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Han Geurdes: on 9/15/14 at 11:34am UTC, wrote Dear Oscar, Thanks. I really like the connection you make to the ballerina...

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Thomas Ray: on 8/27/14 at 18:07pm UTC, wrote The co-domains are covariant, I mean.

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Blogger Oscar Dahlsten wrote on Aug. 17, 2014 @ 15:28 GMT
Figure 1
When a ballerina does a pirouette she must escape the friction of the ground in order to get the freedom to move. (Figure 1: Photo by Michael Garner, courtesy of English National Ballet.) She does this by restricting her contact with the ground to a point. In a recent paper I and my collaborators Andrew Garner and FQXi's Vlatko Vedral show that quantum theory in a very similar way escapes a fundamental constraint on movement by accepting uncertainty.

Quantum systems are associated with states which encode the statistics of future possible measurements. The collection of such states may be represented as a geometric shape. In the smallest possible quantum systems, single qubits (quantum bits), this shape is a sphere, called the Bloch sphere.

For example, think about a property of a qubit, such as its position: the qubit could be associated with two possible positions, A and B, say, or it can be in a fuzzy superposition where it exists in both of these mutually incompatible states simultaneously, before being observed. If it's in a superposition then although experimenters cannot know with certainty what position they will find it in when they make a measurement, they will have some sense of the probability of getting a certain outcome. The Bloch sphere helps to visualise this odd feature and the probabilistic nature of quantum mechanics. In the example, a vector pointing to the north pole of the sphere could represent position A, while the south pole represents position B. (In a classical system, this would represent the only two options available for a binary digit, or bit, to access). However, a qubit can also be represented by a vector pointing elsewhere on the surface of the sphere, corresponding to the fuzzy in-between states.

Figure 2
The maximal state space conceivable would actually be the cube outside of the sphere, as shown in figure 2. The quantum state space is the sphere, but if there were no uncertainty principle all states in the outer cube could be allowed. In this case certain measurements could all have predictable outcomes at the same time, in violation of the quantum uncertainty principle.

One may ask why quantum theory is restricted to the sphere, and accordingly to having the uncertainty principle.

We came across an intriguing answer when thinking about how the cube state space would handle an interferometer. In an interferometer the particle or photon is firstly placed in a superposition of being in two places and then operations are done on each site. Now when you have two different sites fundamental locality restrictions come into play. In particular, we point out that if a system has 0 probability of being found on site B, then an operation on site B must leave the state of the system invariant. Otherwise we could do action at a distance. Contrary to some popular science depictions, quantum theory does not allow action at a distance. The universe would be almost inconceivably odd and complicated if action at a distance were possible. We would not be able to make a statement about an individual system without taking into account what happens everywhere else in the world.

On the Bloch diagram, state transformations move points around, e.g. by rotating the shape. So, if one accepts that this locality restriction holds, it turns out that operations on site B must leave all points (states) on the lower plane of the cube invariant. It is like the points are stuck by total friction between the shape and the lower plane. As a result the cube has a big disadvantage over the sphere because if the entire square face touching the ground is restricted, then the whole cube gets stuck and no states can change.

But now imagine metamorphosing the cube into a sphere, or indeed something else with only one point on the lower plane, like how the ballerina goes up on one toe. Then the shape, with all the quantum states in it, can move. The quantum sphere has the advantage over the cube that it can rotate even if there is full friction with the lower (and/or upper) plane, just as the ballerina accepts the uncertainty of only having a point in contact with the ground in return for the ability to pirouette.

One may say that uncertainty, rather than being just limiting, liberates quantum states to change.

--

Oscar Dahlsten is affiliated with Oxford University

The paper appears in Nature Communications.

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Joy Christian wrote on Aug. 17, 2014 @ 16:47 GMT
"Contrary to some popular science depictions, quantum theory does not allow action at a distance. The universe would be almost inconceivably odd and complicated if action at a distance were possible. We would not be able to make a statement about an individual system without taking into account what happens everywhere else in the world."

The above statements are grossly misleading, if not outright wrong.

To be sure, one cannot send a signal violating special relativity with the so-called quantum non-locality, but nonetheless quantum theory is not a locally causal theory.

But here is a good news: Quantum theory can be completed (in a manner espoused by Einstein) into a locally causal theory, as done, for example, in this paper and this simulation.

For a more compete discussion, see also this page of my blog, or this discussion of mine.

