Spin difference between entangled and non-entangled

[Alien8]

That number itself is an arbitrary one, nothing fundamental about it. Prior to Bell type inequalities, I am not aware of any specific measures of quantum non-locality. I guess you could say the perfect correlations mentioned a la EPR fit the bill. I can't think of any specific early points at which someone was saying "aha, look how non-local QM is." They were, however, saying that it was non-realistic (observer dependent). This was EPR's chief objection to QM.

Yeah, it all started with uncertainty and non-reality, but somehow ended up with non-locality. What's the connection?

 

[Alien8]

Well, the informal "recipe" for using quantum mechanics is explicitly nonlocal and instantaneous:

1. Describe the initial state by some wave function $\Psi$
2. Later perform a measurement corresponding to operator $O$.
3. Get a value $\lambda$
4. For future measurements, use $P_{O,\lambda} \Psi$, where $P_{O,\lambda}$ is the projection operator that projects onto the subspace of wave functions that are eigenstates of $O$ with eigenvalue $\lambda$

This recipe is explicitly instantaneous and nonlocal, since a measurement here causes the wave function describing distant phenomena to change instantly. Of course, many people didn't think of that as really nonlocal, because the wave function was regarded (at least by some) as reflecting our knowledge of the distant phenomena, rather than anything objective about that phenomena.

But where does it say a single wave function can be applied to two separate photons interacting with two separate polarizers?

 

[atyy]

That's what I'm talking about. It says a lot about how locality is supposed to fail, but little about how non-locality is supposed to work. I've learned in the other thread how to derive CHSH local prediction: $1/2 * cos^2(a-b)$ from Malus' law, but I am yet to hear what law is QM prediction: $cos^2(a-b)$ based on. It seems it has to do with uncertainty principle, but I don't see uncertainty can explain or justify non-locality, at all.

The interesting thing is that there are at least two notions of locality. The first is the notion is called "local causality" and can be built up from local deterministic theories, and is the notion addressed by Bell's inequality. A wider notion of locality is called "relativistic causality" and means that we cannot send messages faster than the speed of light. Although QM violates local causality, it is consistent with the wider notion of relativistic causality.

 

[DrChinese]

It says a lot about how locality is supposed to fail, but little about how non-locality is supposed to work. I've learned in the other thread how to derive CHSH local prediction: $1/2 * cos^2(a-b)$ from Malus' law, but I am yet to hear what law is QM prediction: $cos^2(a-b)$ based on. It seems it has to do with uncertainty principle, but I don't see uncertainty can explain or justify non-locality, at all.

That is because no one knows anything deeper about quantum non-locality. It may not be a non-local force in the the sense of "something" moving faster than c. Or maybe the Bohmians have it right. At this point, there is no candidate theory which is local realistic due to experimental failure, and the interpretations of QM cannot currently be distinguished on the basis of experiment. So your choice of the available QM interpretations is as good as anyone's.

 

[stevendaryl]

But where does it say a single wave function can be applied to two separate photons interacting with two separate polarizers?

Quantum mechanics describes any collection of particles by a single wave function (or, more generally, a density matrix). There is no way to describe the interaction of two particles, or two subsystems without using a single, composite wave-function (or density matrix).

 

[DrChinese]

But where does it say a single wave function can be applied to two separate photons interacting with two separate polarizers?

Ah, but it does just that! Check out (1) and (3) at the following excellent reference:

http://arxiv.org/abs/quant-ph/0205171

It is actually called the EPR state.

 

[DrChinese]

Although QM violates local causality, it is consistent with the wider notion of relativistic causality.

Which is wider and which is narrower? :) I can't tell anymore!

 

[Alien8]

Quantum mechanics describes any collection of particles by a single wave function (or, more generally, a density matrix). There is no way to describe the interaction of two particles, or two subsystems without using a single, composite wave-function (or density matrix).

