Meaning of the word 'instantaneous'

[DrChinese]

I don't see a problem here. By symmetry both Alice and Bob's result happen on each particle independently of the other.

No! That is exactly my point. Because you are not studying the non-aligned cases closely, you are missing this point entirely. There can't be symmetry you imagine. Look at the 0/120/240 cases. There cannot be the symmetry you anticipate because the relationship does not work for the pairs 0/120, 120/240, and 240/0 at the same time (equally) as would be necessary for the symmetry. It's AS IF Bob needs to know which selection Alice is going to make before he decides which to make.

What you are describing is a normal hidden variable model. If you look at specific trials you will see this.

Alice: 0/120/240 Bob: 0/120/240
+/+/- ___ +/+/- : 1/3 are matches at different angles, 100% are matches at the same angles
-/+/- ___ -/+/- : 1/3 are matches at different angles, 100% are matches at the same angles
+/-/+ ___ +/-/+ : 1/3 are matches at different angles, 100% are matches at the same angles

Write out 10 or so of these - it doesn't really matter how many, actually the 3 above should be enough. Be sure to keep it so that at the same angle, there is a match (as I have above). Average out the matches when the angles are different for Alice and Bob. Do this for every possible pairing (all permutations). When you are done, calc the average number of matches. It will not be less than 33% correlation. Keep in mind you are the one selecting the outcomes, and you can make them be anything you like.

But the actual experimental outcome, however, will be close to 25% when the angles are different even though there is perfect correlation at the same angle. That is because the system "knows" which pairing is being selected. (Of course, I have no idea how this happens.) There is no way that the stats work out using your idea unless you know both the choices of measurement angles. There is a clear bias at work in actual cases, and the results cannot match the kind of symmetry you imagine.

Again, I urge you to quit waving your hands, and study the actual situation more closely. If you like, forget Bell. The entire proof is in this post already, I have simply restated it. The true quantum context includes Alice's measurement choice AND Bob's, in addition to the entangled system itself. There is NOT independence, and what you see is defined as quantum nonlocality. The time and distance interval is not constrained by c in the normal sense (looking forward in time).

 

[Paul Colby]

No! That is exactly my point. Because you are not studying the non-aligned cases closely, you are missing this point entirely. There can't be symmetry you imagine. Look at the 0/120/240 cases. There cannot be the symmetry you anticipate because the relationship does not work for the pairs 0/120, 120/240, and 240/0 at the same time (equally) as would be necessary for the symmetry.

What do you mean by "at the same time?"

 

[Paul Colby]

What you are describing is a normal hidden variable model. If you look at specific trials you will see this.

Since I'm basing my arguments on QM only this is amusing. You can talk about all the correlations you want I still don't see how one may claim Bob's measurements are modifying Alice's outcome history for each particle? Correlation is not causation.

 

[DrChinese]

Since I'm basing my arguments on QM only this is amusing. You can talk about all the correlations you want I still don't see how one may claim Bob's measurements are modifying Alice's outcome history for each particle? Correlation is not causation.

As I keep saying, your words do not match what QM says. The fact there is dependence on both Alice and Bob's measurement choices is expressed in the formula itself.

If you are not going to work through example I presented, I really can't help you further.

 

[Paul Colby]

If you are not going to work through example I presented, I really can't help you further.

Fair enough. I understand the examples will work just the way you say they will. Still doesn't address the point I was making which is correlation doesn't imply a causation. Bob can't send a message to Alice no matter how hard he flogs on his particles. He's merely mining information about Alice's measurements. No I can't explain "how he knows" beyond the usual formalism or provide a non-existent classical picture of the correlation. How can I claim this correlation exists prior to their choices? Don't know but that is the way of the world and a fundamental nature of QM ensemble. How can I claim there is no cause in light of this? Well, that's a much harder sell and out of place in this discussion.

