Murray Gell-Mann on Entanglement

[atyy]

But, isn't this whole disagreement about 'collapse' and what it means all a bit academic? There are no predicted incorrect experimental consequences (as far as I'm aware) from adopting a traditional 'collapse' picture - just as there are no predicted incorrect experimental consequences from rejecting this view and doing things without explicitly thinking of collapse in this fashion.

Yes, that's why it's a conceptual error to reject collapse on grounds of locality. Collapse is consistent with locality.

 

[vanhees71]

Ad #140:

Note that only Clive knows that particles 1 and 4 are entangled (it's called "entanglement swapping"). This makes the scheme consistent with relativistic QFT (implying the linked-cluster theorem).

I do not use the word collapse for the very reason you see in this thread. NB: The interactions in relativistic QFT are never instantaneous, which includes the interactions of the measured system with the measurement device. To the contrary, all you can say are transition probability rates (S-matrix elements), which describe the transition from an asymptotic free initial to an asymptotic free final state. What to make of the transient state, the theory is silent. So I disagree very strongly with any statement telling a physical process connected with measurement is instantaneous. The collapse assumption usually implies such instantaneous transitions, but that's the very point where this assumption contradicts the very foundations of the theory (relativistic QFT) that it pretends to interpret. Why should I adapt an inconsistent interpretation that I don't need.

 

[vanhees71]

Yes, that's why it's a conceptual error to reject collapse on grounds of locality. Collapse is consistent with locality.

But if there is no need for the collapse assumption (which indeed is inconsistent with the locality of interactions between the measured system and the measurement apparatus!) I better don't use it, because it provides all these problems. An interpretation should be as minimal and consistent as possible, although it's only the "soft part" of the theory!

 

[Demystifier]

But if there is no need for the collapse assumption (which indeed is inconsistent with the locality of interactions between the measured system and the measurement apparatus!) I better don't use it

But you use the update, which is what atyy really means by "collapse".

So you and atyy agree on physics and on interpretation. You only disagree on the language. Atyy refuses to call it "update", and you refuse to translate his "collapse" as "update".

If fights over correct interpretation are not much relevant to physics, fights over correct language are even less relevant to physics.

 

[Simon Phoenix]

This makes the scheme consistent with relativistic QFT

As it must be :-)

The question I suppose I was really getting at is whether we view entanglement as a physical property of things or not.

In the 'collapse' picture we would say that Clive's measurement projects (instantaneously) particles 1 and 4 into an entangled state.

But you would say, I think, that this is inconsistent with QFT (and quite reasonably so too).

The issue I have is that if we're going to view entanglement as having at least some objective physical character - then when, in QFT, would we say that particles 1 and 4 actually become entangled?

Of course, Alice and Bob don't know this until Clive tells them - so no violation of locality etc. But are we justified in saying that the particles are in an entangled state - or is this a meaningless question? Do the particles become 'entangled' only when Alice and Bob receive the supplementary information, or is entanglement an objective physical property?

I don't think there's any great sin in thinking in terms of 'collapse' in the traditional way - after all no experimental contradictions will ensue - but I accept that this is a somewhat uncomfortable perspective for an explicitly local theory. I tend not to worry too much about interpretations (except on forums where it's great to get the ideas of others) and just view 'collapse' (i.e. projective measurements) as a convenient tool.

 

[vanhees71]

But you use the update, which is what atyy really means by "collapse".

So you and atyy agree on physics and on interpretation. You only disagree on the language. Atyy refuses to call it "update", and you refuse to translate his "collapse" as "update".

If fights over correct interpretation are not much relevant to physics, fights over correct language are even less relevant to physics.

Here I disagree again. Correct language is mandatory for all science. You cannot do science without a concise expression of the meaning of words.

 

[ddd123]

The prejudice I refer to is not the common cause principle, it is the idea that the common cause principle has anything to do with parts of a system. A common cause can just as easily be holistic, so rejection of parts, and the locations of parts, and the influences between parts, is 100% independent of a common cause principle. The preparation of the Bell state can simply be the common cause.

Anyway there is a line of research that shows a different link between entanglement and spatial issues: http://www.nature.com/news/the-quantum-source-of-space-time-1.18797 .

Unfortunately I don't understand a single word of it.

 

[Demystifier]

Here I disagree again. Correct language is mandatory for all science. You cannot do science without a concise expression of the meaning of words.

Fair enough! But do you agree with me that the dispute between you and atyy is not about physics, and not about interpretation, but only about language?