As far I know wave function is shared only between two interacting entities, like electron - proton interaction, it doesn't say what other electrons might be doing with some other protons. Wave function can be collective as an average, say a light beam interacting with a polarizer, but that again doesn't say what some other light beam is supposed to be doing with some other polarizer.

Is there any other example where a single wave function is applied to two separate systems and two pairs of interacting entities instead of a single system and two interacting entities?

 

[stevendaryl]

Quantum mechanics always uses a single wave function to describe all particles and subsystems of interest. If the subsystems don't interact very strongly, it is possible to get a good approximation in some circumstances by only analyzing the subsystems separately, but that's always only a matter of convenience and making the analysis simpler.

 

[atyy]

Which is wider and which is narrower? :) I can't tell anymore!

The notion of relativistic causality (no signalling) is wider than local causality (local determinism or Bell nonlocality). The idea was that although quantum mechanics is nonlocal, it is still surprisingly consistent with special relativity. So people began to wonder whether QM is the maximal amount of nonlocality that is permitted by relativity. The surprising answer was that relativity is consistent with even more nonlocality than QM.

This isn't the peer-reviewed version of Popescu and Rohrlich's paper, which doesn't seem to be on the arXiv, but it sketches the idea: http://arxiv.org/abs/quant-ph/9508009.

There's also a schematic in Fig. 2 of the Brunner et al review: http://arxiv.org/abs/1303.2849.

 

[atyy]

You are correct that "realism" is not mentioned. This definitely follows Norsen's reasoning. I am surprised to see Cavalcanti in the list of authors, as he had recently written about "local realism" in the same vein as I. So you may be correct that the tide has changed.

Actually, even Norsen himself argues in his paper that a particular notion of 'realism' is required for Bell's theorem; that is, the notion of "metaphysical realism" or the existence of an external world “out there” whose existence and identity is independent of anyone’s awareness:

http://arxiv.org/pdf/quant-ph/0607057v2.pdf

I think there is actually hardly any disagreement, mostly just a change in language. There seem to be two major meanings of "realism".

The first is what Norsen calls "metaphysical realism". This is needed for a Bell test in the sense that one must agree that results at spacelike separation are real. So this meaning of "realism" is a prerequisite for local determinism, and in this sense "local realism" is redundant. Apart from Norsen, I found agreement also in http://arxiv.org/abs/0706.2661 (footnote 16), http://arxiv.org/abs/0911.3814 (p12). There's a similar idea in http://arxiv.org/abs/quant-ph/0509061 (p11).

The second is what DrChinese is calling "counterfactual definiteness". I think everyone also agrees that the class of theories that pass a Bell test can be completely generated from local deterministic theories, so that proving the inequality for a counterfactual definite theory proves it for the entire class. The only reason one might not like this terminology is that there isn't enough consensus on what "counterfactual definiteness" means to agree on whether the local stochastic theories that pass a Bell test are also "counterfactual definite". Nonetheless, it is agreed that the local deterministic theories are key to defining this class, since the "local polytope" and whether a Bell inequality is tight or not all depend on local deterministic theories. An example of the local polytope generated by local deterministic theories is drawn in Fig. 1 of http://arxiv.org/abs/1405.7321. Then the only controversy is whether one wants to consider as "real" the local deterministic theories that can in principle underlie a local stochastic theory, which is why Gill writes uses language like "in a mathematical sense" and "or at least may be constructed" when describing realism in http://arxiv.org/abs/1207.5103. So if we define "realism" as "may be constructed from local deterministic theories", Norsen would like to be able to say that this realism may or may not be real. Is that's a sweet concession from a realist? :)

 

[stevendaryl]

I don't like any of these characterizations of "realism". I think it's perfectly clear what kind of theory that Bell's inequality applies to, and I don't see how the words "realism" or "counterfactual definiteness" help in the characterization.

It seems to me that the kind of theory that Bell's inequality applies to is a theory where the state of the universe is completely defined by the state of every small region making up the universe, and the evolution of the state of the universe is completely determined by the evolution of the individual regions.The evolution of each region depends only on the state of that region, and neighboring regions. In a locally realistic theory, the result of measurement in a particular region is simply revealing a fact about the state of that region.