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Thomas Howard Ray replied on Aug. 17, 2014 @ 17:48 GMT
Joy, I think in a certain technical sense, "action at a distance" -- nonlocal causality -- can be differentiated from nonlocality, without raising the issue of local causality at all.

As is well known to everyone here, I agree with you without reservation that quantum mechanics is not a locally causal theory. Events that are local and causal, however, do not obviate events that are non-local and non-causal (metaphysically real) which quantum theory would conventionally have us believe is locally real by normalization of the metric -- a mathematical kludge based in observation, with no theoretical support.

It is notable that the author invokes the cube (Bloch sphere) in order to instantiate the sphere -- mathematically, this admits linear superposition as foundational, carrying with it, assumptions of quantum entanglement and probabilistic measure schema. These assumptions are sufficient, though not necessary -- and they entirely obviate a geometrical description of quantum foundations, by nothing more than fiat.

OTOH, a foundations model based on the topological generalization of the Euclidean sphere admits no such assumptions and no such measurement schema -- under the simple condition that the topology is simply connected. I think very few understand that your framework is truly analytical, classically-based and completely local realistic.

All best,

Tom

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Blogger Oscar Dahlsten replied on Aug. 19, 2014 @ 14:15 GMT
Hi Joy,

When I say action at a distance I do mean it in the operational sense, i.e. that doing something on site A does not instantaneously lead to changes in probabilities of measurement outcomes on site B. (I have the phrase 'spooky action at a distance' in mind as the popular science description that is dangerous.)

The real vectors here represent the probabilities of outcomes of possible measurements. (They are a generalisation of the quantum density matrix.) The 'freezing' of one plane thus corresponds to the measurement statistics of the states in question being invariant under the given set of operations.

I moreover do think we would have serious problems with predicting anything if there were action at a distance in this operational sense.

Best,

Oscar

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Blogger Oscar Dahlsten replied on Aug. 19, 2014 @ 14:18 GMT
Hi Tom,

Thanks for the post. I don't know Joy's approach but I think I agree with the spirit of what you say.

Best

Oscar

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Steve Agnew wrote on Aug. 17, 2014 @ 17:56 GMT
The good thing about quantum action is that any number of complete or even mostly complete basis sets are solutions to the reality of the Schrödinger equation. The bad thing about quantum action is that any number of complete or even mostly complete basis sets are solutions to the reality of the Schrödinger equation.

Just as the Heisenberg approach is equivalent to Dirac's approach, science can and does argue endlessly about which basis set or approach or interpretation is better. These kinds of arguments are really not that all that useful since action is all about the Schrödinger equation, not really about which basis set you choose or which approach or which interpretation. But the arguments about basis sets go on and on and on...

After all, it is not even necessary to consider action in space as a priori since matter action as exchange is a complete basis for quantum action without space. Locality is simply a convenient and very intuitive representation of phase for the quantum action of matter waves and there are any number of equivalent ways to deal with the issue of locality in quantum action.

The really important goal in all of this is not to to find a better basis set for quantum action, the goal for science is to find a gravity exchange force that scales in a very nice way from that of the quantum action of charge force.

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Thomas Howard Ray replied on Aug. 17, 2014 @ 18:44 GMT
" ... the goal for science is to find a gravity exchange force that scales in a very nice way from that of the quantum action of charge force."

I agree, Steve. And wouldn't that necessarily entail a field theory in a continuum of spacetime over an n-dimension Hilbert space model of probabilistic measure?

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Thomas Howard Ray replied on Aug. 17, 2014 @ 20:47 GMT
Jeez, I should have said " rather than ... an n-dimension Hilbert space ..." instead of "over," which has obvious and unintended mathematical implications.

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Steve Agnew replied on Aug. 17, 2014 @ 22:29 GMT
I would say not necessarily. Science has a great field theory for charge force in spacetime that fails for gravity force. Stringy guys add extra dimensions, quantum loopists add little twirly thingys everywhere, multiversey guys seem to explain anything and nothing at the same time, and arithmetics forgo pdf's and just redefine everything with equations and constants everywhere.

I would say that this arena is definitely a mess and it is no wonder that science has not figured this stuff out with such a cacaphony. I think that you could explain just about anything with an n-dimensional Hilbert space...we could also just assign a constant for n particles in the universe and be done with it.

Some say that models must be first of all falsifiable and that is quite important, but really models must first of all be useful and new models must be much more useful than old models in order to even be considered. Fringe physics necessarily and rightly so has an uphill battle and new models must show utility over all else.

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Robert H McEachern wrote on Aug. 17, 2014 @ 18:01 GMT
"One may ask why quantum theory is restricted to ... having the uncertainty principle."

