MWI as applied to measurement of spin 1/2 entangled particle pair

[cianfa72]

TL;DR Summary

About the measurement of spin 1/2 entangled particle pair using Pauli operator matrices

Hello,
consider a pair of 1/2 spin entangled system of particles A and B given in the basis of eigenvectors of Pauli operator $\sigma_z$ as $$\ket{\psi} = \frac {1} {\sqrt (2)} \left ( \ket {+z} \otimes \ket {-z} - \ket {-z} \otimes \ket {+z} \right )$$
A measurement of particle A's spin along z-axis is given from the self-adjoint operator $\sigma_z \otimes I$ acting on the (overall) system entangled state.

From a formal point of view, how does one get either the state $\ket{+z} \otimes \ket{-z}$ or $\ket{-z} \otimes \ket{+z}$ upon the spin measurement of A sub-system along z-axis ?

Thanks.

 

[Nugatory]

From a formal point of view, how does one get either the state $\ket{+z} \otimes \ket{-z}$ or $\ket{-z} \otimes \ket{+z}$ upon the measurement ?

That’s just the collapse postulate, if I’m understanding your question properly.

 

[cianfa72]

That’s just the collapse postulate, if I’m understanding your question properly.

Yes, formally the writing $( \sigma_z \otimes I) \ket{\psi}$ as
$$ ( \sigma_z \otimes I) \frac {1} {\sqrt (2)} \left ( \ket {+z} \otimes \ket {-z} - \ket {-z} \otimes \ket {+z} \right )$$ gives either $\ket{+z} \otimes \ket{-z}$ or $\ket{-z} \otimes \ket{+z}$ ?

 

[PeterDonis]

Yes, formally the writing $( \sigma_z \otimes I) \ket{\psi}$ as
$$ ( \sigma_z \otimes I) \frac {1} {\sqrt (2)} \left ( \ket {+z} \otimes \ket {-z} - \ket {-z} \otimes \ket {+z} \right )$$ gives either $\ket{+z} \otimes \ket{-z}$ or $\ket{-z} \otimes \ket{+z}$ ?

Only if you apply the collapse postulate. If you don't, the state doesn't change at all, since both terms in it are eigenstates of the operator being applied and the operator is linear. (Do the matrix multiplication and see.)

Note also that by asking how to get either $\ket{+z}$ or $\ket{-z}$ as a state by applying the operator you describe to the state you describe (instead of just asking what the probabilities are for the up and down measurement results), you are implicitly assuming that "measurement" physically changes the state of the individual system being measured in that way. That is interpretation dependent; not all QM interpretations work that way.

 

[cianfa72]

Only if you apply the collapse postulate. If you don't, the state doesn't change at all, since both terms in it are eigenstates of the operator being applied and the operator is linear. (Do the matrix multiplication and see.)

Ok, since $\sigma_z \otimes I$ is linear and $$\sigma_z \otimes I ( \ket {\pm z} \otimes \ket {\mp z} ) = \sigma_z (\ket {\pm z} ) \otimes I (\ket {\mp z}) = \ket {\pm z} \otimes \ket {\mp z}$$

Note also that by asking how to get either $\ket{+z}$ or $\ket{-z}$ as a state by applying the operator you describe to the state you describe (instead of just asking what the probabilities are for the up and down measurement results), you are implicitly assuming that "measurement" physically changes the state of the individual system being measured in that way. That is interpretation dependent; not all QM interpretations work that way.

In other words there are QM interpretations in which there is not a collapse of the QM state/wavefunction when measurement of an observable is performed (here with individual system being measured you mean the overall system composed from the two entangled particles).

 

[PeterDonis]

Ok, since $\sigma_z \otimes I$ is linear and $$\sigma_z \otimes I ( \ket {\pm z} \otimes \ket {\mp z} ) = \sigma_z (\ket {\pm z} ) \otimes I (\ket {\mp z}) = \ket {\pm z} \otimes \ket {\mp z}$$

Yes.

