Collapse and Peres' Coarse Graining

[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 $R$.

Before tackling decoherence in the Bell case, I would first still like to understand how sequential measurements are treated here. In the scheme you outlined, there is a first measurement that takes place on a measurement apparatus that is later traced out. The measurement of the local observable on the apparatus will produce the same statistics p(M at t1) as if the measurement had taken place on the electron. Then the apparatus and other regions are traced out to show that a second local measurement will indeed produce the same statistics p(N at t2) on the state, as if collapse had taken place. But how do you treat the correlations between the first and second measurement? If the first measurement detects that a spin is up, the second measurement will also detect that the spin is up. For this we need something like p(M at t1, N at t2) or p(N at t2|M at t1), but the Born rule doesn't say anything about what happens for measurements at different times.

The problem can be evaded by saying that since the apparatus only briefly interacts with the electron, we can measure later on the apparatus p(M at t2) = p(M at t1), which is what I meant by deferring the measurement so that there are no sequential measurements. My reason for going to the Bell test was to force a sequential measurement in some frame. But if we accept that we can measure M at t1, then I don't understand how the Born rule without collapse can produce p(M at t1, N at t2).

 

[rubi]

My understanding is that the quantum optics and the collider people are using the same theory, just that the collider people never test the collapse postulate, and the quantum optics people never test all the decays. So the wave function and collapse is different for each frame of reference. The wave functions of different observers are not related by unitary transformation after a collapse. However, collapse is consistent with Lorentz invariance, since all predictions of the theory are Lorentz invariant. For example:
http://arxiv.org/abs/quant-ph/9906034
http://arxiv.org/abs/1007.3977

These are just some recent references, but it was figured out quite some time ago that collapse doesn't cause any problems for relativity. So my understanding is that the standard model of particle physics does include the collapse postulate (I think one has to use Landau-Lifshitz-Weinberg style quantum mechanics, unless one uses consistent histories, because there is no consensus whether BM works for the standard model because of the chiral fermion problem, and I think there is still no consensus on whether MWI works even for non-relativistic QM in all technical details).

My point is that if wave-functions of individual observers aren't related by unitary transformations, then the postulate of having a unitary representation of the Poincare group on the Hilbert space is not really justified, because our reason to postulate it in the first place is to make the theory independent of the choice of frame. I feel that the success of that postulate suggests that we should seek to describe the collapse in such a way that unitary equivalence is always obeyed. That would also make the quantum state a coordinate-independent object like a tensor in general relativity. (As a person coming from the quantum gravity camp, I'm basically obliged to prefer such a point of view. :D)

Before tackling decoherence in the Bell case, I would first still like to understand how sequential measurements are treated here. In the scheme you outlined, there is a first measurement that takes place on a measurement apparatus that is later traced out. The measurement of the local observable on the apparatus will produce the same statistics p(M at t1) as if the measurement had taken place on the electron. Then the apparatus and other regions are traced out to show that a second local measurement will indeed produce the same statistics p(N at t2) on the state, as if collapse had taken place. But how do you treat the correlations between the first and second measurement? If the first measurement detects that a spin is up, the second measurement will also detect that the spin is up. For this we need something like p(M at t1, N at t2) or p(N at t2|M at t1), but the Born rule doesn't say anything about what happens for measurements at different times.

The problem can be evaded by saying that since the apparatus only briefly interacts with the electron, we can measure later on the apparatus p(M at t2) = p(M at t1), which is what I meant by deferring the measurement so that there are no sequential measurements. My reason for going to the Bell test was to force a sequential measurement in some frame. But if we accept that we can measure M at t1, then I don't understand how the Born rule without collapse can produce p(M at t1, N at t2).

I would calculate all the individual probabilities using $\mathrm{Tr}(\rho(t)P)$, where $\rho(t)$ is the density matrix for the different times and $P$ is the projection you are interested in. Then I would calculate the conditional probabilities using the rule for conditional probabilities from probability theory. Note that I use the projection operators only as a tool to calculate the probabilities. I'm not actually applying it to the state afterwards.

 

[atyy]

I would calculate all the individual probabilities using $\mathrm{Tr}(\rho(t)P)$, where $\rho(t)$ is the density matrix for the different times and $P$ is the projection you are interested in. Then I would calculate the conditional probabilities using the rule for conditional probabilities from probability theory. Note that I use the projection operators only as a tool to calculate the probabilities. I'm not actually applying it to the state afterwards.

But how do I get the conditional probabilities or joint probabilities between observables that are measured at different times? The Born rule only gives probabilities and conditional probabilities between observables measured at the same time.

 

[rubi]

But how do I get the conditional probabilities or joint probabilities between observables that are measured at different times? The Born rule only gives probabilities and conditional probabilities between observables measured at the same time.

I probably don't understand what the problem is, but here is an example: 50% of all particles leave the SG apparatus as spin up at time $t_1$ and maybe 25% of the particles leave the apparatus as spin up at $t_1$ and are scattered into a solid angle $\Omega$ at time $t_2$. Then the conditional probability would be $\frac{1}{4}/\frac{1}{2} = \frac{1}{2}$. The Born rule is not modified if I reject the projection postulate.

 

[atyy]

I probably don't understand what the problem is, but here is an example: 50% of all particles leave the SG apparatus as spin up at time $t_1$ and maybe 25% of the particles leave the apparatus as spin up at $t_1$ and are scattered into a solid angle $\Omega$ at time $t_2$. Then the conditional probability would be $\frac{1}{4}/\frac{1}{2} = \frac{1}{2}$. The Born rule is not modified if I reject the projection postulate.

Can the Born rule calculate that 25% of the particles leave the apparatus as spin up at $t_1$ and are scattered into a solid angle $\Omega$ at time $t_2$? That uses two different times, so if a probability is calculated using $\mathrm{Tr}(\rho(t)P)$, what $t$ is being plugged into the formula: $t_1$ or $t_2$?

The problem I'm having is that if I have:
$P(M(t_1)) = \mathrm{Tr}(\rho(t_1)M)$
$P(N(t_2)) = \mathrm{Tr}(\rho(t_2)N)$,

how can I calculate $p(N(t_2)|M(t_1))$?

From Bayes's rule, I have $P(N|M) = P(M,N)/P(M)$, but although $P(M)$ is known, neither $P(N|M)$ nor $P(M,N)$ are known.

 

[strangerep]

Try Ballentine, section 9.6.

 

[atyy]

Try Ballentine, section 9.6.

Ballentine assumes the collapse hypothesis in his Eq 9.28.

 

[strangerep]

Ballentine assumes the collapse hypothesis in his Eq 9.28.

No, he doesn't. He's using a filtering operator which maps one state into another. But we've been over this before, so I won't retrace that journey.

 

[Jazzdude]

[...]. I feel that the success of that postulate suggests that we should seek to describe the collapse in such a way that unitary equivalence is always obeyed. That would also make the quantum state a coordinate-independent object like a tensor in general relativity. (As a person coming from the quantum gravity camp, I'm basically obliged to prefer such a point of view. :D)

I think this will not work out, even if you don't have any form of collapse. The multi-particle spaces used in quantum theory are not unitarily equivalent for different rest frames. A simple example like a 2-electron Hilbert space already demonstrates the problem, and introducing a general Fock space doesn't make it any simpler.

So before we discuss the general covariance of a possible collapse mechanism, we'd have to find an alternative for the state space.

 

[Dale]

This thread has well outlived its usefulness