Collapse and Peres' Coarse Graining

[billschnieder]

However, in the full formulation of general relativity, test particles are not fundamental. So quantum mechanics is different from classical physics in needing to specify measurement as a fundamental concept.

Measurement is needed to "actualize" one of the "possibilities" (or a small set of the non-mutually exclusive possibilities -- aka commuting observables). Without measurement, there is nothing actual to talk about, just possibilities. Measurement will not be so important in a theory that does not rely on a device which represents simultaneously all the "possible realities", like general relativity. In probability theory however, measurement is very important.

I am using a particular interpretation to define QM, but it is the minimal interpretation. The measurement problem is that we have to put this classical/quantum cut to define the minimal interpretation.

We do not. In the minimal interpretation, the classical/quantum cut is simply an unnecessary fiction, which only appears once you choose to interpret the wave function as a real physical thing. But the minimal interpretation is that it is a device for cataloging information about possible states within an ensemble.

Another way of stating the measurement problem, is that if everything is quantum and we have a wave function of the universe, how can we make sense of such an idea?

Understanding the wave function as a catalog of information about possible realities of the universe, there is no difficulty to make sense of. It already makes sense, in the minimal interpretation. No classical quantum cut is needed. MWI and BM are attempts to solve a problem introduced because the proponents insist on interpreting the wavefunction as a real physical thing. MWI by suggesting that the possibilities are all actualities (including the mutually exclusive ones), BM by suggesting that the possibilities exist as "guiding waves" to orchestrate observations.

 

[atyy]

Understanding the wave function as a catalog of information about possible realities of the universe, there is no difficulty to make sense of.

If one actually tries to construct the catalogue of all the possible realities in the wave function, one ends up with the consistent histories approach.

 

[zonde]

Say we have two orthogonal polarizers and we shine a light trough them. There is practically no light passing through them.
Then we put in additional polarizer at 45 deg. between first two and we get quarter of light trough.
Considering this I don't understand how one can view measurement as information update.

 

[atyy]

@zonde and @billschneider, could you start a new thread about whether collapse is update or physical? I agree it's an interesting question, but not so relevant at the moment. I would like to focus on the hard science question I wrote in post #47. Is Weinberg's Eq 2.1.7 in Vol 1 of his QFT text part of quantum theory? If it is, is it postulated, or derivable from only {unitary evolution + Born rule without collapse}? If you'd like to discuss whether Eq 2.1.7 represents information update or a physical process in this thread, why don't we wait a bit until we understand vanhees71's view of Weinberg's Eq 2.1.7?

 

[Ilja]

In classical probability, a probability distribution represents alternate "possibilities". A measurement "actualizes" a sub-ensemble. Would you say there is collapse involved?
... However, if probability distributions are understood as information and not physical, there is absolutely no need to invent a concept of "collapse".

In this case, of course, there would be no need for a physical process identified with a collapse. But this picture is incompatible with the violation of Bell's inequalities.

And interpreting something as "information" always requires an answer to the question "information about what?".

 

[Ilja]

Well, it's at least not a physical process, as claimed by some collapse proponents in the case of quantum theory. It's then even less needed in classical than in quantum theory.

Indeed, in classical statistics we can explain the uncertainty of the statistics as completely being a problem of insufficient information. Getting more information about the real process does not mean that the process is physically influenced.

In quantum theory such an explanation is impossible.

You have an eigenstate of A with eigenvalue a1. You can repeat the measurement of A as much as you like, the result is always a1, never a2, a3, ...

Now you measure some non-commuting B. Then, you choose the subgroup of those with result b1. This operation differs from choosing a subgroup in classical statistics given some additional information that B has value b1, as can be easily seen: Measuring A gives now, with nonzero probability, other values than a1. Instead, restricting to a subensemble of those with value b1 would not modify the result of A being a1.