What it means that the evolution of one region cannot depend on distant regions is that they evolve independently. If region $R_1$ can make a transition from state $S_1$ to state$S_1'$, (under certain assumptions about neighboring regions) and region $R_2$ can make a transition from state $S_2$ to state $S_2'$ (under certain assumptions about neighboring regions), then they can both make those transitions, provide the assumptions about neighboring regions hold. In contrast, entanglement such as in the EPR involves a case where something is possible for Alice (Alice measures spin-up along some axis), and something is possible for Bob (Bob measures spin-up along that axis), but the combination is not possible (they both measure spin-up along that axis).

Classical mechanics is a theory of this type. So is a cellular automaton model of the universe (as t'Hooft describes in some recent paper). Classical probability theory is not a theory of this type, but it can be understood as a subjective theory based ignorance of the true state of the universe, where the true state of the universe is described by classical mechanics. Quantum mechanics is not a theory of this type, and cannot be interpreted as a subjective theory based on ignorance of the true state (or true dynamics), if the true state and dynamics are described by a theory of this type.

Determinism and realism and counterfactual definiteness seem too fuzzy for characterizing the theories that Bell's inequalities rule out.

 

[DrChinese]

1. I think there is actually hardly any disagreement...

2. The second is what DrChinese is calling "counterfactual definiteness". I think everyone also agrees that the class of theories that pass a Bell test can be completely generated from local deterministic theories, so that proving the inequality for a counterfactual definite theory proves it for the entire class. The only reason one might not like this terminology is that there isn't enough consensus on what "counterfactual definiteness" means to agree on whether the local stochastic theories that pass a Bell test are also "counterfactual definite".

I agree with the first :) , but am confused by the second. There are no local deterministic theories, stochastic or otherwise, that can pass a Bell test - that I know of anyway. All have been refuted and shown to be non-local or otherwise flawed (I am thinking of the various models by Marshall and Santos as examples).

 

[DrChinese]

I don't like any of these characterizations of "realism". I think it's perfectly clear what kind of theory that Bell's inequality applies to, and I don't see how the words "realism" or "counterfactual definiteness" help in the characterization.
...
Determinism and realism and counterfactual definiteness seem too fuzzy for characterizing the theories that Bell's inequalities rule out.

And yet we have to label it as something, if nothing else so we can deny its existence. :) :) :)

 

[atyy]

I agree with the first :) , but am confused by the second. There are no local deterministic theories, stochastic or otherwise, that can pass a Bell test - that I know of anyway. All have been refuted and shown to be non-local or otherwise flawed (I am thinking of the various models by Marshall and Santos as examples).

By "pass a Bell test" I mean it backwards from the usual meaning, ie. to use the test to detect local determinism, ie. not violate the inequality. So all local deterministic theories like the classical Maxwell's equations will "pass a Bell test".

 

[DrChinese]

By "pass a Bell test" I mean it backwards from the usual meaning, ie. to use the test to detect local determinism, ie. not violate the inequality. So all local deterministic theories like the classical Maxwell's equations will "pass a Bell test".

Ah, good, I wondered given the context. :)

 

[bohm2]

Yeah, it all started with uncertainty and non-reality, but somehow ended up with non-locality. What's the connection?

I've read a few papers suggesting that the non-local implications of QM was what bothered Einstein from the get-go. Einstein's 'telepathy' comments is used by Maudlin to argue this point:

“It seems hard to sneak a look at God’s cards, but that he plays dice and uses “telepathic” methods is something I cannot believe for a moment.”

