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Jochen,

As for "What does measurement of |0> in the {|>,|->}-basis yield?". Definitely 50/50, but we have to be careful in how our actual experimental setup translates into nice mathematical spaces with basis.

Let's consider measurement of spin along z-axis. Up and down are values, or 0 and 1. Now further measurement in ,- what is it? do you mean x-axis, or y-axis measure? Etc. Or do you mean any measurement on which 0th state will yield 50/50? I bet, the later question is a proper one.

So, let's stick to the way math has been formed in 1920's. We call a setup as 0,1, in which we measure, and we take state. It is a renaming of your thing, but it makes clear that your lab is never in superposition relative to you. In this way, we make sure that math doesn't carry you too far away. And yes, you get 50/50.

There's no entanglement between the beam splitter and the photon, since the beam splitter does not change its state during the interaction. The action of the beam splitter is (denoting the state of the beam splitter by |B>, the photon moving 'up' as in the picture by |U>, and the photon moving to the right by |B>):

[math]|B\rangle\otimes|D\rangle\to|B\rangle\otimes\frac{1}{\sqrt{2}}(|D\rangle i|U\rangle)[/math]

You have entanglement only if you cannot write the state in product form, but in this case, since the beam splitter is in the same state before and after the interaction, and only the photon enters a superposition, this is possible. This is something of an idealization, true, essentially modelling the beam splitter as perfectly rigid so it does not receive any 'kick' from the photon, but it's true to very good accuracy.

But the real problem is that this experiment shows that a superposition is not a state of 'not knowing'; it's observably different from a state of not knowing, and that observation is exactly the presence of interference. If the state of the photon were definite at all times, and we simply don't know which it is, then there would not be any interference. You say as much yourself when you say that this is only possible with a pure state; but if you then say that superposition is just the result of not knowing, then that's the definition of a mixed state---a mixed state is one in which you don't know the actual state.

Likewise with entanglement. If, as you say, you always get 50/50 results if measuring the state |0> in the {|>,|->}-basis (and note that I've explicitly defined what I mean by that basis in a previous post), then if your postulate 3 ensures that an 'entangled' system is always in a state of |00> or |11>, then Alice and Bob will obtain absolutely uncorrelated results upon their {|>,|->}-measurement, in contradiction with quantum mechanics and experiment.

Re-reading your posts, I think the best way I can reconstruct your approach is as essentially a modal interpretation. In such an interpretation, the idea is to have an 'apparent' state of the system, which is given by the usual quantum state, say an arbitrary bipartite state

[math]|\psi\rangle_{AB}=\sum_{ij}c_{ij}|i\rangle|j\rangle[/math]

where the |i> and |j> form an orthonormal basis for the A and B systems, respectively.

However, there also is a physical state, which amounts to just one element of the above superposition, i.e. a state |i>|j> for some specific i and j. Upon measurement, one learns the physical state. This physical state corresponds to your 'confined' one, while the quantum state is what the 'external system' 'sees' before it has interacted with the AB system---this is basically how I understand your postulate 4.

The idea here is that the quantum state gives the possibilities ('modalities', hence the name) for the physical state to be, while the physical state realizes one of those possibilities. The physical state then also is a hidden variable, analogously to the particle position in Bohm theory (which can be seen as a kind of modal interpretation).

Such a proposal needs to be augmented by rules which states are to be physical, or equivalently, which observables can take definite values, which you don't give; this leads to the trouble with interference and entanglement. Several possibilities for this have been proposed in the literature: specifying a preferred factorization of Hilbert space, a preferred decomposition (and hence, basis) of the state vector (given by the Schmidt biorthogonal decomposition theorem), or by specifying a preferred observable R, or rules according to which one can choose such, by appealing to the Clifton-Bub uniqueness theorem for such 'no-collapse' theories. However, without any such specification, you simply can't reach empirical adequacy.

Even with such a specification, modal interpretations are fiendishly hard to make consistent: lots of proposals have fallen prey to Kochen-Specker type contradictions, and others have blatantly counterintuitive properties (such as a composite system having different definite properties from each of its subsystems). There are also issues to be addressed about the evolution of the definite quantities/physical state, etc. A model here is the evolution of the particle positions in Bohm theory, which is determined by the wave function in a very peculiar, but consistent, way; but I think few modal interpretations have been developed to this point.