 

[vanhees71]

As it must be :-)

The question I suppose I was really getting at is whether we view entanglement as a physical property of things or not.

Sure, since it leads to observable consequences (as the violation of Bell's inequality).

In the 'collapse' picture we would say that Clive's measurement projects (instantaneously) particles 1 and 4 into an entangled state.

But you would say, I think, that this is inconsistent with QFT (and quite reasonably so too).

The issue I have is that if we're going to view entanglement as having at least some objective physical character - then when, in QFT, would we say that particles 1 and 4 actually become entangled?

I say that the projection describes the update of the state by Clive in the sense of a v. Neumann filter measurement. States refer to ensembles, and filtering out part of the ensemble is a preparation procedure. To evaluate what this means for the far-distantly registered particles 2 and 3 you have to use the corresponding measurement protocol. The observers of particles 2 and 3 alone cannot notice that anything changed, because they have no means to apply the filter. For that they need Clive's results. You can choose the corresponding ensemble a posteriori, and you can have things like the quantum eraser of which-way information which restores an interference pattern for such a partial ensemble. Nowhere in all these ensembles is it necessary to assume any instantaneous interaction at a distance, faster-than-light signal propagation (or even retrocausal inferences) and the like esoterics. That's for the science fiction market but not for science!

Of course, Alice and Bob don't know this until Clive tells them - so no violation of locality etc. But are we justified in saying that the particles are in an entangled state - or is this a meaningless question? Do the particles become 'entangled' only when Alice and Bob receive the supplementary information, or is entanglement an objective physical property?

Sure, the particles are in an entangled state, and usually they are entangled in the very beginning when they are created in a local interaction (like parametric down conversion where a laser em. wave leads to the production of two polarization-entangled photons). Also in your case of entanglement swaping you need the two entangled particle pairs, and then the correlations are there due to this preparation procedure and also the entangledment of particles (1,4) for the corresponding partial ensemble chosen due to Clive's measurements are finally caused by the initial entanglement of the two entangled pairs you begin with. Thus indeed you can have strong correlations (stronger than any correlations possible within any deterministic local hidden-variable model) of far distant parts of a quantum system without violating relativistic causality.

I don't think there's any great sin in thinking in terms of 'collapse' in the traditional way - after all no experimental contradictions will ensue - but I accept that this is a somewhat uncomfortable perspective for an explicitly local theory. I tend not to worry too much about interpretations (except on forums where it's great to get the ideas of others) and just view 'collapse' (i.e. projective measurements) as a convenient tool.

As long as you take it as heuristic, it's fine.

 

[stevendaryl]

Let's go one step further and suppose that Clive sends the information about his result on to Alice and Bob. When they receive this update - they can now assign a more appropriate state (a specific pure entangled state). So as soon as they receive this information their state assignation has 'collapsed' from mixed to pure - but nothing has changed, only their knowledge.

Well, sort of. A feature of quantum density matrices that complicates figuring out what is only a matter of knowledge is the non-uniqueness of writing a density matrix as a mixed state. The "improper" mixed state that you get by tracing over irrelevant components of a composite pure state is identical to the "proper" mixed state that you get by collapse (when you don't know which state it has collapsed into). So the weird nonlocality of the quantum mechanics of entangled particles has the effect of turning one kind of mixed state into another, where they both have the same density matrix.

So Alice initial computes a density matrix for her subsystem, $\rho$. This can be written in many different ways as a mixed state; for example, it can be written as: $\sum_\alpha p_\alpha |\alpha\rangle \langle \alpha|$ or as $\sum_\beta p_j |j\rangle \langle j|$, where $|\alpha\rangle$ and $|j\rangle$ are two different bases. If Alice later finds out that Clive has performed a measurement collapsing her system to pure state $|k\rangle$ (one of the states of the basis $|j\rangle$), Alice can retroactively view this as an update to her knowledge:

$p_j \Rightarrow \delta^k_j$

If instead, she finds out that Clive has collapsed her state into pure state $|\beta\rangle$ (one of the states of the basis $|\alpha\rangle$, she can also retroactively view this as an update to her knowledge:

$p_\alpha \Rightarrow \delta^\beta_\alpha$

So it seems to me that it is the ambiguity of writing a density matrix as a mixture, together with the existence of two kinds of mixed states (proper and improper), that makes it where Alice can always retroactively view wave function collapse as ordinary updating of knowledge based on new information. She can always retroactively view collapse as an analogy to Bertelmann's Socks (Bell's example of classical "collapse" that doesn't involve any superluminal interactions).

