can describe this entirely with quantum theory is the Stern-Gerlach experiment. In this example there's then no collapse but just the ignorance of all particles with the spin component not in the desired state. The "collapse" in the Copenhagen doctrine is nothing but the choice of the subensemble by the experimenter through filtering out all undesired complementary subensembles. There's no spooky non-quantum action-at-a-distance magic called "collapse"!

The reason that this is not preparation only is that it is preparation based on the outcome of the preceding measurement. The new quantum state is prepared based on the previous outcome that $R$ takes a value in an interval $\Delta_a$, so it is a conditional preparation, and is a conditional state. By saying that the new state is prepared by conditioning on the preceding outcome, one has in effect generalized the Born rule for the preceding outcome to the new state.

 

[vanhees71]

Sure, it's a conditional preparation. That's what all preparations are. I think it's implied by the Born rule, but as I said, in the case of the Stern-Gerlach experiment, you can derive it from the dynamics. You just consider only a partial beam after one Stern Gerlach apparatus splitting up the incoming beam into partial beams with determined magnetic quantum number of its spin.

 

[kith]

atyy, my interest in this debate has receded a bit but in our recent discussion of the double slit you linked to a nice computer simulation which seems relevant to me.

A gaussian wave packet hits the double slit. There, a part of the wave packet is transmitted and propagated to the screen and a part is reflected. The reflected part never hits the screen. So for a position measurement at the screen, we get the same probabilities for two different states: (a) the superposition of the transmitted and the reflected part of the wave packet and (b) the transmitted part of the wave packet alone.

Would you say that QM needs a postulate for the case that we chose description (b) for calculations because we don't want to calculate things which aren't relevant for the measurement outcome?

 

[vanhees71]

No, you just only consider that part of particles hitting the screen. There's no addition to standard QT needed whatsoever to understand this experiment.

 

[atyy]

Sure, it's a conditional preparation. That's what all preparations are. I think it's implied by the Born rule, but as I said, in the case of the Stern-Gerlach experiment, you can derive it from the dynamics. You just consider only a partial beam after one Stern Gerlach apparatus splitting up the incoming beam into partial beams with determined magnetic quantum number of its spin.

That's the question: is it implied by the Born rule? In quantum theory in the minimal interpretation there are no sub-ensembles for a pure state before measurement, because sub-ensembles would mean definite trajectories. So at the moment of measurement, two sets of sub-ensembles appear (1) the measurement outcome (2) the quantum state, from which you can select and filter. The appearance of the sub-ensembles of quantum states and the link between the sub-ensembles of measurement outcomes and the sub-ensembles of quantum states is beyond the Born rule (without state reduction), because the Born rule only tells you about the sub-ensembles of measurement outcomes.

So the use of the conditional state to get joint probabilities between sequential measurements is beyond the Born rule.

Ballentine says on p248 "If the operators R and S do not commute, then (9.26) does not apply. We can still use (9.22) as a definition of the joint probability Prob(A&B|C), since the factors on the right hand side are both well defined, in principle." However it is not true that both factors are well-defined unless Eq 9.28 is taken as postulate for the conditional state.

 

[vanhees71]

That's not what I mean. A subensemble is just a partial ensemble the experimenter chooses with his setup. For me (9.28) is the definition in the quantum-theoretical formalism for this choice. Do you think, one would have to derive this somehow? If so from which assumptions? Also what should it have to do with the Born rule?

 

[atyy]

atyy, my interest in this debate has receded a bit but in our recent discussion of the double slit you linked to a nice computer simulation which seems relevant to me.

A gaussian wave packet hits the double slit. There, a part of the wave packet is transmitted and propagated to the screen and a part is reflected. The reflected part never hits the screen. So for a position measurement at the screen, we get the same probabilities for two different states: (a) the superposition of the transmitted and the reflected part of the wave packet and (b) the transmitted part of the wave packet alone.

Would you say that QM needs a postulate for the case that we chose description (b) for calculations because we don't want to calculate things which aren't relevant for the measurement outcome?