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Jochen,

On "if your postulate 3 ensures that an 'entangled' system is always in a state of |00> or |11>" yes but it is a state that qubits see! Its like Everett's RELATIVE state argument. The key word here is RELATIVE. Cause qubits see each other as either 0 or 1, but Alice and Bob, being EXTERNAL systems, have a DIFFERENT perspective, and see pair entangled, which shall decohere on them at further interaction.

Do not miss this little detail, of whose perspective you are at. Cause information happens to be local, and moves according to equation 6 (postulate 4). And each player in a quantum game has different info, and different view of things.

Now on your believing that this is a modal-like interpretation. No. I went now to http://plato.stanford.edu/entries/qm-modal/ and it shows that this modal things start from having Hilbert spaces. We start from physical essence, notion of system, and how it is routed in particles, with which CERN people play. Secondly, unitarity is kinda assumed, and not clear why. We let nature tell us about information, or how it moves around, in interaction confinement. And in our conceptually clean approach you see that nothing else is needed. No need to fight mathematical artifacts that are taken onboard wholesome with upfront introduction of Hilbert spaces.

On "This is something of an idealization, true, essentially modelling the beam splitter as perfectly rigid so it does not receive any 'kick' from the photon, but it's true to very good accuracy." Yep. More so, do not forget that all of these are at all effective systems made up of many-many particle events. And we tend to idealise things a little. And Hilbert spaces math is one of those idealisations, very useful for calculations, but it hides the essence which Wheeler wanted to flesh out.

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Leo,

I like the fact that we are able in essay to say that particle events define points. It is Liebnitz' relational view of space and time, as opposed to absolute, or box-like view.

When I said Higgs mechanics, I mean not necessarily exactly higgs bosons, may be, but not necessarily. It should just be something of this sort computationally. I do not want to speculate here, and do not want to cut possible avenue of search, simply because I have no grounds to do so. All I see is this time dilation, like in QZE. And Einstein's own take on gravity was realization that time dilates in gravity field, therefore, curved metric is needed, etc. I am trying to stand on giant's shoulders :) .

Jochen

"Paul, regarding superpositions..."

As per a post on your essay blog, the point is that the nature of physical existence comes first, ie generically, what can it be and how can that occur. If some 'theory' or experimentation then contradicts that, then there is something wrong with those. Indeed it is likely that the flaw lies in not following the one definitive physically existent state at a time rule, failing to differentiate the existential sequence from the existential representation thereof, or not understanding the reference used in timing.

In respect of SR, here is what Einstein (not Reed) defined it as. My point being that SR is irrelevant. The underlying concept of relativity is what is important, and that boils down to (although he did not express it this way because of the mistakes he made): there is a time differential in physical reality.

Einstein 1916 section7

"At this juncture the theory of relativity entered the arena. As a result of an analysis of the physical conceptions of time and space, it became evident that in reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically rigid theory could be arrived at. This theory has been called the special theory of relativity to distinguish it from the extended theory, with which we shall deal later."

Einstein 1916 section 18

"provided that they are in a state of uniform rectilinear and non-rotary motion...The validity of the principle of relativity was assumed only for these reference-bodies, but not for others (e.g. those possessing motion of a different kind). In this sense we speak of the special principle of relativity, or special theory of relativity."

Einstein 1916 section 22

"A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light)."

Einstein 1916 section 28

"The special theory of relativity has reference to Galileian domains, ie to those in which no gravitational field exists. In this connection a Galileian reference-body serves as body of reference, ie a rigid body...In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity."

Einstein 1921 para 11:

"The development of the special theory of relativity consists of two main steps, namely, the adaptation of the space-time "metrics" to Maxwell's electro-dynamics, and an adaptation of the rest of physics to that altered space-time "metrics."

So, special relativity, as defined by Einstein, involves:

-only motion that is uniform rectilinear and non-rotary

-only fixed shape bodies

-only light which travels in straight lines at a constant speed

It is special because there is no gravitational force, or more precisely, no differential in the gravitational forces incurred.

Paul

PS: as per a post on your essay blog, in order to try and resolve this Einstein argument which keeps arising I have put up the first section of a paper. My essay blog, me posting 24/4 04.19. Sorry I do not know how to do links.

But this 'state that qubits see' then either has no effect on interaction with the external systems---then, you don't solve the measurement problem (as the qubits must interact as if they are in superposition). Or, it is the state that is accessed in measurement of the qubits---then, you don't get the proper correlations for entanglement.