 

[votingmachine]

Fair enough. I understand the examples will work just the way you say they will. Still doesn't address the point I was making which is correlation doesn't imply a causation. Bob can't send a message to Alice no matter how hard he flogs on his particles. He's merely mining information about Alice's measurements. No I can't explain "how he knows" beyond the usual formalism or provide a non-existent classical picture of the correlation. How can I claim this correlation exists prior to their choices? Don't know but that is the way of the world and a fundamental nature of QM ensemble. How can I claim there is no cause in light of this? Well, that's a much harder sell and out of place in this discussion.

He is not "sending" information. There is a correlation that can only be arrived at if the particles measured at Alice have some connection with those measured at Bob.

What you keep saying is that the measurements at Bob allow a prediction of Alice. And therefore they are independent. But I can predict the correlations and statistics, right now. I can tell you the results of correlations at Bob and Alice IN ADVANCE. I can tell you the results of a ball dropped from the Empire State building next year, IN ADVANCE. That does not mean there is any connection between me and the ball, or me and the experiments.

On the other hand, you keep seeming to say that there is nothing startling that implies a connection in an EPR experiment. Dr Chinese keeps saying that there is. Well, more accurately he is saying there is a dependence in the results at Alice, on the results at Bob. There is some semantic difference that puzzles me about what you are saying. The two entangled particles behave as one connected PAIR, and measurements are always on the PAIR. That is not cause and effect, where there is a transmission of information, but cause and effect where there is a loss of information. We know from the measurements that the 3 angle properties exist, and exist for the pair as equal and opposite at any angle. And we know that a measurement of one of the two at one angle CAUSES the other particle to lose properties at the other two angles. That loss of properties shows up in the reduced correlation.

EPR experiments showthat the particles cannot have defined and complete properties for every selected angle of measurement, even though when two measurements at the same angle are taken, they always have the complete information for that angle, no matter what it is (which implies they have complete properties for every angle). If Bob measures at angle 0-degree, Bob can predict a measurement by Alice at 0-degree will perfectly correlate, and that a measurement by Alice at 120-degree or 240 degree will 25% correlate. That prediction is straightforward. But it does not prove that the two measurements are independent, but in fact shows that they are dependent.

Knowledge at a distance does not REALLY bother anyone. As I said, I can predict the future of that ball dropped from the Empire State building. Knowledge and predictions are not bounded by the light cone that limits facts. In EPR, it is that the particles seem to share a fact that is outside of the light cone of information limits. I think (and I am puzzled by a lot of what you say) that you are equating that knowledge at a distance with the facts at a distance.

If I watch a telegraph operator push the button, I "mine" the information about the dots and dashes at the other end. There is a connection. If I open a box with half a deck of cards in it, I can predict the contents of the box with the other half with precision, even at a distance. There is no connection. I see the difference between these two, but I cannot see how the paired particles can be consistent with the second, non-connection system.

 

[Paul Colby]

I think (and I am puzzled by a lot of what you say) that you are equating that knowledge at a distance with the facts at a distance.

Forget non-locality, forget Bell, forget EPR for a second. Measurement of a pure non-eigenstate for a 2 state system separates the ensemble into 2 ensembles, one corresponding to each eigenvalue of the measuring device. Let's say we detect a +1/2 particle. For most in this forum the measuring device has caused a "collapse of the wave function" with a resounding and violent crash. The device has physically ripped the hapless particle out of it's previous pure state and jammed it forcibly into another at complete random. If you view the formalism of QM in this way you will likely not follow what I'm saying at all.

A quantum ensemble has a fundamental property of superposition which means one may view a state as existing[1] along multiple other states. I've scratched my head for a number of days trying to find a realistic way of running this process in reverse. I am very certain I don't know how to construct an example using SG and spin 1/2 particles because one needs to be able to set that relative phase of two independent beams. If I hallucinate for a moment and assume that I could prepare a $\vert +z\rangle$ beam and a separate $-\vert -z\rangle$ beam then mixing that would cause a pure state, $\vert -x\rangle$, with the spins aligned along an axis orthogonal to the original one. Independent particles would come together from independent sources to form a new pure state orthogonal to the first. If one could do such an experiment then I could rightly claim that in a very real sense (real as I can do it in the lab) I could view this new pure state, $\vert -x\rangle$, as a collection of particles that very much preexisted in different states. One may be able to do this with photons from lasers that are phase locked. One can certainly do it with RF oscillators.