In other words there are QM interpretations in which there is not a collapse of the QM state/wavefunction when measurement of an observable is performed (here with individual system being measured you mean the overall system composed from the two entangled particles).

There are interpretations where the QM state represents the physical state of individual systems, but there is no collapse on measurement--e.g., the MWI.

But there are also interpretations where the QM state doesn't represent the physical state of individual systems at all--e.g., the statistical/ensemble interpretation as described in Ballentine, where the state represents either an abstract ensemble of systems all prepared by the same preparation procedure and with the same distribution of probabilities for all observables, or the preparation procedure itself. In such interpretations it doesn't even make sense to ask what happens to the QM state of an individual system when it is measured; the QM state doesn't represent the individual system to begin with.

If you want to discuss interpretations, we should move this thread to the interpretations subforum.

 

[cianfa72]

There are interpretations where the QM state represents the physical state of individual systems, but there is no collapse on measurement--e.g., the MWI.

Sorry, by the physical state of individual systems, in the case of entangled particle pair A and B, do you actually mean the state of the "composite" system A + B, right ?

If you want to discuss interpretations, we should move this thread to the interpretations subforum.

How can I move it to another subforum ?

 

[PeterDonis]

Sorry, by the physical state of individual systems, in the case of entangled particle pair A and B, do you actually mean the state of the "composite" system A + B, right ?

Sure. But just that particular pair. Not an abstract ensemble of pairs all produced by the same preparation procedure.

How can I move it to another subforum ?

Do you want to discuss intepretations in this thread? If so, I'll move it.

 

[cianfa72]

Do you want to discuss intepretations in this thread? If so, I'll move it.

I'd like to discuss the MWI interpretation for such entangled particles system.

 

[PeterDonis]

I'd like to discuss the MWI interpretation for such entangled particles system.

Ok. Thread has been moved to the interpretations subforum. I have also edited the thread title to clarify the topic.

 

[PeterDonis]

From a formal point of view, how does one get either the state $\ket{+z} \otimes \ket{-z}$ or $\ket{-z} \otimes \ket{+z}$ upon the spin measurement of A sub-system along z-axis ?

Since you have now clarified that you want to discuss the MWI as it applies to this scenario, the response to this is simple: you don't get either one of those states as a result of a z spin measurement on one particle. What you do get is a state that now has the two-particle system entangled with the measuring device and its environment, like so:

$$
\frac{1}{\sqrt{2}} \left( \ket{z+} \otimes \ket{z-} \otimes \ket{\text{measured particle 1 spin z up}} + \ket{z-} \otimes \ket{z+} \otimes \ket{\text{measured particle 1 spin z down}} \right)
$$

 

[cianfa72]

Thank you. I read that MWI interpretation is based on dechoerence. From my understanding that means the enviroment became entangled with the system. In other words, starting from separate wavefunctions for the system and the enviroment, there will be only one wavefunction/state for the overall entangled system.

 

[PeterDonis]

I read

Where? You should know by now that you need to give a specific reference.

MWI interpretation is based on dechoerence.

More precisely, decoherence helped greatly to clarify the MWI. But decoherence theory wasn't developed until two decades after the MWI was discovered.

that means the enviroment became entangled with the system.

The measuring device and its environment. That's obvious from the state I wrote down in post #11.

 

[cianfa72]

The measuring device and its environment. That's obvious from the state I wrote down in post #11.

Ah ok, so it is actually the enviroment of the measuring device.

From the entangled state in your post #11, are the two terms in it the parts of wavefunction/state that actually exist in parallel universes of MWI interpretation?

 

[PeterDonis]

From the entangled state in your post #11, are the two terms in it the parts of wavefunction/state that actually exist in parallel universes of MWI interpretation?

The two terms are referred to as "worlds" in the "many worlds" interpretation, yes.