 

[vanhees71]

Fair enough! But do you agree with me that the dispute between you and atyy is not about physics, and not about interpretation, but only about language?

It would be only a discussion about language, if atyy hadn't claimed in the very beginning of this thread that Gell-Mann's view is wrong. I think it's very clear from the mathematics of QED that there is no instantaneous influence on B's photon by A's measurement of hers.

 

[Demystifier]

I think it's very clear from the mathematics of QED that there is no instantaneous influence on B's photon by A's measurement of hers.

Ah, this is a much more serious and subtle issue. I don't think that this is clear from mathematics of QED. See my post #93 on this thread!

 

[Ken G]

Anyway there is a line of research that shows a different link between entanglement and spatial issues: http://www.nature.com/news/the-quantum-source-of-space-time-1.18797 .

Wow, that's really a nice article, thank you. The way I interpret what it is saying is that it takes the AdS/CFT duality (which says that gravity in a 3D "bulk" under some conditions will act like a conformal field theory (without any gravity) on the 2D boundary of the bulk) and extends it to saying that the connectivity of space in the bulk is dual to the entanglement in the CFT on the boundary. If we think of connectivity of space as important for concepts like nonlocality, and if we interpret entanglement as fundamentally an issue about the holism of systems, then the article seems to be suggesting that our discussion about nonlocality vs. holism has an interesting duality as well: nonlocality in the bulk is dual to holism on the boundary. Put differently, the idea in the article is that the space in the bulk is in some sense "due to" entanglement on the boundary, such that space is "made of entanglement", as they like to say. That certainly puts our discussion into an interesting new light-- it would say that nonlocality and holism, though not the same thing any more than gravity and quantum mechanics are the same thing, do share a deep connection in that we get one when we look in the bulk, and we get the other when we look in the boundary.

The way I interpret all this is that the concept of the "parts" of a system is not a fundamental truth of how systems work, rather it is an idealization that works well when the entanglement in the holistic system is decohered enough to get away with treating the system as if it were made of parts. Similarly, the concept of "locality", i.e., that the parts can only enforce correlations in measurements by propagating subluminal signals, is also not a fundamental truth, it is an idealization that works well when, again, the entanglement on the boundary is sufficiently decohered. What all this means is, the limitation of not going faster than c, and the distinctions between spacelike and timelike separations, are not fundamental laws, they are just what happens in the bulk when entanglements on the boundary get decohered.

The key point of all this, in this interpretation, is that we could frame Gell-Mann's objection in the following terms. The reason it is "wrong" to say that entanglement is enforced by influences between the parts of the system is that it reverses the way we should be thinking about it. We should be saying that the idealization that systems are made of parts, and the idealization that those parts have different locations in spacetime, both work when entanglement breaks down. As such, entanglement does not need to be explained, it is the default state of things-- it is the concepts of parts and propagating influences that need to be explained, and those concepts only emerge when entanglement is broken. Perhaps you can think of "parts" and "influences" as the kind of behaviors that emerge when the symmetry associated with entanglement is broken, a symmetry about the unity (or holism) of systems. It really turns entanglement, and holism, on its ear to say that it is their absence that requires understanding, not their presence. So we don't need to explain why all electrons are indistinguishable, that's a symmetry-- we need to explain why we get away with imagining that they are different, and that they occupy different locations in space. It seems that somehow, all the motion we perceive through space might be traced back to changing coherences in the entanglements on the boundary. It's then not the nonlocality of entanglement that we need to understand, it is the origin of locality as that entanglement evolves that we need to understand.

 

[vanhees71]

Ah, this is a much more serious and subtle issue. I don't think that this is clear from mathematics of QED. See my post #93 on this thread!

Sure, what you say in #93 is part of the formalism. The states have probabilistic meaning and only probabilistic meaning (i.e. a statistical meaning for ensembles, i.e., repeated experiments with equally prepared systems). Of you somehow make states "more ontic" in some sense you are in serious trouble concerning causality!

 

[stevendaryl]

It would be only a discussion about language, if atyy hadn't claimed in the very beginning of this thread that Gell-Mann's view is wrong. I think it's very clear from the mathematics of QED that there is no instantaneous influence on B's photon by A's measurement of hers.

If there were a definitive notion of the state of Bob's photon, then you could definitively prove that Alice has no effect on Bob's photon. But quantum mechanics doesn't actually give a definitive notion of the state of a single particle; it only has a notion of the state of the entire system.