[1] It's quite fair to ask what the hell I mean by exist. I likely don't have a good answer. Whatever it is it isn't EPRs answer. I can only point to the example above and wave my hands.

 

[DrChinese]

Fair enough. I understand the examples will work just the way you say they will. Still doesn't address the point I was making which is correlation doesn't imply a causation. Bob can't send a message to Alice no matter how hard he flogs on his particles. He's merely mining information about Alice's measurements. No I can't explain "how he knows" beyond the usual formalism or provide a non-existent classical picture of the correlation. How can I claim this correlation exists prior to their choices? Don't know but that is the way of the world and a fundamental nature of QM ensemble. How can I claim there is no cause in light of this? Well, that's a much harder sell and out of place in this discussion.

No one knows what causes what, so no one is asserting X causes Y. Keep in mind that there is no obvious sense in which Alice does anything more to Bob than Bob does to Alice. However, what has been proved is that the correlations are NOT spurious and cannot be independent while entanglement is present. No one knows the precise underlying mechanism.

Therefore, the example you give where Bob measures and the Alice reads her particle is somewhat skewed in the sense that it implies something that may or may not be present. It certainly is not present in the formalism. I will again remind you that particles can be entangled AFTER they are measured, and they do not need to come into contact to become entangled. That alone should dash your mental picture of what is happening. The mechanism for doing that is called entanglement swapping.

 

[Paul Colby]

No one knows what causes what, so no one is asserting X causes Y. Keep in mind that there is no obvious sense in which Alice does anything more to Bob than Bob does to Alice. However, what has been proved is that the correlations are NOT spurious and cannot be independent while entanglement is present. No one knows the precise underlying mechanism.

Agreed. Assuming this, how do you address the question asked by the OP? Do you consider the EPR example as "something happening" instantaneously? If so, can you define "something" and "happening" in this context?

 

[DrChinese]

Agreed. Assuming this, how do you address the question asked by the OP? Do you consider the EPR example as "something happening" instantaneously? If so, can you define "something" and "happening" in this context?

I am not sure "what" is happening. Since the system has both a spatial and temporal extent that does not follow any regular pattern, I don't even know if the word "instantaneous" is appropriate. Or even "faster than light". I don't even know if it has a direction in time or space. Very confusing. But in my mind, the observer(s) is part of the system.

 

[Paul Colby]

I am not sure "what" is happening. Since the system has both a spatial and temporal extent that does not follow any regular pattern, I don't even know if the word "instantaneous" is appropriate. Or even "faster than light". I don't even know if it has a direction in time or space. Very confusing. But in my mind, the observer(s) is part of the system.

The "observer", which in every case is an inanimate piece of lab equipment, is part of every QM measurement extended or not local or not. This is not new and I never (intentionally) meant to indicate otherwise. No matter how one rails on the correlations being observer dependent I still believe there is no "cause" in the sense that people normally use the term outside of that inherent in the initial preparation of the entangled system. To make any headway with this view, I need to chip away at the usage of the word cause as applied to QM measurements. When one measures a +z prepared spin with a x-directed SG, does the SG cause the individual spins to flip? From the discussions so far I'm convinced that this is not the case.

 

[Jilang]

Spins precess in magnetic fields. This may help.
http://www.tcm.phy.cam.ac.uk/~bds10/aqp/lec6_compressed.pdf

 

[zonde]

Agreed. Assuming this, how do you address the question asked by the OP?

OP was no asking about EPR.

Do you consider the EPR example as "something happening" instantaneously? If so, can you define "something" and "happening" in this context?