See recent Maudlin paper/slides on this argument:

Note the second part of Einstein’s concern: not merely that God plays dice but that he “uses ‘telepathic’ methods”. This is, of course, the “spukhafte Fernwirkung” (“spooky action-at-a-distance”) that Einstein is also known to have railed against. A careful reading of Einstein makes clear that it is the spooky action-at-a-distance, i.e. the non-locality, implicit in the standard account of quantum theory that bothered him, not the indeterminism per se. Einstein did not look for a deterministic underpinning of quantum mechanical predictions because he was wedded to determinism, he did so because he was wedded to locality, and he was the first to recognize that in quantum theory indeterminism can further imply non-locality.

http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/Maudlin_Sesto_2014.pdf
http://arxiv.org/ftp/arxiv/papers/1408/1408.1826.pdf

 

[Alien8]

I've read a few papers suggesting that the non-local implications of QM was what bothered Einstein from the get-go. Einstein's 'telepathy' comments is used by Maudlin to argue this point:

There was no any non-locality experiments or Bell inequalities known at that point in time, and the term "entanglement" had not even been invented yet. What could Einstein possibly be referring to?

See recent Maudlin paper/slides on this argument:
http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/files/Maudlin_Sesto_2014.pdf
http://arxiv.org/ftp/arxiv/papers/1408/1408.1826.pdf

I'm looking at EPR paper itself:
http://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777

I see in the first part they are talking about one single particle and two of its properties (location & momentum), operators of which either commute or not. The conclusion is that if the operators corresponding to two physical quantities do not commute then the precise knowledge of one of them precludes such a knowledge of the other, whatever is that supposed to mean.

In the second part they repeat the same thing only this time it's about interaction between two particles. I don't see any part of it is referring to anything like "entangled state", to anything like "interaction over distance", or to any interaction between more than two particles/systems, like this:

1. local interaction: particle_A <-> particle_B

Bell test experiments on the other hand go like this:

1. local interaction: particle_A <--> polarizer_A
2. non-local interaction: particle_A <-/-> particle_B
3. local interaction: particle_B <--> polarizer_B

EPR paper is about quantum uncertainty and existence of definite reality. Bell tests and inequalities are about non-locality. One is about local uncertainty, the other is about non-local certainty. Those look like two very different concepts to me, and I don't see any way uncertainty can justify non-locality, or any connection between them at all.By the way, what do they mean in EPR paper when they say: "if the operators corresponding to two physical quantities do not commute then the precise knowledge of one of them precludes such a knowledge of the other"? How about tennis balls, do their position and momentum commute or not? What would be some other examples of commuting and non-commuting properties?

 

[Nugatory]

There was no any non-locality experiments or Bell inequalities known at that point in time, and the term "entanglement" had not even been invented yet. What could Einstein possibly be referring to

Although the term was not yet in widespread use, the phenomenon it describes (that in some quantum systems, the result of a measurement in one location can be correlated with the results of measurements in other locations) had been well known for more than a decade. So even if he didn't use the word, he was talking about entanglement.

In the second part they repeat the same thing only this time it's about interaction between two particles. I don't see any part of it is referring to anything like "entangled state", to anything like "interaction over distance", or to any interaction between more than two particles/systems, like this:

Any time that we're talking about two particles, we're talking about non-locality - the particles aren't in the exact same place so any correlation between measurements of them is either the result of some non-local effect or the non-local result of their local common origin (as is the case with Bertelmann's socks).

By the way, what do they mean in EPR paper when they say: "if the operators corresponding to two physical quantities do not commute then the precise knowledge of one of them precludes such a knowledge of the other"? How about tennis balls, do their position and momentum commute or not? What would be some other examples of commuting and non-commuting properties?

Some other non-commuting observables: The polarization of a photon measured on one angle will not commute with the polarization measured along any other angle; the spin of a particle measured on one angle will not commute with the spin measured on any other angle.

And no, the position and momentum of a tennis ball do not commute. It is impossible in principle to put a tennis balls into a state such that I can predict the exact value of a position measurement AND the exact value of a momentum measurement. For a macroscopic object like a tennis ball, the uncertainty is negligibly small, but if you could find sufficiently precise measuring instruments you would find it.