 

[zonde]

@zonde and @billschneider, could you start a new thread about whether collapse is update or physical? I agree it's an interesting question, but not so relevant at the moment. I would like to focus on the hard science question I wrote in post #47. Is Weinberg's Eq 2.1.7 in Vol 1 of his QFT text part of quantum theory? If it is, is it postulated, or derivable from only {unitary evolution + Born rule without collapse}? If you'd like to discuss whether Eq 2.1.7 represents information update or a physical process in this thread, why don't we wait a bit until we understand vanhees71's view of Weinberg's Eq 2.1.7?

Born rule without collapse could work only if filtering measurement represents information update. So it's relevant to your question. Besides didn't vanhees71 express already his viewpoint in post #45?

 

[TrickyDicky]

In classical physics (Newtonian physics, special and general relativity), measurement is not a fundamental concept. Historically, Einstein did postulate measurement as fundamental in special relativity: the speed of light measured by any inertial observer is the same. However, we have removed that, and nowadays we say that special relativity means the laws have Poincare symmetry. Historically, measurement was also important in the genesis of general relativty: test particles follow geodesics. The test particle is a sort of measurement apparatus that is apart from the laws of physics because it does not cause spacetime curvature, in contrast to all other forms of matter. However, in the full formulation of general relativity, test particles are not fundamental. So quantum mechanics is different from classical physics in needing to specify measurement as a fundamental concept.

I am using a particular interpretation to define QM, but it is the minimal interpretation. The measurement problem is that we have to put this classical/quantum cut to define the minimal interpretation. The other interpretations are then approaches to solving the measurement problem by removing the need for measurement to be a fundamental concept in the mathematical specification of a theory. Examples of such interpretations are consistent histories (flavour of Copenhagen), hidden variables (generally predicting deviations from QM), or Many-Worlds.
Another way of stating the measurement problem, is that if everything is quantum and we have a wave function of the universe, how can we make sense of such an idea? The
minimal interpretation cannot make sense of such an idea, and always needs a classical/quantum cut. Bohmian Mechanics and Many-Worlds are two approaches to solving the measurement problem, in which the wave function of the universe is proposed to make sense.

But you don't seem to be talking about what is usually known as the minimal or ensemble interpretation, you are giving too much reality to the quantum state, not even Copenhagen is so realistic about the state, it only considers it well defined in the context of observation, that's why they need a well defined collapse of the wave function at the moment of measurement. If you don't consider physical states at all then collapse doesn't arise in such explicit form because there is no wave function to collapse in any actual sense to begin with, but of course you still have irreversibility and the Born rule in your post 47 form, by definition the basic math of QM must be the same for all interpretations, otherwise it would be a different theory.
This doesn't mean the ensemble interpretation solves the measurement problem anymore than magical collapse, or decoherence or many-worlds do, they all leave the preferred-basis side of it basically untouched, but as we saw recently in the thrashing thread's paper, that problem is inherent to the first postulate so it can't be cured just looking at measurement.

 

[vanhees71]

Collapse is the statement of the Born rule in the form $P(\phi) = |\langle \phi | \psi \rangle|^{2}$, which is what happens in a filtering measurement. This is Eq 2.1.7 in volume 1 of Weinberg's QFT text. I think you agreed that measurement can be used as a means of state preparation, so in that sense I thought you said that collapse exists. If collapse does not exist, then are you saying that Weinberg's Eq 2.1.7 does not exist? If collapse does exist, then it seems you are saying that collapse can be derived, ie. Weinberg's Eq 2.1.7 can be derived.

The only thing I need is the statement that, if a quantum system is prepared in a state, represented by a normalized vector $|\psi \rangle$, the probability (density) to find the value $a$ of the observable $A$ in the discrete (continuous) part of the spectrum of its representing self-adjoint operator $\hat{A}$, is given by
$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$
where $\beta$ labels a complete set of orthonormalized eigenvectors of $\hat{A}$ to the eigenvalue $a$. Of course, $\beta$ can also be continuous. Then the sum has to be substituted with the corresponding integral.

Where do I need a collapse for Born's postuate?