Quantum Theory has an uncertainty principle, because it employs Fourier Transforms to describe systems. Since the uncertainty principle is a property of a Fourier Transform, it is inevitable that any system, being described via Fourier Transforms, will also exhibit the uncertainty principle. Regardless of whether or not the uncertainty principle is a property of the system itself, it is a property of the chosen means for describing systems; hence it will inevitably exist, within the description of the system.

Rob McEachern

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Christophe Galland replied on Aug. 17, 2014 @ 20:14 GMT
I agree that uncertainty relationships are intrinsic to Fourier transforms, yet in quantum mechanics it seems to be a more fundamental feature independent of the calculation tool.

In the paper, the key property of the operators that is used is not uncertainty per se, but non-commutativity.

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Thomas Howard Ray replied on Aug. 17, 2014 @ 20:38 GMT
" ... the key property of the operators that is used is not uncertainty per se, but non-commutativity."

That's a great point. Non-commutative quantities imply uncertainty in a precise manner. If you don't mind indulging me, the attached work in progress explains why I think so.

attachments: 1_The_CHSH_result_is_free_of_context.pdf

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Robert H McEachern replied on Aug. 17, 2014 @ 23:19 GMT
"yet in quantum mechanics it seems to be a more fundamental feature independent of the calculation tool"

It may indeed "seem" that way, but it is not.

It is fundamental, if and only if, there is a single particle being observed. But as soon as one attempts to observe more than one particle, as in any attempt to observe an "interference" pattern, then it is not fundamental at all.

The quantity that is fundamental to an observation, is the absolute limit of an observation's recoverable information content; one cannot recover less than a single bit of information from an observation, and still claim to have made an "observation".

In the case of a single particle observation, Shannon's Capacity for the amount of recoverable information, equaling one bit, yields the uncertainty principle. But when more than one identical particle can be measured, it is possible to recover more than one bit of information, and hence "violate" the uncertainty principle. But using Fourier Transforms is an inappropriate means for obtaining such results.

Rob McEachern

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Akinbo Ojo wrote on Aug. 17, 2014 @ 18:07 GMT
North and South poles are relative positions. Somewhere, A and Nowhere, B are absolute positions. From cosmology, our universe seems to tell us that both states, A and B can be occupied, viz. 'Before' Big bang (Nowhere), the current epoch with 'Somewhere' increasing in size, and a Big crunch back to Nowhere (equal to non-existence).

I am not a professional Quantum Machinist, but if it can be contemplated on this blog and stated that, "For example, think about a property of a qubit, such as its position: the qubit could be in two different positions, A and B", why must the choice of positions only be relational and not absolute, when a geometric system as big as the Universe, and which itself is a collection of all that exists, including '…the smallest possible quantum systems, single qubits (quantum bits)', can occupy the two absolute positions, A and B, how much more a qubit? Does anything prevent a qubit from disappearing to Nothing (Nowhere) and appearing from Nothing (Somewhere)? Are virtual particles not said to behave similarly?

Akinbo

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Blogger Oscar Dahlsten replied on Aug. 19, 2014 @ 14:42 GMT
Hi Akinbo and thanks for the post.

I think you touch on some very profound issues. I did not understand all your questions, but I note that for the paper in question we describe the experiments, and the two positions, from the perspective of some given observer who is the same every time the experiment is ran. But it may be very interesting to see what one can get out of applying consistency conditions between different observers, that they should all agree on the measurement statistics (objective reality) even if they disagree on what to call different positions involved.

Best

Oscar

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Akinbo Ojo replied on Aug. 21, 2014 @ 09:40 GMT
Thanks Oscar for your reply. Not just 'profound'. In my opinion, foundational in keeping with the original motivation of FQXi, not the current dabbling into politics and social science.

If you do not understand some of the questions, I in turn can understand your non-comprehension. It appears 'bizarre', but till it can be falsified logically or experimentally it must be on the table,...

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Thomas Howard Ray wrote on Aug. 17, 2014 @ 23:20 GMT
To make a point that mathematically astute readers already know, that I nevertheless think deserves singling out:

Comparing the Bloch model to a framework of generalized Euclidean spheres (known as the science of topology) -- be reminded that regardless of the "cubifying" of the Bloch sphere, it is still in Euclidean terms, a 2-sphere, i.e., a three dimensional object. The faces of the...

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Jonathan J. Dickau replied on Aug. 18, 2014 @ 12:23 GMT
Regarding 'cubifying' the sphere...