In the first case, any external system will attribute the state |00> + |11> to the qubits, and any interaction will proceed as if this is the case; in particular, I can do interference measurements to detect the off-diagonal terms of the density matrix. But then, the only possible conclusion is that according to any external system, there's no fact of the matter regarding that the qubits are 'actually' in the state |00> or |11>, since that wouldn't produce any interference. But then, any regular measurement---being after all an interaction like any other---should leave the measurement apparatus in a state entangled with the qubits, thus failing to record a definite measurement outcome. If you say that now, the measurement apparatus is 'part of' the system and thus 'sees' the true state, then this is just the projection postulate in a different guise, because the system manifestly did not interact with itself before as if it possessed that state (it self-interfered).

In the second case, if the actual state of the system is |00> or |11>, then measurement by Alice and Bob will yield uncorrelated results; because if it is in the state, say, |00>, and thus, if Alice had measured in that basis, she would have determinately received 0, and so would Bob, then it is also in the state |++> - |+-> - |-+> + |--> (the two are just different mathematical representations for the same object), and Alice and Bob's measurements in that basis would be absolutely uncorrelated. In order to get the requisite correlations, you'd have to stipulate that the 'true' state is simultaneously, say, |00> and |++>; but that's mathematically inconsistent.

Added to this are the worries that your postulate 3 explicitly invokes nonunitary evolution, and there does not seem to be a way to derive the Born probabilities.

Jochen,

Please take a look at Everett, Hugh (1957). "Relative State Formulation of Quantum Mechanics". Reviews of Modern Physics 29: 454-462.

There you will find a concept of relative states. When you make interaction between two qubits, they "measure" each other, because any measurement is just an interaction! But you, an external system, will see on experiment these qubits as entangled. Again, take a coffee, or two, make yourself relaxed, and try to absorb Everett's idea about relative state.

We have this concept of relative state here as well, that is why there is no fundamentally special interaction called "measurements". Yet, we go a little further then Everett, with interaction confinement. And since we do not assume Hilbert spaces from the start, we do not have to hide them, like many-worlds need to.

I quote you from 2nd paragragh: "If you say that now, the measurement apparatus is 'part of' the system and thus 'sees' the true state ..." Have I ever pronounced a word combination "true state"? It is anathema in relative states world. Think about it. Information confinement tells us that info is local. So, info distribution will characterize a state of the thing, you have this information about. This maid Wheeler say that information defines state, or that two are born in the union; you know, this whole it-vs-bit charade :) .

Again, do not rush. Look through Everett. Read Wheeler's reference. Absorb this relative state concept. And go throw essay slow. The short length requires you write like Dirac -- every word has precise, measured meaning.

I am thankful to you for this discussion, though. We showed that interaction confinement can be seen as statement about nature of information. Somene can explore this in a PhD thesis alone. I've also developed a coin analogy, and a clear explanation of pure vs mixed state. I am pleased.

I'm quite familiar with Everett's formulation, thank you (if you look at my essay, I cite Saunders' intriguing version, which is in part the inspiration to my concept of relative facts). Everett's relative states are e.g. of the form |"0">|0> |"1">|1>, where the quotation marks indicate for instance a measurement result; hence, the result is "0" relative to the state being |0>, and "1" relative to the state being |1>. In the 'many worlds' reading (which I consider very problematic), this means that in one world, 1 is observed, while in the other, 0 is observed.

The reason this account is possible is essentially the linearity of quantum mechanics: you can now continue the discussion 'as if' for instance only the |"1">|1> term were actual, thus producing the appearance of collapse, since nothing that happens to the |"0">|0> term will introduce any problems; any operator acting on the total state can be considered to act on either term separately. But if the two terms are recombined coherently, the usual quantum behaviour is observed.

Your account, however, is very different. Since upon interaction, two qubits in your picture evolve to a state that's either |00> or |11>, and not to a state in which the state of the second qubit is merely |0> relative to the state of the first one being |0>, and |1> relative to the second one's being |1>, you explicitly break unitary evolution, and the resulting states cannot be recombined coherently. If you intended to extend Everett's proposal, you have extended it too far: your resulting state, while it may be 'relative' in the sense that 'inside' and 'outside' systems consider it to be different, is absolute in the sense that to the two qubits, it's either |00> or |11>. But that's enough to destroy the quantum effects.

Ultimately, it comes really down to this: if the state the two qubits see (where usual quantum mechanics would consider them to be in the entangled state |00> |11>) is |00> or |11>, and measurement in the {|0>,|1>}-basis then reveals this state, thus solving the measurement problem, then by the same logic, measurement in the {|>,|->}-basis must be completely uncorrelated, since as you say, each of Alice and Bob upon measuring their qubit will find either |> or |-> with 50% probability. But this means that your correlations are observably different from those predicted by quantum mechanics and observed in nature.