 

[atyy]

The only thing I need is the statement that, if a quantum system is prepared in a state, represented by a normalized vector $|\psi \rangle$, the probability (density) to find the value $a$ of the observable $A$ in the discrete (continuous) part of the spectrum of its representing self-adjoint operator $\hat{A}$, is given by
$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$
where $\beta$ labels a complete set of orthonormalized eigenvectors of $\hat{A}$ to the eigenvalue $a$. Of course, $\beta$ can also be continuous. Then the sum has to be substituted with the corresponding integral.

Where do I need a collapse for Born's postuate?

So you disagree with Weinberg that his Eq 2.1.7 is a postulate?

 

[dextercioby]

What is equation 2.1.7?

 

[atyy]

What is equation 2.1.7?

Here is my transcription of Weinberg's Eq 2.1.7 in http://books.google.com/books?id=doeDB3_WLvwC&source=gbs_navlinks_s (p50).

$P(\mathscr{R} \rightarrow\mathscr{R_{n}}) = |(\Psi,\Psi_{n})|^{2}$

It is the probability that a system prepared in state $\mathscr{R}$ is found in state $\mathscr{R_n}$, if a test is done to find out whether the system is in one of several orthogonal states $\{ \mathscr{R_1}, \mathscr{R_2}, .. \}$, and $\Psi$ is a (unit) vector representing the state $\mathscr{R}$.

 

[dextercioby]

Oh, you mean his QFT book, I thought his regular QM book. OK, that makes sense now, even though it's odd it appears in his QFT book.

 

[microsansfil]

Here is my transcription of Weinberg's Eq 2.1.7 in http://books.google.com/books?id=doeDB3_WLvwC&source=gbs_navlinks_s (p50).

$P(\mathscr{R} \rightarrow\mathscr{R_{n}}) = |(\Psi,\Psi_{n})|^{2}$

It is the probability that a system prepared in state $\mathscr{R}$ is found in state $\mathscr{R_n}$, if a test is done to find out whether the system is in one of several orthogonal states $\{ \mathscr{R_1}, \mathscr{R_2}, .. \}$, and $\Psi$ is a (unit) vector representing the state $\mathscr{R}$.

Is it applicable for a dissipative system, that is put in a mixed quantum state ?

Patrick

 

[atyy]

Is it applicable for a dissipative system, that is put in a mixed quantum state ?

When the state is mixed, it has to be represented by a density operator. The Born rule, and collapse or state reduction for a density operator is given in
http://arxiv.org/abs/1110.6815 (Eq II.3, II.4 on p9)
http://arxiv.org/abs/0706.3526 (Eq 2, 3 on p4)

 

[vanhees71]

I guess, you mean Eq. (2.1.7) in Quantum Theory of Fields Vol. 1? Then, of course it's Born's rule and thus one of the postulates of quantum theory. I still don't see, where you need a collapse for its statement.

There's no difference when you are in a general (mixed ) state. Also there no collapse is needed. You have a mixed state, whenever in addition to the irreducible indeterminacy of observables (which are present even when the state of the system is completely determined) there's also incomplete knowledge about in which state the system is prepared.

A pure state is of course also uniquely and equivalently described by a statistical operator, not only a ray in Hilbert space. A statistical operator represents a pure state if and only if it is a projection operator.

 

[atyy]

I guess, you mean Eq. (2.1.7) in Quantum Theory of Fields Vol. 1? Then, of course it's Born's rule and thus one of the postulates of quantum theory. I still don't see, where you need a collapse for its statement.

Yes, in Weinberg's QFT volume 1. Eq 2.1.7 is the definition of collapse, because it gives the probability for the system to jump from state R to Rn. For example, given a Bell state |00> + |11>, when Alice gets the measurement outcome "0", the state will transition to |00>. That is the collapse, just a mathematical statement that is postulated.