In the simplest version of the double slit, there is only one measurement, which is registered when the particle hits the screen. Collapse is only needed for calculating the joint probabilities of two or more sequential measurements. So there is no collapse in the simplest version of the double slit.

 

[atyy]

That's not what I mean. A subensemble is just a partial ensemble the experimenter chooses with his setup. For me (9.28) is the definition in the quantum-theoretical formalism for this choice. Do you think, one would have to derive this somehow? If so from which assumptions? Also what should it have to do with the Born rule?

For just choosing a sub-ensemble, and calculating the probabilities of future measurements on the sub-ensemble, there is no need for collapse. In this case, it is indeed fine to consider the procedure one does as simply state preparation.

However, for calculating the probability of one measurement $a$, selecting a sub-ensemble based on the first measurement outcome then doing measurement $b$, collapse is needed to calculate $P(A,B)=P(B|A)P(A)$. What collapse does is define $P(B|A)$ via $P(\rho'|A)$.

 

[vanhees71]

I still don't understand this argument. In the example of the Stern-Gerlach experiment with measuring the magnetic spin quantum number of a neutral particle, which is the paradigmatic filter measurement, you just have the following:

(a) a unpolarized beam of neutral silver atoms, originating from a little oven by letting it out of a little hole in this oven
(b) the beam is directed through an inhomogeneous magnetic field with a large homogeneous component in $z$ direction, which entangles the magnetic spin quantum number $\sigma_z \in \{\pm 1/2 \}$ with the position of the silver atom. The magnetic field is taylored such that you get two well-separated partial beams with defined $\sigma_z$. This is described by unitary time evolution.
(c) The partial beam with, say, $\sigma_z=1/2$ is directed through a 2nd magnetic field with a large homogeneous component in $x$ direction and so $\sigma_z$ is measured via the resulting position-$\sigma_z$ entanglement. Again this is described by unitary time evolution.

The only non-unitary thing in (9.28) is the renormalization in the denominator, but that's simply, because you want to consider only the partial beam with $\sigma_z=+1/2$ prepared with step (b). That's just a calculational convenience and has nothing to do with a physical collapse. You could as well choose to normalize the total probability to 42, which is never done in practice, because we define probabilities to add up to 1.

At least for such an idealized von Neumann Filter measurement (which I'd rather call preparation for clarity), you don't need a collapse, and only for such a type of filter measurement you need it in that flavor of the Copenhagen doctrine.

In most other cases, you don't care about what happens after the measurement. You don't think much about the switched off proton or heavy-ion beam at the LHC ending quite abruptly in the beam dump. In principle you "measure" position of the particles in the beam, because you know it ends somewhere at the beam dump, but you cannot say that afterwards the particles have a somehow "collapsed" new state localized in the beam dump, because most of them are just gone by some annihilation process or something else. I consider the collapse hypothesis as quite empty and more confusing than helpful, because it has very serious issues with relativistic causality, as was pointed out by EPR in their famous paper (although Einstein himself didn't like it too much, and he has written a much better version by himself alone (in German):

A. Einstein, Quanten-Mechanik und Wirklichkeit, Dialectica 2, 320 (1948)
http://dx.doi.org/10.1111/j.1746-8361.1948.tb00704.x

 

[kith]

In the simplest version of the double slit, there is only one measurement, which is registered when the particle hits the screen.

There's also the preparation by the double slit. You often associate collapse with Ballentine's equation 9.28 which can be applied here: after the wave packet hit the slit, we project on the transmitted part because we don't care about the reflected part. Since you think that equation 9.28 needs to be derived or postulated, I'd like to know what you think about this concrete example.

 

[atyy]

I still don't understand this argument. In the example of the Stern-Gerlach experiment with measuring the magnetic spin quantum number of a neutral particle, which is the paradigmatic filter measurement, you just have the following:

(a) a unpolarized beam of neutral silver atoms, originating from a little oven by letting it out of a little hole in this oven
(b) the beam is directed through an inhomogeneous magnetic field with a large homogeneous component in $z$ direction, which entangles the magnetic spin quantum number $\sigma_z \in \{\pm 1/2 \}$ with the position of the silver atom. The magnetic field is taylored such that you get two well-separated partial beams with defined $\sigma_z$. This is described by unitary time evolution.
(c) The partial beam with, say, $\sigma_z=1/2$ is directed through a 2nd magnetic field with a large homogeneous component in $x$ direction and so $\sigma_z$ is measured via the resulting position-$\sigma_z$ entanglement. Again this is described by unitary time evolution.