 

[DrChinese]

There was no any non-locality experiments or Bell inequalities known at that point in time, and the term "entanglement" had not even been invented yet. What could Einstein possibly be referring to?
I'm looking at EPR paper itself:
http://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777

I see in the first part they are talking about one single particle and two of its properties (location & momentum), operators of which either commute or not. The conclusion is that if the operators corresponding to two physical quantities do not commute then the precise knowledge of one of them precludes such a knowledge of the other, whatever is that supposed to mean.

In the second part they repeat the same thing only this time it's about interaction between two particles. I don't see any part of it is referring to anything like "entangled state", to anything like "interaction over distance", or to any interaction between more than two particles/systems, like this:

1. local interaction: particle_A <-> particle_B

Bell test experiments on the other hand go like this:

1. local interaction: particle_A <--> polarizer_A
2. non-local interaction: particle_A <-/-> particle_B
3. local interaction: particle_B <--> polarizer_B

EPR paper is about quantum uncertainty and existence of definite reality. Bell tests and inequalities are about non-locality. Those look like two very different concepts to me, and I don't see any way uncertainty can justify non-locality, or any connection between them at all.

By the way, what do they mean in EPR paper when they say: "if the operators corresponding to two physical quantities do not commute then the precise knowledge of one of them precludes such a knowledge of the other"? How about tennis balls, do their position and momentum commute or not? What would be some other examples of commuting and non-commuting properties?

OK, there are a lot of issues with what you are saying above.

EPR is about entanglement, although that word itself is not mentioned as entanglement was a very new concept at that time. The word was coined that year I believe. Instead, the particles are allowed to interact in a manner that has a more classical meaning (with the assumption that the exact mechanism could be filled in later).

Re "if the operators corresponding to two physical quantities do not commute then the precise knowledge of one of them precludes such a knowledge of the other": this is simply a restatement of the Heisenberg Uncertainty Principle (HUP). While it may not seem to be relevant to a discussion of entanglement, it is. If the HUP did NOT apply even across a spatially separated entangled system, then you COULD beat (violate) the limits of the HUP with entangled particles. That does not happen though. The position and momentum of a tennis ball absolutely do not commute, and there is uncertainty in that as a result. It is *very* small, however.

You are absolutely correct that EPR is about "definite reality". In the language of the day, that was more clear than it is today. EPR argued there MUST be a deeper reality. They did NOT seriously consider that there was spooky action at a distant but did mention it as a possibility.

 

[atyy]

There was no any non-locality experiments or Bell inequalities known at that point in time, and the term "entanglement" had not even been invented yet. What could Einstein possibly be referring to?

Entanglement is related to superpositions of product states. The idea of superpositions of product states comes very early in quantum mechanics as formulated by Heisenberg and Schroedinger. Superpositions of product states are needed to deal with two particles like the electrons in a helium atoms. This is why Schroedinger was able to formulate the idea of remote steering using entanglement, which is definitely a nonlocal idea, and EPR were able to consider action at a distance. Of course, putting these ideas into a tight framework only came much later. Now we know that not all entangled states are nonlocal in the sense of Bell, so entanglement is something that only partially overlaps with Bell's nonlocality. (However, we do know that all *pure* entangled states can violate a Bell inequality.)

 

[Alien8]

Any time that we're talking about two particles, we're talking about non-locality - the particles aren't in the exact same place so any correlation between measurements of them is either the result of some non-local effect or the non-local result of their local common origin (as is the case with Bertelmann's socks).

EPR paper is talking about direct, close proximity interaction, like collision between photon and polarizer or electron-proton interaction in hydrogen atom. The distances between entities in these "collision" sort of like direct interactions are within reach and power of electric and magnetic fields, therefore they are local interactions. Do you see any paragraph in EPR paper is referring to to any interaction between two entities which at more than atomic distance apart?

Some other non-commuting observables: The polarization of a photon measured on one angle will not commute with the polarization measured along any other angle; the spin of a particle measured on one angle will not commute with the spin measured on any other angle.