 

[vanhees71]

The formulation in Weinberg may be a bit misleading, but there's no collapse implied as far as I understand him. Also one should be more careful in formulating Born's rule to make it compatible with the dynamics. You have to distinguish between the kets representing the state (rays in Hilbert space) and the (generalized) eigenvectors of observables. In a general picture (Dirac picture) of time evolution they evolve with different unitary time-evolution operators
$$|\psi,t \rangle=\hat{C}(t) |\psi,0 \rangle, \quad |\vec{a},t \rangle=\hat{A}(t) |\vec{a},0 \rangle.$$
Here $\vec{a}$ is in the common spectrum of a complete compatible set of observables, and the time-evolution operators obey the equations of motion
$$\frac{\mathrm{d}}{\mathrm{d} t} \hat{C}(t)=-\mathrm{i} \hat{H_1}(t) \hat{C}(t), \quad \frac{\mathrm{d}}{\mathrm{d} t} \hat{A}(t)=\mathrm{i} \hat{H_2}(t) \hat{C}(t)$$
with the initial conditions
$$\hat{C}(0)=\hat{A}(0)=\hat{1}.$$
The operators $\hat{H}_1$ and $\hat{H}_2$ are local in $t$ and self-adjoint operators, related to the Hamiltonian of the system by
$$\hat{H}=\hat{H}_1+\hat{H}_2.$$
Then the probability (density) to find the outcome $\vec{a}$ at time $t$ when measuring the complete set of observables on the system, prepared in the state represented by $|\psi,t \rangle$ is
$$P(\vec{a},t|\psi)=|\langle \vec{a},t|\psi,t \rangle|^2.$$
By construction the time evolution of this probability (density) is independent of the choice of the picture of time evolution.

This does not imply that the quantum system automatically somehow collapses into a state represented by $|\vec{a},t \rangle$ at the time of the measurement. In the case of observables with continuous spectrum this even contradicts the "kinematical" part of the quantum postulates, because then this is not a normalizable state in Hilbert space but a generalized state in the dual space of the nuclear space, where the observable operators are densely defined. Already this formal argument shows that the collapse hypothesis makes no proper sense.

 

[stevendaryl]

The only thing I need is the statement that, if a quantum system is prepared in a state, represented by a normalized vector $|\psi \rangle$, the probability (density) to find the value $a$ of the observable $A$ in the discrete (continuous) part of the spectrum of its representing self-adjoint operator $\hat{A}$, is given by
$P(a)=\sum_{\beta} |\langle a,\beta|\psi \rangle|^2,$
where $\beta$ labels a complete set of orthonormalized eigenvectors of $\hat{A}$ to the eigenvalue $a$. Of course, $\beta$ can also be continuous. Then the sum has to be substituted with the corresponding integral.

Where do I need a collapse for Born's postuate?

I think we already went around once about this. The question is: How does measurement constitute a preparation procedure for a particular state? Take the simplest example of preparing an electron in the state with spin-up in the z-direction. The usual assumption is that measuring the spin, and finding it to be spin-up means that afterward, the electron is in the spin-up state. Isn't that basically the collapse hypothesis? Without some assumption along those lines, how would you prepare an electron in the spin-up state?

 

[vanhees71]

I think we already went around once about this. The question is: How does measurement constitute a preparation procedure for a particular state? Take the simplest example of preparing an electron in the state with spin-up in the z-direction. The usual assumption is that measuring the spin, and finding it to be spin-up means that afterward, the electron is in the spin-up state. Isn't that basically the collapse hypothesis? Without some assumption along those lines, how would you prepare an electron in the spin-up state?

I answered this already, but again:

Take a Stern-Gerlach (SG) apparatus to "measure" the spin-z direction. A particle running through this apparatus is deflected in one of two possible directions. After this deflection, which is described by unitary time evolution for a particle with spin running through an appropriate inhomogeneous magnetic field (with a large homogeneous component in z direction, which sorts out the spin-z components as measured observables), particles at the respective are with in principiple arbitrary accuracy prepared in the corresponding spin state. Nowhere do you need a collapse. It's simply unitary time evolution, in this example even of a single-particle Schrödinger-Pauli equation.