Some of this is sorted out by David M. Keirsey here:

Section 2.1.3 on 'complicating' measures and also the following section (2.1.4) on how we define measure, number, and space.

Keirsey has been working independently on developing information-theoretic measures, and this work would probably be of interest. It is notable here that this brings up a point I've made several times already; while in hypercubic measures, which I think refers to Hilbert spaces as well, the assumption is that volume increases with n, the number of dimensions, with a hyperspherical measure there is a volume and area peak.

It would appear that the rule pertaining to Quantum Mechanics must be the one for higher-dimensional spheres, rather than cubes, and that the difference is often overlooked or ignored for the sake of linearization. As Keirsey explains, the Ricci flow maps out the areas and extent of disagreement or frustration, between the two measures - round and flat. But what Dahlsten writes about above is intuitively obvious to me. My thought is that this work by Dahlsten, Garner, and Vedral is only an advancement because people are so strongly invested in a sort of cubism in the Math of Physics.

All the Best,

Jonathan

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Thomas Howard Ray replied on Aug. 18, 2014 @ 15:36 GMT
Thanks, Jonathan -- this should spark some important discussion. As an aside, I hadn't realized that this Keirsey is the son of the late David W. Keirsey of "Temperament Sorter" fame. Having been curious for a while, I took the test and (not surprisingly) came up INTP, my career as a technical writer being one of the top 10 career choices of the type. Impressive.

Back to the mathematics, after digesting.

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Blogger Oscar Dahlsten replied on Aug. 19, 2014 @ 14:54 GMT
Hi Tom and Jonathan,

Just to emphasise that these shapes correspond to different measurement statistics being allowed. Consider for simplicity systems that have three measurements that are incompatible in the sense that by assumption we cannot measure them simultaneously. We may label these, in analogy with the quantum Pauli operators, X, Y and Z and take their outcome to be +1 or -1. The state vector is then represented as [E(X), E(Y) ,E(Z)] where E(.) is the average of (.). (we choose that arbitrary ordering of the letters for certain reasons). Now quantum theory says that (for pure states) E(X)^2+E(Y)^2+E(Z)^2=1. If you think about it that gives you a sphere as the state space, and moreover the operational meaning is that of an uncertainty relation, because if E(X)=1 say, X is predictable, but then E(Z)=E(Y)=0 which means +1 or -1 are equally likely for those, they are uniformly random. Knowing X means Y and Z are unknown.

However the cube allows for e.g. [1, 1, 1], they can all be predictable at the same time. It does not respect the uncertainty principle.

Another point you may find interesting is that one can get higher dimensional spheres in other theories, notably quaternionic quantum theory where the analogue of the qubit is a 5-dimensional sphere.

Best wishes

Oscar

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John Brodix Merryman wrote on Aug. 19, 2014 @ 11:39 GMT
As a sort of meta-observation, might it be worth considering the dichotomy of information and energy, in that because information is static, we assume it must have some Platonic permanence, yet when dynamic conditions are being described statically, the potential information goes to infinity. Now our neural functions are designed to extract and process information, so this creates an infinite feedback loop, but if we want to put it in context, then we need to acknowledge that dynamical basis.

Writing this on a phone, so pardon run on sentences.

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John Brodix Merryman replied on Aug. 19, 2014 @ 12:21 GMT
PS,

While this might not be the focus of particular qm debates, it very much goes to the relationship of order and chaos/complexity and from there to explaining various political dynamics and the breakdown of civil and social order occurring around the world, which will eventually affect even academia.

That being the inherent expansion of energy and consolidation/contraction of order. Which would also explain why these debates invariably go to very focused points of conflict.

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Blogger Oscar Dahlsten replied on Aug. 19, 2014 @ 14:59 GMT
Hi John,

Ah, defining energy and information, this is worthy of another post and debate at least.

Best

Oscar

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John Brodix Merryman replied on Aug. 19, 2014 @ 18:41 GMT
Thanks Oscar,

Given the fact energy transmits information and, conversely, information defines energy, it is surprising the relationship doesn't elicit more discussion

As biological organisms, we have evolved a central nervous system to process information and the digestive, respiratory and circulatory systems to process energy. While there is an obvious intellectual bias toward consideration of information, while it seems energy gets dismissed as "undefined," it is evident that the physical properties of energy very much set the limits of what can be done with information.