 

[Demystifier]

Of you somehow make states "more ontic" in some sense you are in serious trouble concerning causality!

Well, you cannot get something for nothing. If gaining ontology means loosing relativistic causality, I can live with that (provided that I don't contradict any existing experiment). It is certainly possible that standard quantum theory in the MEI form is not the final theory. It is also possible that a better theory has more ontology and less relativistic causality than standard quantum theory.

Or perhaps your main problem is to understand why would one even want more ontology? Well, if MEI is perfect for you, then you can't understand it. Only if you can see an interesting question not answered by MEI, you can start to appreciate other (more ontological) interpretations of quantum theory.

 

[vanhees71]

The state of Bob's photon is given by the reduced state
$$\hat{\rho}_{B}=\mathrm{Tr}_A \hat{\rho}_{AB}$$
with
$$\hat{\rho}_{AB} = \frac{1}{2} (|HV \rangle-|VH \rangle)(\langle HV|-\langle VH |).$$
The partial trace is
$$\hat{\rho}_B=\sum_{i,j,k=HV} |j \rangle \langle ij|\hat{\rho}_{AB}|ik \rangle \langle k|=\frac{1}{2} \hat{1}.$$
It's all very well defined in the formalism, and clearly Bob has unpolarized photons.

 

[stevendaryl]

Sure, what you say in #93 is part of the formalism. The states have probabilistic meaning and only probabilistic meaning (i.e. a statistical meaning for ensembles, i.e., repeated experiments with equally prepared systems). Of you somehow make states "more ontic" in some sense you are in serious trouble concerning causality!

To me, the claim that states only have probabilistic meaning is not completely coherent. You prepare a system in some state. You use some device to measure a property of that system--for simplicity, a boolean-valued property--and the result is either that the dial points to "yes", with probability $P_{yes}$, or it points to "no", with probability $P_{no}$. Say the dial points to "yes". Would you then say that it is in a particular state (namely, one with the dial pointing to "yes")? For macroscopic objects, is definitely not the case that states only have probabilistic meaning. But since a macroscopic object is (presumably) just a complicated arrangement of elementary particles, that would seem to mean that combinations of particles can have states that have more than just probabilistic meaning.

 

[vanhees71]

This alone doesn't determine the state completely. See Ballentine, Quantum Mechanics for a detailed discussion on "state determination".

 

[stevendaryl]

The state of Bob's photon is given by the reduced state
$$\hat{\rho}_{B}=\mathrm{Tr}_A \hat{\rho}_{AB}$$
with
$$\hat{\rho}_{AB} = \frac{1}{2} (|HV \rangle-|VH \rangle)(\langle HV|-\langle VH |).$$
The partial trace is
$$\hat{\rho}_B=\sum_{ij=HV} |j \rangle \langle ij|\hat{\rho}_{AB}|ij \rangle \langle j|=\frac{1}{2} \hat{1}.$$
It's all very well defined in the formalism, and clearly Bob has unpolarized photons.

As I said to Simon, it's complicated to figure out what is subjective and what is objective in QM. Bob can describe his photon using such a density matrix. Later on, but before Bob has measured his photon, he may find out the result of Alice's measurement, and so he would update his density matrix to that of a pure state, $|H\rangle\langle H|$, maybe. It seems that Bob can understand this in two ways:

1. It was always in that pure state, but he didn't know it, and his density matrix reflected his ignorance.
2. It was once in an unpolarized state, but later changed its state to a polarized state.

It seems to me that both possibilities lead to difficulties, and it seems that one or the other must be true. Option 1 is a hidden-variables theory, ruled out by Bell, while option 2 requires a nonlocal interaction between Alice and Bob.

 

[stevendaryl]

This alone doesn't determine the state completely. See Ballentine, Quantum Mechanics for a detailed discussion on "state determination".

Just a gentle reminder: It would be helpful to include in your posts what statement you are responding to.

 

[vanhees71]

As I said to Simon, it's complicated to figure out what is subjective and what is objective in QM. Bob can describe his photon using such a density matrix. Later on, but before Bob has measured his photon, he may find out the result of Alice's measurement, and so he would update his density matrix to that of a pure state, $|H\rangle\langle H|$, maybe. It seems that Bob can understand this in two ways:

1. It was always in that pure state, but he didn't know it, and his density matrix reflected his ignorance.
2. It was once in an unpolarized state, but later changed its state to a polarized state.