You are missing the first measurement. You must make a direct or indirect measurement of the spin after the first apparatus, before it enters the second apparatus. If that is not part of your experiment, there is only one measurement, and collapse is not needed.

 

[atyy]

There's also the preparation by the double slit. You often associate collapse with Ballentine's equation 9.28 which can be applied here: after the wave packet hit the slit, we project on the transmitted part because we don't care about the reflected part. Since you think that equation 9.28 needs to be derived or postulated, I'd like to know what you think about this concrete example.

You must make a first measurement of the position (or some other observable) at the slits, otherwise there is only one measurement, not the minimum of two measurements needed to discuss sequential measurement. The important point about Eq 9.28 is not Eq 9.28 alone, but that it is conditional on the previous measurement outcome.

 

[kith]

You must make a first measurement of the position (or some other observable) at the slits, otherwise there is only one measurement, not the minimum of two measurements needed to discuss sequential measurement. The important point about Eq 9.28 is not Eq 9.28 alone, but that it is conditional on the previous measurement outcome.

Do you consider a polarization filter or a single slit to be an appropriate measurement device?

 

[atyy]

Do you consider a polarization filter or a single slit to be an appropriate measurement device?

Do you get a definite measurement outcome at a particular time after using these devices?

 

[kith]

Do you get a definite measurement outcome at a particular time after using these devices?

What do you mean by outcome? I do get a state of definite polarization resp. position immediately after the filter resp. slit. Regarding the time of the measurement, the situation is analogous to the measurement at the screen.

 

[vanhees71]

You must make a first measurement of the position (or some other observable) at the slits, otherwise there is only one measurement, not the minimum of two measurements needed to discuss sequential measurement. The important point about Eq 9.28 is not Eq 9.28 alone, but that it is conditional on the previous measurement outcome.

No, why? The 1st SG device separates the particles spatially according to $\sigma_z$. I just place the 2nd SG device in the partial beam providing particles with determined $\sigma_1=1/2$. Then I count the particles with determined $\sigma_x=\pm 1/2$ by just setting counters at the appropriate places. There the particles are absorbed and the detector counts them. The result, according to QT, is of course $P(\sigma_x=1/2|\sigma_z=1/2)=P(\sigma_x=1/2|\sigma_z=1/2)=1/2$, and that's what comes out from actual measurements as far as I know.

 

[vanhees71]

Do you consider a polarization filter or a single slit to be an appropriate measurement device?

In my opinion both are both measurement devices (for polarization or position, respectively) and filter-preparation devices (for the corresponding quantities).

 

[atyy]

No, why? The 1st SG device separates the particles spatially according to $\sigma_z$. I just place the 2nd SG device in the partial beam providing particles with determined $\sigma_1=1/2$. Then I count the particles with determined $\sigma_x=\pm 1/2$ by just setting counters at the appropriate places. There the particles are absorbed and the detector counts them. The result, according to QT, is of course $P(\sigma_x=1/2|\sigma_z=1/2)=P(\sigma_x=1/2|\sigma_z=1/2)=1/2$, and that's what comes out from actual measurements as far as I know.

If one assigns the particle a definite spin or position without making a measurement then one is assuming these quantities exist without measurement. It is analogous to assuming that particles have definite trajectories, which are hidden variables.

 

[vanhees71]

It exists due to the preparation procedure, because after the 1st SG apparatus $\sigma_z$ and position are entangled. There's a 100% correlation between the value of $\sigma_z$ and the partial beams coming out of the 1st SG apparatus!

 

[atyy]

What do you mean by outcome? I do get a state of definite polarization resp. position immediately after the filter resp. slit. Regarding the time of the measurement, the situation is analogous to the measurement at the screen.