 

[stevendaryl]

I answered this already, but again:

Take a Stern-Gerlach (SG) apparatus to "measure" the spin-z direction. A particle running through this apparatus is deflected in one of two possible directions. After this deflection, which is described by unitary time evolution for a particle with spin running through an appropriate inhomogeneous magnetic field (with a large homogeneous component in z direction, which sorts out the spin-z components as measured observables), particles at the respective are with in principiple arbitrary accuracy prepared in the corresponding spin state. Nowhere do you need a collapse. It's simply unitary time evolution, in this example even of a single-particle Schrödinger-Pauli equation.

I'm not sure about that. If you have a single electron, and you send it through a stern-gerlach device, then afterward, the electron is describable as a superposition of a left-going spin-up electron and a right-going spin-down electron. How do you then get a prepared state $|U\rangle$ consisting of only a spin-up electron?

 

[vanhees71]

I just take into account only electrons in that region of space, where they have the desired spin-z component and ignore the ones in the other region. That's the paradigmatic example for a filter measurement. Of course you are right, after the SG apparatus the state is
$$|\Psi \rangle=|\Phi_{\vec{x}},\sigma_1=1/2 \rangle+|\Phi_{\vec{y}},\sigma_z=-1/2 \rangle,$$
where $\Phi_{\vec{x}}$ and $\Phi_{\vec{y}}$ are states corresponding to wave packets located around $\vec{x}$ and $\vec{y}$, respectively, i.e., the SG apparatus entangles the position of the electron with its $\sigma_z$ value.

In terms of wave mechanics (i.e., the spin-position representation) this state reads
$$\Psi(\vec{r})=\begin{pmatrix}
\Phi_{\vec{x}}(\vec{r}) \\ \Phi_{\vec{y}}(\vec{r})
\end{pmatrix}.$$

 

[stevendaryl]

I'm not sure about that. If you have a single electron, and you send it through a stern-gerlach device, then afterward, the electron is describable as a superposition of a left-going spin-up electron and a right-going spin-down electron. How do you then get a prepared state $|U\rangle$ consisting of only a spin-up electron?

Just to reiterate: Suppose you have some source of electrons in an unknown spin state. You would describe this as the density matrix:

$\rho = \frac{1}{2} ( |U\rangle \langle U | + |D\rangle \langle D | )$

What could you do to the electron to put it into the pure spin-up state $|U\rangle \langle U|$ using only unitary evolution? I think that you can't.

 

[stevendaryl]

I just take into account only electrons in that region of space, where they have the desired spin-z component and ignore the ones in the other region. That's the paradigmatic example for a filter measurement. Of course you are right, after the SG apparatus the state is
$$|\Psi \rangle=|\Phi_{\vec{x}},\sigma_1=1/2 \rangle+|\Phi_{\vec{y}},\sigma_z=-1/2 \rangle,$$
where $\Phi_{\vec{x}}$ and $\Phi_{\vec{y}}$ are states corresponding to wave packets located around $\vec{x}$ and $\vec{y}$, respectively, i.e., the SG apparatus entangles the position of the electron with its $\sigma_z$ value.

In terms of wave mechanics (i.e., the spin-position representation) this state reads
$$\Psi(\vec{r})=\begin{pmatrix}
\Phi_{\vec{x}}(\vec{r}) \\ \Phi_{\vec{y}}(\vec{r})
\end{pmatrix}.$$

That doesn't prepare a pure spin-up state. I suppose you could say that because the experiment involves a region of space in which the spin-down component is negligible, then effectively, we can pretend that there is only a spin-up component. That's fine, but then the notion of preparing a system in a particular state is a shorthand for something more complicated.

 

[vanhees71]

Why is it too complicated to just consider only the electrons in a certain region of space?