It seems to me that both possibilities lead to difficulties, and it seems that one or the other must be true. Option 1 is a hidden-variables theory, ruled out by Bell, while option 2 requires a nonlocal interaction between Alice and Bob.

Bob cannot but he must describe his photon by this statistical operator given that the pair is prepared in the state given above. Further, if Bob can find out A's result, then there is no problem with causality, because then his measurement is time-like separated and in the future lightcone of A's measurement event. Of course, B's photon is for B not in the pure state but in the state of an unpolarized photon before the measurement and before he as either gotten the information about A's outcome or has made his own measurement. Contrary to what you say the assignment of states is objective, it's defined by equivalence classes of preparation procedures.

Finally, 2. is the correct answer, but it doesn't need nonlocal interactions between A and B. B can only take notice of A's measurement result via a maximally luminally propagating signal and thus in this case his measurement is in the future light cone of A's measurement event, i.e., then his update of his knowledge to the pure state is objective and precisely not obtained in an instantaneous or nonlocal interaction.

 

[Demystifier]

The states have probabilistic meaning and only probabilistic meaning

Are you familiar with the PBR theorem?
https://en.wikipedia.org/wiki/PBR_theorem
This theorem seems to prove that state contains something more than mere probability.

 

[stevendaryl]

Bob cannot but he must describe his photon by this statistical operator given that the pair is prepared in the state given above. Further, if Bob can find out A's result, then there is no problem with causality, because then his measurement is time-like separated and in the future lightcone of A's measurement event. Of course, B's photon is for B not in the pure state but in the state of an unpolarized photon before the measurement and before he as either gotten the information about A's outcome or has made his own measurement. Contrary to what you say the assignment of states is objective, it's defined by equivalence classes of preparation procedures.

Once again, could you please quote the relevant statement(s) that you are responding to?

As I said, your point of view seems slightly incoherent to me. You seem to be wanting to say both that

• Bob's photon is objectively in an unpolarized state, initially.
• After receiving information from Alice, he updates its state to a pure state (horizontally polarized, for example).

If the state is updated based on new information, that makes it subjective. That's sort of the definition of subjective probability, that it reflects the (lack of) information of the observer.

 

[vanhees71]

Are you familiar with the PBR theorem? This theorem seems to prove that state contains something more than mere probability.

No. I've found the Nature Physics paper via google. What's the relation to my statement above? It seem that this is at least one of the final possibilities left by the paper, which I have to study first, to make up my mind about it of course. The final words in the paper read:

For these reasons and others, many will continue to view the
quantum state as representing information. One approach is to
take this to be information about possible measurement outcomes,
and not about the objective state of a system 23 . Another is to
construct concrete models of reality wherein one or more of
our assumptions fail.

 

[vanhees71]

Once again, could you please quote the relevant statement(s) that you are responding to?

As I said, your point of view seems slightly incoherent to me. You seem to be wanting to say both that

• Bob's photon is objectively in an unpolarized state, initially.
• After receiving information from Alice, he updates its state to a pure state (horizontally polarized, for example).

If the state is updated based on new information, that makes it subjective. That's sort of the definition of subjective probability, that it reflects the (lack of) information of the observer.

I don't know, how else I can refer to what I'm answering to than quoting the posting, which I did by hitting the reply button.

There is no contradiction. 1. Bob's photon is objectively in an unpolarized state because the state of the two-photon system is objective due to the preparation procedure (e.g., via parametric down conversion). 2. is also clear, because when you get new knowledge you update your probabilities according to this knowledge. Where is the contradiction?

 

[stevendaryl]

This theorem seems to prove that state contains something more than mere probability.

As I said to Vanhees71, I think that the idea that states only have a probabilistic meaning is incoherent. To make sense of quantum probabilities, you have to compile statistics of measurement results. But a measurement result is just a state of an observer, a persistent record, or a measurement device. So macroscopic objects must have a notion of "state" that is nonprobabilistic. Presumably, since macroscopic objects are made of microscopic objects, this means that a nonprobabilistic notion of state must somehow emerge from the microscopic notion of state.

 

[stevendaryl]

There is no contradiction. 1. Bob's photon is objectively in an unpolarized state because the state of the two-photon system is objective due to the preparation procedure (e.g., via parametric down conversion). 2. is also clear, because when you get new knowledge you update your probabilities according to this knowledge. Where is the contradiction?

The contradiction is that the first notion of "state" is objective, while the second notion is subjective.

 

[vanhees71]

The contradiction is that the first notion of "state" is objective, while the second notion is subjective.