A definite measurement outcome is not a quantum state, it is a macroscopic "classical" event, like the registration of particle position on the screen. A definite outcome is whatever one predicts using the Born rule (without state reduction).

 

[kith]

A definite measurement outcome is not a quantum state, it is a macroscopic "classical" event, like the registration of particle position on the screen. A definite outcome is whatever one predicts using the Born rule (without state reduction).

I see this as an argument for using the term "measurement" more restrictively but what are sequential measurements then? Textbook examples like multiple polarization filters, SG experiments, etc. don't produce intermediate outcomes.

 

[atyy]

It exists due to the preparation procedure, because after the 1st SG apparatus $\sigma_z$ and position are entangled. There's a 100% correlation between the value of $\sigma_z$ and the partial beams coming out of the 1st SG apparatus!

This is different from what I am talking about. In order to discuss collapse, there have to be two measurements, so by avoiding two measurements, it isn't clear that one needs collapse.

 

[atyy]

I see this as an argument for using the term "measurement" more restrictively but what are sequential measurements then? Textbook examples like multiple polarization filters, SG experiments, etc. don't produce intermediate outcomes.

The polarization filter is an interesting case I don't understand well. Regardless, the textbook treatment of a polarizer is indeed very similar to a measurement followed by collapse since the Born rule is applied to the quantum state for describing the action of a polarizer. My guess is that there is a deterministic unitary description of a polarizer, but it's not immediately obvious to me.

However, to make things easy to discuss, one can follow the polarizer with a detection with a definite outcome and time stamp, as is done in Bell tests.

 

[vanhees71]

A polarizer is usually effectively described as a projection operator and as such you can argue that this is not a description in terms of a unitary time evolution. Of course this description is an effective and idealized one. A full microscopic theory of the interaction of the (quantized) electromagnetic field with the material of the polarizer is described by a unitary time evolution and you don't need any collapse anymore. Collapse is a FAPP description of preparation procedure in terms of filter measurements. As such I buy it, but not the claim that there's instantaneous collapse over the entire universe by a local measurement/interaction procedure. It's also not clear, where there should be made a cut between quantum and classical description in principle. Of course FAPP often the (semi-)classical description of macroscopic objects is precise enough, as in this example of a polarizer.

The same holds true, by the way, for other optical elements like quarter-wave plates or lenses, which are described by unitary operators. They are also effective descriptions of the much more complicated microscopic theory. You get it by coarse graining over the irrelevant microscopic degrees of freedom, where "irrelevant" refers to the macroscopic scale whose resolution is sufficient to describe phenomena.

 

[atyy]

A full microscopic theory of the interaction of the (quantized) electromagnetic field with the material of the polarizer is described by a unitary time evolution and you don't need any collapse anymore. Collapse is a FAPP description of preparation procedure in terms of filter measurements. As such I buy it, but not the claim that there's instantaneous collapse over the entire universe by a local measurement/interaction procedure. It's also not clear, where there should be made a cut between quantum and classical description in principle.

There is no claim that collapse is not FAPP, but that is because in the orthodox Copenhagen interpretation, the wave function and its time evolution are all FAPP.

Anyway, the polarizer is tricky. And yes, in discussing Ballentine's 9.28 we do need two measurements, since that is what he is discussing. Eq 9.28 is conditioned on the outcome of a previous measurement, so there must be a measurement after the first SG device that produces a definite outcome. Eq 9.28 is exactly the collapse, and as far as I can tell, there is no known way to derive it from unitary evolution and the Born rule.

To be clear, the situation in 9.28 is that there is measurement $a$ followed by measurement $b$. A measurement is something which produces a definite macroscopic outcome. The probability of the outcome $P(A)$ is given by the Born rule, and the probability of the outcome $P(B)$ is given by the Born rule. However, experimentally there is also the joint probability $P(A,B)$ and $P(B|A)$, neither of which are given by the Born rule and unitary evolution, because the Born rule only applies to measurements at a single time. So when 9.28 is used to calculate $P(B|A)$, it is a postulate beyond unitary evolution and the Born rule.