Of course, also if you initial state is in a mixture, a Stern-Gerlach apparatus can be used as described. In the original experiment Stern and Gerlach used thermal silver atoms from a oven extracting a beam by keeping a small opening. That's clearly a mixture of the type you've written down.

 

[stevendaryl]

Why is it too complicated to just consider only the electrons in a certain region of space?

I'm just pointing out that when people say that they are starting with electrons in some particular state $|\psi\rangle$, that is not what they are literally doing, if you don't invoke a collapse hypothesis. You don't get to pick the starting state, if you are only allowed unitary evolution.

 

[vanhees71]

But in my gedanken experiment I simply take a particle in a certain region of space after it has run through a SG apparatus sorting the particles according to their spin-z components in different regions. I could set a "beam dump" to absorb the particles except the ones in the region associated with the wanted $\sigma_z$ value. Then, of course, the time evolution of the system as a whole is not described by unitary time evolution, if you don't take the whole beam dump into account. What's wrong with that?

 

[stevendaryl]

But in my gedanken experiment I simply take a particle in a certain region of space after it has run through a SG apparatus sorting the particles according to their spin-z components in different regions. I could set a "beam dump" to absorb the particles except the ones in the region associated with the wanted $\sigma_z$ value. Then, of course, the time evolution of the system as a whole is not described by unitary time evolution, if you don't take the whole beam dump into account. What's wrong with that?

I didn't say there was anything wrong with it, only that it isn't accurate to describe such a thing as "preparing the system in state $|\psi\rangle$". In another post, I suggested an alternative description in terms of histories of observations. You use unitary evolution, plus a generalization of the Born rule to give the relative probability of getting history $H$ given that the initial part of the history is $H_0$ (some initial segment of history $H$).

 

[vanhees71]

Ok, so how would you describe the working of the SG apparatus?

 

[stevendaryl]

Ok, so how would you describe the working of the SG apparatus?

I don't have a suggestion for the right way to describe things, I'm only questioning what are the implications of describing things in terms of "preparing a system in a particular initial state". If you assume "collapse of the wave function", then you can describe things this way: If you measure the position of the electron, then afterwards, the electron is in a state in which position is localized. Then you pass that electron through a Stern-Gerlach device, and the spin becomes correlated with position. So if you later measure its position, then the wave function collapses to a state of localized position and definite spin. Now you've prepared an electron in the state spin-up in whatever direction.

If you don't assume collapse of the wave function, then it's less clear to me exactly how things should be described. Possibly, as I suggested, it should be described in terms of the relative probabilities of histories of observations, rather than "prepare in state $|\psi\rangle$, measure observable $O$, get eigenvalue $\lambda$ with probability such and such computable using $\psi$, $O$, $\lambda$, and unitary evolution". It could be that the latter description is derivable from the description in terms of histories.

 

[rubi]

I think vanhees is right. The projection postulate is not needed here. If you use the SG apparatus to spatially separate the different spin particles, you end up with a mixed state $\rho_{SG} = \left|x_1,\uparrow\right>\left<x_1,\uparrow\right|+\left|x_2,\downarrow\right>\left<x_2,\downarrow\right|$ (the environment has already been traced out and the small off-diagonal terms have been neglected). Assume you want to do scattering experiments with the spin up particles by a potential $V(x)$, which is supported in a bounded region $R$. You would arrange the SG apparatus in such a way that the spin down particles end up in a different region ($x_2\notin R$), while $x_1\in R$. Now you would choose a basis for $L^2(R)$ and calculate the partial trace $\rho_R=\mathrm{Tr}_R\rho_{SG}=\left|x_1,\uparrow\right>\left<x_1,\uparrow\right|$. If you only want to measure observables in $R$, the states $\rho_{SG}$ and $\rho_R$ are indistinguishable for you. The whole system is still in a mixed state (or even in a pure state, if you include the environmental degrees of freedom) and no collapse has ever happened, but you have isolated a state $\rho_R$ that is indistinguishable from a hypothetically collapsed, pure state $\left|x_1,\uparrow\right>$ for experimenters who only measure in the region $R$.