"something happening" is concept directly related to physical reality but pure QM formalism is only loosely connected to physical reality. So if you want to model how "something happens" physically you have to pick some interpretation of QM.
Basically "instantaneous" is meaningless within pure QM formalism.

 

[nortonian]

But in my mind, the observer(s) is part of the system.

The "observer", which in every case is an inanimate piece of lab equipment, is part of every QM measurement extended or not local or not. This is not new and I never (intentionally) meant to indicate otherwise.

I am confused about the difference between "state" and "wave function" as they are used in EPR type experiments. Assume an entangled state consisting of a pair of polarized photons with Bob's measuring instrument located close to the experiment and Alice's far away. Bob measures first and records the photon's spin and its polarization. According to the Copenhagen interpretation the wave function includes aspects of both the photon and apparatus so there is a collapse of the wave function but apparently the state is still entangled. At a later point in time Alice measures the second photon's properties and collapses the wave function determined by her instrument and the second photon, but she also causes the state to become unentangled. So if there are two wave functions, one for each photon, then why is there a question about actions occurring faster than light? We can simply say that when the photon pair is created the state is divided and two wave functions come into existence. Then instantaneous collapse can be said to refer to the state but not the wave function. How can the photon pair be assigned a wave function immediately upon creation before they are measured, that is, before instruments are able to detect them?

 

[DrChinese]

I am confused about the difference between "state" and "wave function" as they are used in EPR type experiments. Assume an entangled state consisting of a pair of polarized photons with Bob's measuring instrument located close to the experiment and Alice's far away. Bob measures first and records the photon's spin and its polarization. According to the Copenhagen interpretation the wave function includes aspects of both the photon and apparatus so there is a collapse of the wave function but apparently the state is still entangled. At a later point in time Alice measures the second photon's properties and collapses the wave function determined by her instrument and the second photon, but she also causes the state to become unentangled. So if there are two wave functions, one for each photon, then why is there a question about actions occurring faster than light? We can simply say that when the photon pair is created the state is divided and two wave functions come into existence. Then instantaneous collapse can be said to refer to the state but not the wave function. How can the photon pair be assigned a wave function immediately upon creation before they are measured, that is, before instruments are able to detect them?

This is exactly the point I was discussing with Paul. Yes, I know perfectly well that yours appears to be a good explanation - at least for the so-called perfect correlations. Those are the cases in which Alice and Bob measure at the same angle. Please note that the outcomes are essentially redundant in that case - and it is a special case. This special case certainly suggests strongly that there is no question about "about actions occurring faster than light". And in fact, this is essentially the premise of EPR (1935).

But all that stops when other angle settings are selected at random by Alice and Bob. Then, the results distinctly show that Alice's results are dependent on Bob's measurement choice, and vice versa. To understand this, either follow the Bell argument (1965) or see the example I provided to Paul. Please read my posts #64 and #71. The upshot is: You can't have outcomes consistent with the predictions of QM at the angle settings I specified (when Alice and Bob randomly select their measurement angles independently).

Also: QM says that the entangled observables are a single combined system (state) and "collapse" at some point into separated systems (states). But no one knows precisely when that happens. Is it when the "first" observation is made? Or the "second"? Or when the observations all become "un-eraseable"? The answer to this is subject to interpretation.

And finally: entanglement of 2 photons does not require them to have interacted in the past. Nor to have interacted with anything that ever interacted in the past. Nor to have ever co-existed. Or even have existed in a common light cone. None of this is consistent with your mental picture of entanglement. Which is: "the photon pair is created the state is divided and two wave functions come into existence". That is not consistent with the QM formalism, which posits a single combined state/WF.

 

[jimgraber]

"And finally: entanglement of 2 photons does not require them to have interacted in the past. Nor to have interacted with anything that ever interacted in the past. Nor to have ever co-existed. Or even have existed in a common light cone. None of this is consistent with your mental picture of entanglement. Which is: "the photon pair is created the state is divided and two wave functions come into existence". That is not consistent with the QM formalism, which posits a single combined state/WF."