 

[cianfa72]

The two terms are referred to as "worlds" in the "many worlds" interpretation, yes.

But...since the two terms are part of the same complete entangled system wavefunction/state does make sense to split them each in a different "world" ?

 

[PeterDonis]

since the two terms are part of the same complete entangled system wavefunction/state does make sense to split them each in a different "world" ?

If you're an MWI proponent, apparently it does. Those who are MWI skeptics are not so sure.

 

[jbergman]

But...since the two terms are part of the same complete entangled system wavefunction/state does make sense to split them each in a different "world" ?

The point is that in the example, the worlds are no longer interacting and are behaving in some sense like separate classical worlds. This is related to the concept of decoherence. And it isn't just the measuring device that ends up entangled with these measured states, it is the observers and the entire environment where the experiment took place. Essentially, what is happening is that the two environments end up with separate evolutions post measurement since things like photons and particles will interact with the measurement outcome differently.

Maybe Peter can state this in a more precise way. :)

 

[cianfa72]

The point is that in the example, the worlds are no longer interacting and are behaving in some sense like separate classical worlds. This is related to the concept of decoherence. And it isn't just the measuring device that ends up entangled with these measured states, it is the observers and the entire environment where the experiment took place. Essentially, what is happening is that the two environments end up with separate evolutions post measurement since things like photons and particles will interact with the measurement outcome differently.

So the dechoerence "forces" the measurement device, the observer and the environment (where the experiment take place) to be entangled with the system of two entangled particles it interacts with.

In other words, the overall "system" composed from the two particles, the measurement device, the observer and the environment that "splits" in two independent/separate worlds.

 

[PeterDonis]

the dechoerence "forces" the measurement device, the observer and the environment (where the experiment take place) to be entangled with the system of two entangled particles it interacts with.

No. The entanglement is produced by interactions between the measured system, the measuring device, the observer, and the environment.

Decoherence occurs when the entanglement has spread among a large number of untrackable degrees of freedom. This makes interference effects effectively impossible to detect and makes the process effectively irreversible.

the overall "system" composed from the two particles, the measurement device, the observer and the environment that "splits" in two independent/separate worlds.

The overall "system" includes everything that becomes involved in the interactions. Ultimately that includes the entire universe.

 

[cianfa72]

Decoherence occurs when the entanglement has spread among a large number of untrackable degrees of freedom. This makes interference effects effectively impossible to detect and makes the process effectively irreversible.

So, let me say, dechoerence comes after the entanglement is produced (as you described) when it has "extended" among a large number of untrackable degree of freedom. The fact that such process is irreversibile effectively implies that the system behaves like a "classic system".

 

[PeterDonis]

dechoerence comes after the entanglement is produced (as you described) when it has "extended" among a large number of untrackable degree of freedom.

Yes.

The fact that such process is irreversibile effectively implies that the system behaves like a "classic system".

If we're talking about the MWI, the point is that decoherence being effectively irreversible is what grounds the claim that each individual term in the total entangled wave function represents a "world". But the overall wave function itself is certainly not "classical".

 

[cianfa72]

If we're talking about the MWI, the point is that decoherence being effectively irreversible is what grounds the claim that each individual term in the total entangled wave function represents a "world".

Consider your overall wavefunction in post #11. Does each term represent the entangled state after the dechoerence has occurred or the entangled wavefunction/state before it has occurred ?

 

[PeterDonis]

Consider your overall wavefunction in post #11. Does each term represent the entangled state after the dechoerence has occurred or the entangled wavefunction/state before it has occurred ?

After. The kets for "measured" include all of the degrees of freedom involved in decoherence.

 

[cianfa72]

After. The kets for "measured" include all of the degrees of freedom involved in decoherence.

Ah ok, so only the kets with "measured particle...." inside include all of the degree of freedom involved in dechoerence.

 

[PeterDonis]

only the kets with "measured particle...." inside include all of the degree of freedom involved in dechoerence.

Yes.