 

[vanhees71]

I only fight against the word "collapse" and want to substitute by what's really done. You shoot a particle in an arbitary (perhaps unknown) state into an SG apparatus for the spin-z component. Experimental evidence tells us (since 1923 when Stern and Gerlach did their experiment for the first time) that it is deflected in such a way that the spin-z component has a well defined value in the corresponding region. Then I let all particles with the unwanted spin states run into the beam dump and I've either no particle or one with the desired spin-z value. That's what's called, in my understanding a state preparation by a filter (von Neumann) measurement. There's nothing in this setup making it necessary to invoke some strange "mechanism", claiming quantum theory to be invalid and call it "collapse".

 

[Demystifier]

There's nothing in this setup making it necessary to invoke some strange "mechanism", claiming quantum theory to be invalid and call it "collapse".

Of course. To repeat the words of Goldstein, "Either the Schrodinger equation is not right or wave function is not all." The collapse hypothesis corresponds to the Schrodinger-equation-not-right option, while you choose (even if you are not always aware* of it) the other, wave-function**-not-all option.

(*Here is the place where you are aware of it: When Quantum Mechanics is thrashed by non-physicists #1 )

(**In this context, the "wave function" means not only wave function in the narrow non-relativistic-QM sense, but also wave functional in QFT, or more abstractly, a ray in the physical Hilbert space.)

 

[atyy]

I think vanhees is right. The projection postulate is not needed here. If you use the SG apparatus to spatially separate the different spin particles, you end up with a mixed state $\rho_{SG} = \left|x_1,\uparrow\right>\left<x_1,\uparrow\right|+\left|x_2,\downarrow\right>\left<x_2,\downarrow\right|$ (the environment has already been traced out and the small off-diagonal terms have been neglected). Assume you want to do scattering experiments with the spin up particles by a potential $V(x)$, which is supported in a bounded region $R$. You would arrange the SG apparatus in such a way that the spin down particles end up in a different region ($x_2\notin R$), while $x_1\in R$. Now you would choose a basis for $L^2(R)$ and calculate the partial trace $\rho_R=\mathrm{Tr}_R\rho_{SG}=\left|x_1,\uparrow\right>\left<x_1,\uparrow\right|$. If you only want to measure observables in $R$, the states $\rho_{SG}$ and $\rho_R$ are indistinguishable for you. The whole system is still in a mixed state (or even in a pure state, if you include the environmental degrees of freedom) and no collapse has ever happened, but you have isolated a state $\rho_R$ that is indistinguishable from a hypothetically collapsed, pure state $\left|x_1,\uparrow\right>$ for experimenters who only measure in the region $R$.

The projection postulate is only needed for sequential measurements. What you are doing here is instead of sequential measurements, you have the system interact with an ancilla, then take first and second measurements at late and delayed time. It is a principle http://en.wikipedia.org/wiki/Deferred_Measurement_Principle that one can always push all measurements to the final step of the analysis, so that there are no sequential measurements, which is why collapse is not needed if one thinks of the SG experiment in this way.

But can one do this in the Bell test?

 

[vanhees71]

Why shouldn't one be able to do that for a Bell test? Usually you just count entangled biphotons in coincidence experiments. So why can't I describe this in this way?

Of course, I'm silent about how the probabilities come about. For me that's just an independent postulate of the formalism, and within quantum theory I cannot answer the question of, how this occurs, i.e., I don't know of any more simple system of postulates that make the Born postulate to a derivable theorem rather than an independent postulate. Weinberg gives in his newest textbook, "Lectures of Quantum Mechanics" a pretty convincing line of arguments that the Born rule indeed cannot be derived but is an independent postulate.

Of course, that doesn't exclude the possibility that somebody finds a more comprehensive theory of some kind that "explains" it from "more simple" postulates, but as long as I don't see such a theory, I'm inclined to live with quantum theory.