Usually it's arithmetic operators like multiplication and addition which are commutative. I still don't get what does it mean for physical properties to commute or not. Does it have anything to do with cause and effect? Anything to do with time at all? Can you define "commutation" of physical properties in general terms and plain language?

 

[atyy]

Usually it's arithmetic operators like multiplication and addition which are commutative. I still don't get what does it mean for physical properties to commute or not. Does it have anything to do with cause and effect? Anything to do with time at all? Can you define "commutation" of physical properties in general terms and plain language?

This is a very important point: non-commutation in QM means we cannot say that position and momentum are definite simultaneous properties of a particle. This is why a quantum particle differs from a classical particle - it does not have simultaneous position and momentum. However, there is no puzzle if we consider "position" and "momentum" to label the outcomes of different experiments. In QM, position and momentum of a particle at a particular time are measured with different apparatuses, and there is no reason for different experiments to give the same results.

 

[Alien8]

You are absolutely correct that EPR is about "definite reality". In the language of the day, that was more clear than it is today. EPR argued there MUST be a deeper reality. They did NOT seriously consider that there was spooky action at a distant but did mention it as a possibility.

I see they are saying the two properties somehow must exist simultaneously, and since their wave function says they don't, EPR conclude the wave function is not their complete description. Admittedly I don't quite get how they arrive to their premise or what are they trying to say with their conclusion, I'm still quite sure I don't see anything that would fit the description of "spooky action at a distance".

 

[DrChinese]

Anything to do with time at all? Can you define "commutation" of physical properties in general terms and plain language?

Yes and yes.

Time is involved because of ordering. In plain language: putting on your shoes before you put on your socks yields a different outcome than putting on your socks before you put on your shoes.

 

[DrChinese]

I see they are saying the two properties somehow must exist simultaneously, and since their wave function says they don't, EPR conclude the wave function is not their complete description. Admittedly I don't quite get how they arrive to their premise or what are they trying to say with their conclusion, I'm still quite sure I don't see anything that would fit the description of "spooky action at a distance".

If their assumptions were accurate, they would have had a really strong argument. But their assumptions were always suspect.

EPR did not seriously consider FTL influences to be a factor. They assume there is "an absence of an interaction between the two systems."

 

[Alien8]

Time is involved because of ordering. In plain language: putting on your shoes before you put on your socks yields a different outcome than putting on your socks before you put on your shoes.

Ok, let's try to boil down the whole EPR paper to a simple two points argument, premise & conclusion, in the most simple and practical terms possible. I'll try to pick out the key points and try to rephrase them as plainly and as directly as I can...

If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

By example: - acceleration due to gravity can be predicted with certainty, therefore gravity field must actually exist at all times regardless of whether anyone is measuring it or if no one is even looking at it? Is that the whole point, is that the whole premise?

(1) the quantum mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding .to two physical quantities do not commute the two quantifies cannot have simultaneous reality. For if both of them had simultaneous reality -- and thus definite values -- these values would enter into the complete description, according to the condition of completeness. If then the wave function provided such a complete description of reality, it would contain these values; these would then be predictable. This not being the case, we are left with the alternatives stated.

So basically the premise is that a description can be complete only if it can make predictions with 100% certainty. And because the wave function doesn't make such deterministic predictions, it therefore can not be a complete description. Is that it? Did I misinterpret anything or missed anything important?

That's kind of incoherent relation between two rather ambiguous concepts, one doesn't quite follow from the other. What am I missing? For example, what would they say about predicting the state of a fair coin, having maximum certainty of only 50% for either heads or tails, would that mean (according to EPR trio) the existence of its two sides is actual and defined at all times, or only when someone is looking, or what?

 

[wle]

I see they are saying the two properties somehow must exist simultaneously, and since their wave function says they don't, EPR conclude the wave function is not their complete description. Admittedly I don't quite get how they arrive to their premise or what are they trying to say with their conclusion, I'm still quite sure I don't see anything that would fit the description of "spooky action at a distance".