Reference https://www.physicsforums.com/threads/meaning-of-the-word-instantaneous.877568/page-5
So is this in conflict with the backward in time explanations?
TIA
Jim Graber

 

[Paul Colby]

You can't have outcomes consistent with the predictions of QM at the angle settings I specified (when Alice and Bob randomly select their measurement angles independently

You make a true statement. Still, I believe there is no instantaneous interaction involved in the way I would define one. Alice and Bob's observables commute and may be simultaneously diagonalized. So Bob or Alice's measurement results in a state of the other's particle which depends on the relative angle settings. Let Alice record the outcome of each of $N_1$ particle detections in turn with her detector at 0. The experiment is done twice. The first time with Bob's detector at say $\alpha_1$ and the second with an angle of $\alpha_2$. We take more data than necessary $N_2>N_1$ so that we may go back and amend history making Alice's results identical to her first on an event by event basis. Bob's perspective is identical(ish) in the two experiments except the event by event record would not be. When Bob consults Alice's data he would see two different partitioning of Alice's two identical results that depends on his relative angle setting. So, as an isolated experimenter, Bob's measurements have no effect (Alice's two results are identical after all) on Alice's measurements.

 

[ftr]

You make a true statement. Still, I believe there is no instantaneous interaction involved in the way I would define one. Alice and Bob's observables commute and may be simultaneously diagonalized. So Bob or Alice's measurement results in a state of the other's particle which depends on the relative angle settings. Let Alice record the outcome of each of $N_1$ particle detections in turn with her detector at 0. The experiment is done twice. The first time with Bob's detector at say $\alpha_1$ and the second with an angle of $\alpha_2$. We take more data than necessary $N_2>N_1$ so that we may go back and amend history making Alice's results identical to her first on an event by event basis. Bob's perspective is identical(ish) in the two experiments except the event by event record would not be. When Bob consults Alice's data he would see two different partitioning of Alice's two identical results that depends on his relative angle setting. So, as an isolated experimenter, Bob's measurements have no effect (Alice's two results are identical after all) on Alice's measurements.

No matter how you word or think about( not especially you), the problem is well established now decades later, otherwise we would not be talking about it

https://en.wikipedia.org/wiki/Quantum_entanglement

"Measurements of physical properties such as position, momentum, spin, polarization, etc., performed on entangled particles are found to be appropriately correlated. For example, if a pair of particles are generated in such a way that their total spin is known to be zero, and one particle is found to have clockwise spin on a certain axis, then the spin of the other particle, measured on the same axis, will be found to be counterclockwise, as to be expected due to their entanglement. However, this behavior gives rise to paradoxical effects: any measurement of a property of a particle can be seen as acting on that particle (e.g., by collapsing a number of superposed states) and will change the original quantum property by some unknown amount; and in the case of entangled particles, such a measurement will be on the entangled system as a whole. It thus appears that one particle of an entangled pair "knows" what measurement has been performed on the other, and with what outcome, even though there is no known means for such information to be communicated between the particles, which at the time of measurement may be separated by arbitrarily large distances."

 

[DrChinese]

You make a true statement. Still, I believe there is no instantaneous interaction involved in the way I would define one. Alice and Bob's observables commute and may be simultaneously diagonalized. So Bob or Alice's measurement results in a state of the other's particle which depends on the relative angle settings. Let Alice record the outcome of each of $N_1$ particle detections in turn with her detector at 0. The experiment is done twice. The first time with Bob's detector at say $\alpha_1$ and the second with an angle of $\alpha_2$. We take more data than necessary $N_2>N_1$ so that we may go back and amend history making Alice's results identical to her first on an event by event basis. Bob's perspective is identical(ish) in the two experiments except the event by event record would not be. When Bob consults Alice's data he would see two different partitioning of Alice's two identical results that depends on his relative angle setting. So, as an isolated experimenter, Bob's measurements have no effect (Alice's two results are identical after all) on Alice's measurements.