 

[vanhees71]

The fact that the beam after the first SG apparatus is split in two partial beams of definite $\sigma_z$ is already a measurement. The particles are macroscopically separated into particles with definite $\sigma_z$. This can be verified by setting a screen to detect these two partial beams, but then you cannot make further experiments. So you take away the screen and take it as a fact that this separation is objective i.e., not due to the presence of the detector but due to the SG apparatus. Then you can do experiments with the partial beams running them through another SG apparatus measuring the same spin (no further split of beams) or another one (further split of beams with certain probabilities depending on the relative orientation of the magnetic field). As far as I know all SG experiments have verified the predictions of standard QT, including Eq. (9.28). Whether you take it as additional postulate or try to derive it (which in my opinion can be done in the simple case of the SG apparatus but of course not in all cases, as the example of the polarizer shows), is not so important. It's just a pretty well verified piece of QT, while the collapse hypothesis has already theoretical problems, let alone its empirical verification. E.g., I haven't ever heard about an experiment proving action at a distance in contradiction to the hypotheses of local relativistic QFT. All experiments so far are explained by local relativistic QFT. So there's no reason to believe in a collapse in clear contradiction to this very successful paradigm.

 

[atyy]

The fact that the beam after the first SG apparatus is split in two partial beams of definite $\sigma_z$ is already a measurement. The particles are macroscopically separated into particles with definite $\sigma_z$. This can be verified by setting a screen to detect these two partial beams, but then you cannot make further experiments. So you take away the screen and take it as a fact that this separation is objective i.e., not due to the presence of the detector but due to the SG apparatus. Then you can do experiments with the partial beams running them through another SG apparatus measuring the same spin (no further split of beams) or another one (further split of beams with certain probabilities depending on the relative orientation of the magnetic field). As far as I know all SG experiments have verified the predictions of standard QT, including Eq. (9.28). Whether you take it as additional postulate or try to derive it (which in my opinion can be done in the simple case of the SG apparatus but of course not in all cases, as the example of the polarizer shows), is not so important. It's just a pretty well verified piece of QT, while the collapse hypothesis has already theoretical problems, let alone its empirical verification. E.g., I haven't ever heard about an experiment proving action at a distance in contradiction to the hypotheses of local relativistic QFT. All experiments so far are explained by local relativistic QFT. So there's no reason to believe in a collapse in clear contradiction to this very successful paradigm.

What you call the splitting of the two beams is not a measurement. One must add a screen or an ancilla to get a measurement outcome to which the Born rule applies. The screen will destroy the particle, preventing a second measurement, but in principle quantum theory allows the coupling of an ancilla, followed by a measurement on the ancilla leaving the system available for a second measurement. If there is no first measurement, then the conditional probability $P(B|A)$ and the joint probability $P(A,B)$ where both $A$ and $B$ are measurement outcomes is pointless from the point of view of the orthodox interpretation, since quantum theory is only a tool to calculate the probabilities of measurement outcomes.

 

[vanhees71]

Then you could never do these SG experiments measuring subsequently first $\sigma_z$ and then $\sigma_x$, which brings me to the question, whether this gedankenexperiment has ever been done in practice.

 

[atyy]

Then you could never do these SG experiments measuring subsequently first $\sigma_z$ and then $\sigma_x$, which brings me to the question, whether this gedankenexperiment has ever been done in practice.

As far as I know, this thought experiment has not been done. The closest I can think of are the Bell tests, which are sequential measurements in some frames of reference. Bu the important point is that in the orthodox Copenhagen interpretation, a measurement is something that produces a definite macroscopic outcome, also called a classical outcome. It is when one has sequential measurements with sequential outcomes and one needs to calculate P(B|A) or P(A,B) that collapse is postulated. It is not necessary to postulate collapse, for example, a formula for P(A,B) can be postulated directly without any collapse (in which case, collapse can be derived). Another alternative is to do what bhobba does and postulate the equivalence of proper and improper mixtures following decoherence (again, collapse can be derived). But as far as I know, there has to be something beyond unitary evolution and the Born rule.