The EPR paper starts by introducing what the authors consider a minimal criterion of reality, which is basically that if you could predict with certainty in advance that a measurement will produce a certain outcome, then there is an "element of reality" associated with that outcome. So for instance if a particle happens to be in an exact position eigenstate $\vert x \rangle$, then EPR would say that, at least in that case, the particle has a real position $X = x$ because you know with certainty in advance that you will find the particle located at $x$ if you measure its position.

Starting on the third page, EPR consider a situation in which two spatially separated systems, I and II, which shouldn't be able to interact after they are separated, each contain a particle with both in a momentum-entangled state of the form $$\lvert \Psi \rangle \propto \int \mathrm{d}p \, e^{i p x_{0}} \, \lvert p \rangle_{\mathrm{I}} \, \lvert -p \rangle_{\mathrm{II}} \,.$$ (This is Eq. (9) in the paper, just expressed in the Dirac bra-ket notation and with unimportant constants removed.) This is a situation that is in principle allowed in quantum mechanics. For this initial state, if you measure the momentum of particle I and find that it has momentum $p$, then according to quantum mechanics the state of particle II is projected onto the momentum eigenstate $\lvert -p \rangle_{\mathrm{II}}$. In this case, EPR would argue that system II has a real momentum $P = -p$, since you could predict with certainty that the result of a momentum measurement will be $-p$.

By a change of basis, the initial entangled state can also be expressed in the position basis as $$\lvert \Psi \rangle \propto \int \mathrm{d}x \, \lvert x \rangle_{\mathrm{I}} \, \lvert x + x_{0} \rangle_{\mathrm{II}} \,.$$ This is exactly the same state written just above, just expressed in the position basis instead of the momentum basis. So if you measured and found that the position of particle I is $x$, then particle II gets projected onto the position eigenstate $\lvert x + x_{0} \rangle_{\mathrm{II}}$. In this case, similar to before, EPR would say that particle II has the real position $X = x + x_{0}$.

From this, in the final paragraphs, EPR consider two possible conclusions based on this example:

1. Particle II must have both a real position and a real momentum simultaneously, in which case the quantum mechanical account is incomplete, or
2. What is real in system II can depend on whether a position or momentum measurement is performed on system I.

Point 1 is the conclusion obviously favoured by EPR. Point 2 is the "spooky action at a distance" that EPR considered completely implausible ("No reasonable definition of reality could be expected to permit this").

 

[DrChinese]

So basically the premise is that a description can be complete only if it can make predictions with 100% certainty. And because the wave function doesn't make such deterministic predictions, it therefore can not be a complete description. Is that it? Did I misinterpret anything or missed anything important?

wle says it well in his summary. Their conclusion would be correct if their assumptions were correct. They are not: "No reasonable definition of reality could be expected to permit this." QM does lead one to an unreasonable definition of reality, that being: "ndeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted." In fact, Bohr and others would insist that, as that is a simple consequence of the HUP.

 

[bohm2]

There was no any non-locality experiments or Bell inequalities known at that point in time, and the term "entanglement" had not even been invented yet. What could Einstein possibly be referring to?

Schrödinger was the first to use the term entanglement at about the time (1935) the EPR paper was written:

Schrödinger coined the term ‘entanglement’ to describe this peculiar connection between quantum systems (Schrödinger, 1935; p. 555):

"When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled."

He added (Schrödinger, 1935; p. 555):

"Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being ‘best possibly known,’ i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself. Attention has recently been called to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it."

http://plato.stanford.edu/entries/qt-entangle/#1

 

[bohm2]

Ok, let's try to boil down the whole EPR paper to a simple two points argument, premise & conclusion, in the most simple and practical terms possible...

You can kinda summarize the EPR in 2 sentences:
1. Either QM is incomplete or if it's complete, it must be nonlocal.
2. Nonlocality is unreasonable, therefore it is incomplete.