I will repeat (because your example fails this important point): Alice and Bob must make their measurement selections independently and randomly (in other words, neither knows what the other will do). Obviously the results of this (as an actual experiment) will match the predictions of QM. But you cannot construct an example picking the values yourself that would (i.e. applying your hypothetical mechanism).

And I would not agree that Alice and Bob's observables commute. For example: Alice measuring at 0 degrees and Bob at 45 degrees does not give the same result as Alice measuring at 45 degrees and Bob at 0 degrees. Any more than Alice measuring at 0 degrees and then at 45 degrees gives the same result as Alice measuring at 45 degrees and then at 0 degrees.

 

[DrChinese]

... any measurement of a property of a particle can be seen as acting on that particle (e.g., by collapsing a number of superposed states) and will change the original quantum property by some unknown amount; and in the case of entangled particles, such a measurement will be on the entangled system as a whole. It thus appears that one particle of an entangled pair "knows" what measurement has been performed on the other, and with what outcome, even though there is no known means for such information to be communicated between the particles, which at the time of measurement may be separated by arbitrarily large distances."

Worth repeating the quote. Note that no one fully understands the mechanism itself. So when the phrases "knows what measurement" and "will change the original quantum property" are used, you might add "as if" so as to get this idea incorporated.

Note that the "arbitrarily large distances" point relates the the "instantaneous" portion of this thread's title. The separation can also be in terms of both time and distance in such a way that "instantaneous" loses all meaning.

 

[Paul Colby]

No matter how you word or think about( not especially you), the problem is well established now decades later, otherwise we would not be talking about it

Yes, I understand all possible issues have been considered, discussed and understood better than I am ever likely to. This is more about me coming to grips with the physics and understand what people are saying, hopefully without pissing them off. Wiki pages are helpful but, like certain religious text, written by people with opinions that must be teased out and separated from the actual facts.

 

[Paul Colby]

And I would not agree that Alice and Bob's observables commute. For example: Alice measuring at 0 degrees and Bob at 45 degrees does not give the same result as Alice measuring at 45 degrees and Bob at 0 degrees. Any more than Alice measuring at 0 degrees and then at 45 degrees gives the same result as Alice measuring at 45 degrees and then at 0 degrees.

You must be operating with a different understanding of what constitutes an observable. My understanding is Bob's observable is $S_{1,\alpha}$ a component of spin for his particle (1) while Alice's is $S_{2,\beta}$ for particle (2). These occupy two separate and independent Hilbert spaces, $H_1$ and $H_2$. Therefore, $[S_{1,\alpha},S_{2,\beta}]=0.$ This remains true when entangled when the Hilbert space is the direct product, $H=H_1\otimes H_2$.

 

[DrChinese]

Wiki pages are helpful but, like certain religious text, written by people with opinions that must be teased out and separated from the actual facts.

Without defending Wikipedia: the usual criticism of the Wikipedia physics pages is not one of bias, but that the matters are presented at an insufficient depth level. I certainly don't think your "religious" comment applies to the section quoted above. This is pretty standard wording for describing entanglement in lay terms.

 

[DrChinese]

My understanding is Bob's observable is $S_{1,\alpha}$ a component of spin for his particle (1) while Alice's is $S_{2,\beta}$ for particle (2). These occupy two separate and independent Hilbert spaces, $H_1$ and $H_2$. Therefore, $[S_{1,\alpha},S_{2,\beta}]=0.$ This remains true when entangled when the Hilbert space is the direct product, $H=H_1\otimes H_2$.

Sorry, not true while they are entangled. There is just one system! If there were a product state (a combination of 2 separable particles) we would not be having this discussion.

 

[Paul Colby]

Sorry, not true while they are entangled. There is just one system! If there were a product state (a combination of 2 separable particles) we would not be having this discussion.

Hard to have a discussion if the rules aren't understood. Every quantum book I've read says the state space for a two particle system is $H_1\otimes H_2$ from which my statement follows. Hard for me to conclude you have a point since my working of the problem would contain this "flaw" from the outset.