The purpose of Physics ought to be concentrated on reality. Reality is unique and it is never in need of an explanation or of repeatable experimentation. Anything that does need an explanation and experimentaion cannot be real. The more the need for an explanation, the greater the unreality.

On "In the 'many worlds' reading (which I consider very problematic), this means that in one world, 1 is observed, while in the other, 0 is observed." Bravo! You definitely see the many-world extension of Everett's as splitting possibilities in a global sense. This is how it is usually described. But it is equation 5, not nature's preference 6. More so, in many-worlds unitarity is just postulated, but equation 5 is inconsistent with it. So, there is a conceptual self contradiction in usual many-worlds, unless you start to complicate world-splitting process, saying that it is local, which noone has ever explicitly done. This problem with Deutsch's many-worlds, however, says nothing bad about relative states concept from Everett's original work! And you'll start to see it, once you stop applying common classical world sense of information to quantum. So, get Everett's article, and the one right before or after it in the same journal's volume. The second article was written by Wheeler himself in support of Everett's piece. Notice also, how easy it is to spot a problem with multi-worlds using language of our postulates, while before I also had this uneasiness, which was impossible to articulate.

So, again. You take two qubits, and send them into unitary gate, which makes 0 for 0, and 1 for 1. Qubits see each other as being in 0 or 1 state. There is no superposition from qubit's PERSPECTIVE. The gate was unitary, which means that this choice was NOT LEAKING into lab, into external system. And from lab's PERSPECTIVE, qubits are in a superposition of entangled states, 00+11, which will decohere for lab when Bob/Alice will do their thing. This difference in PERSPECTIVE is not illusionary, but physical, cause it is governed by equation 6, producing real difference, which can be appreciated at least in Deutsch algorithm, where quantum computer takes less steps to get to answer, than is classically possible.

On "Your account, however, is very different. Since upon interaction, two qubits in your picture evolve to a state that's either |00> or |11>, and not to a state in which the state of the second qubit is merely |0> relative to the state of the first one being |0>, and |1> relative to the second one's being |1>, you explicitly break unitary evolution" Equation 3 deals with only two systems. To add another perspective, we add third system, and form equations 5 & 6 as possibilities of how OR in postulate 3 spreads onto other systems. This produces postulate 4. I am trying here again to tell you, that OR there is not your classical OR. That is a point of info being local, not global like with classical OR. That is why postulate 4, when we use language of Hilbert spaces, is equivalent to unitarity.

To sum it up. Brush up on the concept of relative state, where there is NO MEASUREMENT PROBLEM. And go through essay again. And when you put + sign in any of your calculation, ask yourself what it means and to which system. Good luck.

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"You definitely see the many-world extension of Everett's as splitting possibilities in a global sense."---I said I consider that reading to be very problematic...?

So you say that the lab sees the state |00> |11>, while the qubits 'see' the state |00> or |11> (this is more or less what a modal interpretation would hold, but no matter). So let's say that the qubits see |11>. What does now happen during measurement? Is this state what will be observed? In other words, Alice sees 1, and Bob sees 1, too?

Bob and Alice will get the usual thing. Let's be more precise, when we collect Bell's statistics on this, it will such that it comes from entangled pair. Definitely, since we have unitarity! Now, for HOW this happens, and we have mentioned this already, its like asking why speed of light is constant. We do not want to construct another "quantum aether" here, as it will always end up with something, governed by equation 5, which is too limiting for what nature does, i.e. equation 6. We just postulate interaction confinement based on experimental evidence.

I think, this is already a repetition of things said above. Give yourself time for this to sink in.

And another thing, any measurement is information transfer, any interaction is information transfer. When we say measurement, we mean interaction. But to drive this point, I prefer dropping word measurement at all in favour of interaction, carefully saying what sort of information gets transferred and to which system. For example, internal interaction is characterized by info flow that does not cross a boundary of a closed system.

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Well, you can't really go and simply assert that 'Bob and Alice will get the usual thing', you'd have to derive this from your formalism. Otherwise, anybody could just write down a few 'axioms', claim that they derive quantum mechanics, and then respond to all questions with 'well, since the axioms derive quantum mechanics, you'll get everything the same way as in quantum mechanics' (and then go on to further claim that the measurement problem is solved in their approach, as well).