All notions of state as I use them are objective. It's given by a preparation procedure. In this case before A's measurement and B's noticing of her result, the state is given due to to the parametric down conversion, and after B's noticing of A's result it's due to that measurement. Here you through away half the ensemble, and this half of the ensemble is described by the corresponding pure state of B's photon.

 

[Demystifier]

It seem that this is at least one of the final possibilities left by the paper

According to the theorem, it is still possible that state is nothing more than a mere probability, provided that even non-entangled spatially separated systems can be statistically dependent. Are you sure that this possibility is more acceptable to you?

Anyway, the PBR theorem is certainly a very serious threat to the minimal ensemble interpretation of QM. So I would highly recommend you to study it and make your own opinion on it.

 

[stevendaryl]

All notions of state as I use them are objective.

Maybe I misunderstood. Bob initially describes his photon as unpolarized. Later, Alice tells him the result of her measurement. Bob afterwards uses the pure state $|H\rangle \langle H|$ to describe his photon's state. The change is a result of Bob acquiring new information, which makes it a subjective change, rather than an objective change. That's the definition of subjective probability--probability that is dependent on the information available to the observer.

"Subjective" doesn't mean "a matter of opinion". It means (in this case) a matter of information.

 

[vanhees71]

Maybe I misunderstood. Bob initially describes his photon as unpolarized. Later, Alice tells him the result of her measurement. Bob afterwards uses the pure state $|H\rangle \langle H|$ to describe his photon's state. The change is a result of Bob acquiring new information, which makes it a subjective change, rather than an objective change. That's the definition of subjective probability--probability that is dependent on the information available to the observer.

"Subjective" doesn't mean "a matter of opinion".

I don't know what you mean by subjective vs. objective here. If a photon is measured to be H-polarized it's polarization state is $|H \rangle \langle H|$, and if A measures V and B gets this information, given that before the two photons were in the polarization singlet state, B objectively knows that his photon is H-polarized, and thus that then his photon's state is described by $|H \rangle \langle H|$.

 

[zonde]

I don't know what you mean by subjective vs. objective here.

Objective is when B gets "click" in H channel of his PBS at particular time window, subjective is when B expects to get "click" in H channel at particular time window.

 

[Simon Phoenix]

I don't think my basic question has been answered (or maybe I didn't understand the answer, which is quite likely )

In the entanglement-swapping scheme we have prepared a state such that (1,2) are maximally entangled and (2,3) are maximally entangled. Given this then we can say for definite that there is initially no entanglement between particles 1 and 4. Now some measurement is performed on particles 2 and 3 (a Bell measurement) and after this measurement particles 1 and 4 are entangled.

If entanglement is a definite objective property then something physical has changed before and after measurement. The measurement part is critical since local unitary transformations on particles (2,3) cannot change the degree of correlation, or entropy of entanglement, between particles 1 and 4 - it requires a non-unitary process.

Another example would be to take a GHZ state of the form |111> + |000> as our initial state (described in the spin-z basis), and give one of the particles to each of Alice, Bob and Clive. Now one party, say Clive, makes a spin-x measurement and the other two particles of Alice and Bob (initially in a correlated but un-entangled state) are now in a maximally entangled state.

So assuming entanglement is a physical property we've (instantaneously?) effected some significant change to the state and the physical properties of 2 particles (non-entangled to entangled) that could be separated by some distance. Furthermore, this change cannot be described by purely unitary processes local to (2,3) - since the degree of entanglement between 1 and 4 is constant under such unitary processes. And similar considerations apply for the GHZ example too.

Maybe it's only me that has a problem with this - but then I don't view QM as applying only to ensembles - I think it works just fine for individual systems too.

 

[stevendaryl]

I don't know what you mean by subjective vs. objective here.

I defined it many times: If the state is updated based on new information available to Bob, then it's subjective. If the state changed from $\frac{1}{2} |H\rangle \langle H| + \frac{1}{2} |V\rangle \langle V|$ to $|H\rangle \langle H|$ through no interaction with the photon, but only by Alice whispering something in Bob's ear, then that means that either:

1. The state is subjective; it reflects Bob's information, or
2. Bob was wrong about the initial state. (This would be the case if Bob initially thought that Alice's pet was a cat, and then Alice informed him that it was a dog. Dog versus cat is an objective difference, and if Bob initially thought it was a cat, then he was mistaken.)

The objective state of an object can only change by something being done to that object. The subjective state can be changed by the observer acquiring new information.