 

[DrChinese]

Hard to have a discussion if the rules aren't understood. Every quantum book I've read says the state space for a two particle system is $H_1\otimes H_2$ from which my statement follows. Hard for me to conclude you have a point since my working of the problem would contain this "flaw" from the outset.

Your statement is true for 2 normal particles. But not true for entangled particles, at least in the space where they are entangled.

I can't stress enough that your point of view was the starting point for EPR, 1935. They looked at entangled particle pairs as belonging to different spaces as an assumption. Unfortunately, they (and everyone else for the next 30 years) missed out on the implications of such an assumption. This was ultimately noted by Bell. Today, we know entangled systems are not separable and they are not represented in that manner.

 

[Paul Colby]

I can't stress enough that your point of view was the starting point for EPR, 1935. They looked at entangled particle pairs as belonging to different spaces as an assumption. Unfortunately, they (and everyone else for the next 30 years) missed out on the implications of such an assumption. This was ultimately noted by Bell. Today, we know entangled systems are not separable and they are not represented in that manner.

Okay, so how do you compute $[S_{1,\alpha},S_{2,\beta}]$?

So, I might add this is an operator expression which holds independent of a give system state.

 

[DrChinese]

Okay, so how do you compute $[S_{1,\alpha},S_{2,\beta}]$?

It's actually nicely discussed in section 4.1 of the reference ftr provided (did you miss this part when you were dismissing the link?):

https://en.wikipedia.org/wiki/Quantum_entanglement#Pure_states

Again: IF you assume that the combined entangled system is in a (separable) Product state, THEN you can prove that the combined entangled system is in a... (separable) Product state. Not much there. Of course you cannot use this assumption AND get agreement with the predictions of QM for entangled systems. As Bell showed us.

 

[Paul Colby]

It's actually nicely discussed in section 4.1 of the reference ftr provided (did you miss this part when you were dismissing the link?):

Sorry, I thought you understood my question. The $S_{1,\alpha}$ and ##S_{2,\beta} operators commute independent of section 4.1 which correctly points out the entangled states are generally not separable into a product. Both these facts are known to me. Because they commute these operators may be simultaneously diagonalized. This yields a perfectly valid view of the entangle system. When expressed in terms of eigenstates of Alice and Bob's observables, one obtains an expression for the entangled $S=0$ state where the coefficients or probability amplitudes are angle dependent in accordance with Bell and the physics. So, each measurement Bob makes determines a single particle state whose coefficients for Alice are dependent on Bob's angle setting.

 

[DrChinese]

Sorry, I thought you understood my question. The $S_{1,\alpha}$ and ##S_{2,\beta} operators commute independent of section 4.1 which correctly points out the entangled states are generally not separable into a product. Both these facts are known to me. Because they commute these operators may be simultaneously diagonalized. This yields a perfectly valid view of the entangle system. When expressed in terms of eigenstates of Alice and Bob's observables, one obtains an expression for the entangled $S=0$ state where the coefficients or probability amplitudes are angle dependent in accordance with Bell and the physics. So, each measurement Bob makes determines a single particle state whose coefficients for Alice are dependent on Bob's angle setting.

You are mixing (often conflicting) subjects within sentences. So no, nothing you are saying makes sense. Your viewpoint simply does not follow standard physics: Entangled systems are NOT created in local separable states as you somehow imagine. Entangled states are in fact not separable, and product statistics (which do commute) do not accurately describe entangled systems. The bottom line is that you want to have your cake and eat it too - QM without quantum nonlocality.

You are welcome to your personal opinion, but this conversation has gone 'round in circles far too long. I will simply say that for anything you state contrary to standard QM (such as I have presented), please provide peer-reviewed or other suitable references to support such. Those are the forum rules in this situation. Readers may otherwise get the idea that there is scientific controversy in this realm, when there is not.

 

[Paul Colby]

You are mixing (often conflicting) subjects within sentences. So no, nothing you are saying makes sense.