 

[Alien8]

From this, in the final paragraphs, EPR consider two possible conclusions based on this example:

1. Particle II must have both a real position and a real momentum simultaneously, in which case the quantum mechanical account is incomplete, or
2. What is real in system II can depend on whether a position or momentum measurement is performed on system I.

Point 1 is the conclusion obviously favoured by EPR. Point 2 is the "spooky action at a distance" that EPR considered completely implausible ("No reasonable definition of reality could be expected to permit this").

I see some faint allusion to it, more like an afterthought than a serious consideration. They seem to be saying they have a straight forward local explanation (1), and explanation (2) is just "far out" to even bother saying anything about it, I guess. They speak of some "condition of completeness", but to me it sounds no more than a sort of "lack of necessity" argument.

Starting on the third page, EPR consider a situation in which two spatially separated systems, I and II, which shouldn't be able to interact after they are separated, each contain a particle with both in a momentum-entangled state of the form $$\lvert \Psi \rangle \propto \int \mathrm{d}p \, e^{i p x_{0}} \, \lvert p \rangle_{\mathrm{I}} \, \lvert -p \rangle_{\mathrm{II}} \,.$$ (This is Eq. (9) in the paper, just expressed in the Dirac bra-ket notation and with unimportant constants removed.) This is a situation that is in principle allowed in quantum mechanics. For this initial state, if you measure the momentum of particle I and find that it has momentum $p$, then according to quantum mechanics the state of particle II is projected onto the momentum eigenstate $\lvert -p \rangle_{\mathrm{II}}$. In this case, EPR would argue that system II has a real momentum $P = -p$, since you could predict with certainty that the result of a momentum measurement will be $-p$.

By a change of basis, the initial entangled state can also be expressed in the position basis as $$\lvert \Psi \rangle \propto \int \mathrm{d}x \, \lvert x \rangle_{\mathrm{I}} \, \lvert x + x_{0} \rangle_{\mathrm{II}} \,.$$ This is exactly the same state written just above, just expressed in the position basis instead of the momentum basis. So if you measured and found that the position of particle I is $x$, then particle II gets projected onto the position eigenstate $\lvert x + x_{0} \rangle_{\mathrm{II}}$. In this case, similar to before, EPR would say that particle II has the real position $X = x + x_{0}$.

Ok. So before Bell came along, what was the excuse to even begin entertaining the idea EPR argument in both cases would not actually hold true?

 

[DrChinese]

Ok. So before Bell came along, what was the excuse to even begin entertaining the idea EPR argument in both cases would not actually hold true?

QM was working well. So opinion split, some thinking QM was "complete" (really as complete as it gets) and others (such as EPR) thinking QM was a stopgap. This debate continued for decades.

 

[Nugatory]

I
Ok. So before Bell came along, what was the excuse to even begin entertaining the idea EPR argument in both cases would not actually hold true?

There was none, but there also was not a lot of interest in the question because no one could imagine any experiment in which it might make a difference. Bohr and Einstein agreed to disagree and the physics community figured that if Einstein was right we'd know when and if the more complete theory was discovered and until then QM was the only game in town. It pretty much stayed that way for the next three decades until Bell came along.

Bell's crucial contribution was to show that there was a way of settling the question by experiment. The most important words in his paper are "The example considered above has the advantage that it requires little imagination to envisage the experiments involved actually being made", and that's what the experimentalists jumped on.

Einstein died shortly before Bell discovered his theorem, and for me one of the most tantalizing unanswerable questions is what Einstein would have done with Bell's theorem if he had been around to see it.

 

[Nugatory]

Ok. So before Bell came along, what was the excuse to even begin entertaining the idea EPR argument in both cases would not actually hold true?

If you can possibly get hold of https://www.amazon.com/dp/1400095263/?tag=pfamazon01-20, do so. It's written for your level of understanding of the underlying physics and it covers the history far more completely than any internet forum thread ever will.