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Vladimir Rogozhin wrote on Aug. 20, 2014 @ 07:52 GMT
Hi Oscar,

"The crisis of representation and interpretation" (T.Romanovskaya) in quantum mechanics - the crisis of the philosophical foundations of the QM and all fundamental physics. It is true, the way to overcome the crisis - is a further deepening of the Geometry, but rather in the "origin of Geometry" (E.Husserl) and the dialectical- ontological unification of matter, search for the absolute foundations of physics and knowledge, the absolute generating structure. Necessary to consider limiting (absolute, unconditional) state of matter: absolute motion (rotation, "vortex", discretuum) + absolute rest (linear state, continuum)) + absolute becoming (absolute wave -"figaro" of states = discretuum + continuum).

Then geometrized basis of QM: "sphere" + "cube" + "cylinder". Each limit (absolute, unconditional) state of its way - the absolute vector, the vector of the absolute state. This "triangle" of absolute states of matter - the ontological representation of the triune foundation - "origin of geometry", the beginning of physics, the beginning, framework and carcas of knowledge. This is what David Gross calls - "general framework structure" (D.Gross, an interview "Iz chego sostoit prostranstvo-vremya/What is in the space-time) the same for the QM and for GM. Today QM and GM are parametrical theories without ontologic justification.

All best,

Vladimir

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Thomas Howard Ray wrote on Aug. 27, 2014 @ 15:33 GMT
This thread is too important to let wither and die.

I think Oscar's parting statement is quite profound:

"One may say that uncertainty, rather than being just limiting, liberates quantum states to change."

The subtraction of a few words, and the addition of one other, however, changes the meaning from uncertain to determinate:

One may say that uncertainty, being...

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Thomas Howard Ray replied on Aug. 27, 2014 @ 18:07 GMT
The co-domains are covariant, I mean.

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Vladimir Rogozhin wrote on Aug. 28, 2014 @ 10:39 GMT
Overcoming the "crisis of understanding" in fundamental physics is only possible through the introduction of standard ontological justifications addition to the empirical standard. Requires broadening and deepening of the limits of knowledge through a deeper ontological interpretation and representation of of the experimental data for the formation of the ontological framework, carcas and foundation of knowledge. "Quantum" is not ontologically grounded concept - this parametric concept. Accordingly, the "quantum theory" - a theory is not ontologically grounded, it is not fundamental in the full sense of the word.

The information revolution requires the introduction of the conceptual structure of the Universum the new ontological concepts that represent the deepest meanings of the "LifeWorld". Otherwise, the "uncertainty" "will hold by a throat" fundamental knowledge, first of all physics and cosmology.. This is indicated by the results of the fifth Contest FQXi. The winner of the fourth contest FQXi Robert Spekkens concluded: «Rest in peace kinematics and dynamics. Long live causal structure! . But where is it - «causal structure» with the ontological justification? I offered "The Absolute generating structure". But where open competition of alternative, ontologically and empirically based "general framework structure" of the Universum and knowledge that will enable to overcome the ontological uncertainty ?

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Han Geurdes wrote on Sep. 15, 2014 @ 11:34 GMT
Dear Oscar,

Thanks. I really like the connection you make to the ballerina and quantum non-locality indeterminacy. You apparently seem to refer to a "Swan lake" type of dance and for instance not to "Le Sacre du Printemps". I might be mistaken there too. Modern dance isn't exactly similar to a Tsjaikovski ballet. But ok.

My question to you relates back to an arXiv paper of mine http://arxiv.org/abs/1409.0740 or\and (much more elaborated) to my loophole CHSH paper Results in Physics 4, (2014), 81-82. I appologize for this self-referring approach, however, I see no other way.

If we have Swan lake escapes from determinism then why can one arrive at the quantum correlation from Bell's correlation formula (maybe that formula is far too primitive to describe what is needed ?) or why is it then possible to see that CHSH holds a probability loophole. The arXiv paper shows a numerical argument for the derivation of a.b = cos(phi_a-phi_b) from Bell's formula.

I think it is time to realize that lhv and go-away-lhv are perhaps both wrong. The lhv camp served to show the way however.

It is concrete mathematical incompleteness that is bothering us here. Discussions in our quarters often degenerate to ordinary exchanges of nothing between, most of the time, adversaries with a hearing problem. Some of the adversaries also show a kind of "Gilles de la Tourette" type of response to criticism on a favourite theorem. I am- and probably Joy also is- sorry for all the inconvenience.

For your convenience to kill me off instantly, I attach the RinP paper (although it is very densly written one can verify the argument easily).

attachments: 1-s2.0-S2211379714000254-main.pdf

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