Fair enough. I'm just trying to understand. Thank you for the help.

 

[Paul Colby]

Entangled states are in fact not separable,

I guess I fail to see where I ever said they are?

When expressed in terms of eigenstates of Alice and Bob's observables, one obtains an expression for the entangled $S=0$ state where the coefficients or probability amplitudes are angle dependent in accordance with Bell and the physics. So, each measurement Bob makes determines a single particle state whose coefficients for Alice are dependent on Bob's angle setting.

So, what exactly is wrong with my statement?

Ah, "in accordance with Bell" could be construed as incorrect.

 

[zonde]

I guess I fail to see where I ever said they are [separable]?

Here:

You must be operating with a different understanding of what constitutes an observable. My understanding is Bob's observable is $S_{1,\alpha}$ a component of spin for his particle (1) while Alice's is $S_{2,\beta}$ for particle (2). These occupy two separate and independent Hilbert spaces, $H_1$ and $H_2$. Therefore, $[S_{1,\alpha},S_{2,\beta}]=0.$ This remains true when entangled when the Hilbert space is the direct product, $H=H_1\otimes H_2$.

 

[zonde]

Still, I believe there is no instantaneous interaction involved in the way I would define one. Alice and Bob's observables commute and may be simultaneously diagonalized.

I don't think it's very smart thing to do - redefine terms just to look still right.

 

[Paul Colby]

Here:

Bob's observable is $\hat{n}_1\cdot S_1$ where $S_1$ is a 3-vector with the Pauli spin matrices as components. The unit vector, $\hat{n}_1$ is a unit vector along a direction chosen by Bob and is the alignment of his SG. This matrix operates on a 2 dimensional Hilbert space $H_1$. Likewise a for Alice. Her observable is $\hat{n}_2\cdot S_2$ which is an operator which lives an independent 2 dimensional Hilbert space, $H_2$. Alice and Bob's observables do in fact commute and may be simultaneously diagonalized. However, this does not imply in anyway that the $S=0$ is separable as DrChines insists[1].

These facts may be checked in virtually every book on QM. That the $S=0$ doesn't factor is trivial to show. Let Bob's SG be along the z-axis to save me some writing. The $S=0$ state is then,

$\frac{1}{\sqrt{2}}(\vert 1\rangle_{zB}\otimes\vert -1\rangle_{zA} - \vert -1\rangle_{zB}\otimes\vert 1\rangle_{zA})$
​

where, the observant reader will instantly accuse me of using Bob's z-axis to expand Alice's vectors. Because I'm free to use Alice's coordinates for her particle I may expand them as such,

$\vert 1\rangle_{zA} = a\vert 1\rangle_{\alpha A} + b\vert -1\rangle_{\alpha A}$ [2]

$\vert -1\rangle_{zA} = c\vert 1\rangle_{\alpha A} + d\vert -1\rangle_{\alpha A}$
​

where $a, b, c$ and $d$ are complex coefficients which very much depend on the choice of $\hat{n}_1$ and $\hat{n}_2$ just as DrChines has pointed out numerous times. Substitution of these expressions into the one for the $S=0$ above yields (a very much not separable) expression for the $S=0$ state in terms of both Bob's and Alice's particle eigenvectors. Before anyone replies asserting that I am claiming this somehow removes all EPR mysteries consider the fact that I do not claim such nor do I ever feel I have in the past. I have always assumed a level of mathematical sophistication which is perhaps unwarranted.

[1] Weather he's insisting $[S_1,S_2]=0$ implies factorability of the $S=0$ state or insisting that I have insisted such is unclear. Both positions are in fact wrong.
[2] My notation here is very confusing. The basis on the right are eigenstates of Alice's SG while those on the left are eigenstates assuming she had aligned hers with Bob's which is along the z-axis. Okay, I've attempted to repair it. I've added a subscript $z$ to denote z-axis eigenvectors and an $\alpha$ to denote eigenvectors for Alice's SG direction which is arbitrary.