So if the qubits see the state |11>, does Alice as well as Bob get 1 upon measurement?

All interpretations have to get Hilbert spaces and unitarity. You've agreed that unitarity produces quantum effects. So in postulates 3 we say that system behave probabilistically. Reference 9 tells how to built mathematical Hilbert spaces, following exactly along reasoning from 1920's, along experiments and nothing else. Postulate 4 gives unitarity. This is it, checkmate. Collapsing a priory postulated mathematical states (or have special interaction named measurement), dropping non-preferred basis of a priory postulated mathematical spaces is not needed in our formulation.

Essay presents an interpretation that produces suggestions for things outside QM experiments, like time and space. So, even if you personally do not care about presented simplicity and close proximity between experiments and quantum postulates, we have a shot at moving onto nature of time and space here.

Yes, unitary evolution in Hilbert space is indeed responsible for the bulk of quantum effects (though some, such as the quantum Zeno effect, require the nonunitarity of measurement). But when you say that postulate 3 is probabilistic, you say that it breaks unitarity: unitary evolution is always deterministic. So, it's far from clear that your framework reproduces quantum effects.

Well, in any case, I guess I've done all I can here. Maybe some time you sit down and actually try to work through the example of entanglement between Alice and Bob using your framework, which so far you can't seem to do. It'll become plain that then, either postulate 3 solves the measurement problem and results in a definite state for the participants to merely uncover, or the state remains in superposition and you get the necessary correlations for entanglement---but not both. Maybe then you can revisit and improve your proposal.

Jochen

"Otherwise, anybody could just write down a few 'axioms', claim that they derive..., and then respond to all questions with 'well, since the axioms derive...., you'll get everything..."

Very true. But have you ever considered that this might apply to QM itself!! With all its constructs and processes which are contradictory to the way in which physical existence must occur. And then there is the inverted snobbery sales pitch, it is counter intuitive, only the cognoscenti understand.

Mikalai

The nature of time and space is easy to discern.

Distance is an artefact of physically existent entities, it being a difference between them in terms of spatial position. Existence necessitates physical space, but that can only be assigned via entities. As yet, nobody has demonstrated the existence of absolutely nothing in any given spatial position at any given time, just different forms of something.

Time relates to the turnover rate of physically existent states, ie it is not a feature of any given one, but a feature of the difference between them. That is, physical existence is purely spatial, with different states of that occurring over time.

Paul

But Paul, nobody wrote down the axioms of quantum mechanics and then said 'this is the new orthodoxy; anybody who disagrees is branded a heretic'. Rather, quantum mechanics was hard won, often against the explicit resistance of those that would then become its 'founding fathers': it was foisted upon them by simple experimental necessity. The classical conception of the world made predictions that could not be brought into agreement with experimental observation; thus, very gradually, very reluctantly, this conception was revised and replaced by what today we know as 'quantum mechanics'. The best thing about this theory is that it makes predictions, that is, you can rigorously derive sentences of the form 'if quantum mechanics is right, experiment x should yield outcome y' from the axioms; and in *all cases* for almost the past 100 years, these predictions have been brought out by experiment, to an accuracy no physical theory ever before enjoyed.

This, of course, does not prove the correctness of quantum mechanics: it is in principle always possible that further observations may disagree with it. And consequently, efforts are constantly underway to find the next even more accurate theory capable of superseding quantum mechanics. But it is important to realize that this does not undo the agreement already reached, any more than Newtonian gravity being superseded by general relativity means that all of a sudden, I can't calculate the curve of a baseball using the former anymore. So in this sense, there's no going back.

Jochen,

On "unitary evolution is always deterministic". Let there be one closed system, and one external to it. According to interaction confinement (postulate 4), although there might be some processes in a closed system, no information leaks to the external system. Yes, it is very deterministic situation, external system predictably gets ZERO information, because there is no interaction by construction. Let's call it trivially deterministic, so that we do not confuse this term with real determinism of classical mechanics.

But, it is usually argued, that wave function can be nicely written, how it spreads in time, and thus, it is determinism. This will be true IF we assume that the wave function is physical, which we DO NOT in essay. Wave function is a mathematical tool for prediction of future outcomes. Function may vary smoothly, or not, in time, but it is not a physical entity, which is closed, in a sense that no information about it state leaking into outside.

I agree, that one has to square indeterminism with determinism in a situation, were wave functions, or Hilbert spaces are assumed to exist. This reminds me a situation with luminiferous aether, which had to assume seemingly opposite characteristics. This never works out cleanly.