To me, the most simple way to think about Bell inequalities is as (hyper-)planes delimiting the set of convex combinations of value-assignments of possible measurements. That is, for e. g. the CHSH-setup, you have four measurements, and hence sixteen value assignments from (0,0,0,0) (or -1, which I use in the paper) to (1,1,1,1). Then, the general state of the system is a 16-dimensional vector of unit 1-norm, i. e. a probability distribution yielding the probability of finding each of the sixteen possible value assignments. The states which have only one entry equal to 1, and the rest equal to 0, then form the vertices of a convex polytope; this convex polytope can equally well be described in terms of its facets, which are the Bell inequalities of this setting.

Given this, I think how Bell inequalities are violated in my setting becomes readily apparent: if all Bell inequalities are obeyed, then you can construct a description in terms of the above, as a convex mixture of fixed value assignments. But the diagonal argument shows precisely that you can't make such an assignment. Hence, in some cases at least, it follows that we can't formulate a description of the system in the above terms; but then, in these cases, some Bell inequality must be violated.

Of course, this doesn't get me anywhere near deriving the Tsirelson bound. Non-computability lurks there, too, as was just recently shown (https://arxiv.org/abs/2001.04383).

As for counterfactual definiteness, I think a strength of my approach is that it gives a straightforward explanation where and when it is applicable---namely, only when reasoning about values explicitly provided by my f(n,k). We can talk counterfactually about the value of the spin (in some particular direction) of a distant particle, reasoning that it would have been the same even had we made a different local measurement, only if there is a definite value provided by the maximum information attainable about the system; but if, for example, that information is instead taken up by yielding a definite value for the correlation between two observables, then such talk becomes meaningless.

So if our knowledge about the system is given by (x-spin 1 is up, x-spin 2 is down), we can consider that x-spin 2 would have been down, even if we had made a different measurement on 1; but if it's instead given by (x-spin 1 is up, x-spin 2 is opposite that of 1), then the fact that the x-spin of 1 is some particular way is a necessary prerequisite for being able to reason about x-spin 2---a prerequisite that we loose if we imagine that we had made some other measurement on 1.

Jochen,

Your chart which I copy below, though it looks clumsy, is a sort of polytope. This is a tesseract in 4-dimensions. The z_A and z_B states form a bipartite entanglement. The same occurs for the states x_A and x_B, where the states z and x are in superposition quantum mechanically. We may then think of some additional spin state that for x_A and x_B these are replaced with the states Z_A and Z_B, say as some needle state. This means the bipartite system is replaced with a qu4-nit or quadpartite entanglement. This 4-entangled system then has the states (±1, ±1, ±1, ±1). A form of quantum teleportation may accomplish this.

With the identification of states that have Hamming distance 1 do not contribute to superposition and for this and with the separable states (-1, -1, -1, -1) and (1, 1, 1, 1) we have the Kirwan polytope. The Kirwan convexity theorem proves that the momentum map has image that is a convex set. This convex set or Kirwan polytope intersects the positive Weyl chamber. I have some more on this in my essay. For the 4-tangle case the states (1011), (1101), (1110) and (0111), where the remainder define with (0000) and (1111) the Kirwan polytope. This has 12 vertices and 12 faces. This has a relationship to the 24-cell of the F4 group.

Of course I have used the idea of quantum teleportation, but we might however drop the idea of this being a quantum teleportation and simply a classical replacement. The result will have a 4-tangle obstruction similar to the 3-tangle obstruction. In that case the cube defines the set of tripartite states and the Hamming 1-distance states removed (110), (101) and (0,1,1) define a double tetrahedron. The 12-cell above is constructed from 4 of these.

Jochen,

Your chart which I copy below, though it looks clumsy, is a sort of polytope. This is a tesseract in 4-dimensions. The z_A and z_B states form a bipartite entanglement. The same occurs for the states x_A and x_B, where the states z and x are in superposition quantum mechanically. We may then think of some additional spin state that for x_A and x_B these are replaced with the states Z_A and Z_B, say as some needle state. This means the bipartite system is replaced with a qu4-it or quadpartite entanglement. This 4-entangled system then has the states (±1, ±1, ±1, ±1). A form of quantum teleportation may accomplish this.

With the identification of states that have Hamming distance 1 do not contribute to superposition and for this and with the separable states (-1, -1, -1, -1) and (1, 1, 1, 1) we have the Kirwan polytope. The Kirwan convexity theorem proves that the momentum map has image that is a convex set. This convex set or Kirwan polytope intersects the positive Weyl chamber. I have some more on this in my essay. For the 4-tangle case the states (1011), (1101), (1110) and (0111), where the remainder define with (0000) and (1111) the Kirwan polytope. This has 12 vertices and 12 faces. This has a relationship to the 24-cell of the F4 group.

Of course, I have used the idea of quantum teleportation, but we might however drop the idea of this being a quantum teleportation and simply a classical replacement. The result will have a 4-tangle obstruction similar to the 3-tangle obstruction. In that case the cube defines the set of tripartite states and the Hamming 1-distance states removed (110), (101) and (0,1,1) define a double tetrahedron. The 12-cell above is constructed from 4 of these.

That this has some relationship to the 24-cell and the F4 group means this argument is similar to the Kochen-Specker theorem for 4-dimensions. So this insight should work.

Cheers LC

State xA zA xB zB P(λi)

λ1 1 1 1 1 p1

λ2 1 1 1 -1 p2

λ3 1 1 -1 1 p3

λ4 1 1 -1 -1 p4

λ5 1 -1 1 1 p5

λ6 1 -1 1 -1 p6

λ7 1 -1 -1 1 p7

λ8 1 -1 -1 -1 p8

λ9 -1 1 1 1 p9

λ10 -1 1 1 -1 p10

λ11 -1 1 -1 1 p11

λ12 -1 1 -1 -1 p12

λ13 -1 -1 1 1 p13

λ14 -1 -1 1 -1 p14

λ15 -1 -1 -1 1 p15

λ16 -1 -1 -1 -1 p16

Jochen,

First off, due to a copy paste error the above does not work well. So here is a better formatted post.

Your chart which I copy below, though it looks clumsy, is a sort of polytope. This is a tesseract in 4-dimensions. The z_A and z_B states form a bipartite entanglement. The same occurs for the states x_A and x_B, where the states z and x are in superposition quantum mechanically. We may then think of some additional spin state that for x_A and x_B these are replaced with the states Z_A and Z_B, say as some needle state. This means the bipartite system is replaced with a qu4-it or quadpartite entanglement. This 4-entangled system then has the states (±1, ±1, ±1, ±1). A form of quantum teleportation may accomplish this.

With the identification of states that have Hamming distance 1 do not contribute to superposition and for this and with the separable states (-1, -1, -1, -1) and (1, 1, 1, 1) we have the Kirwan polytope. The Kirwan convexity theorem proves that the momentum map has image that is a convex set. This convex set or Kirwan polytope intersects the positive Weyl chamber. I have some more on this in my essay. For the 4-tangle case the states (1011), (1101), (1110) and (0111), where the remainder define with (0000) and (1111) the Kirwan polytope. This has 12 vertices and 12 faces. This has a relationship to the 24-cell of the F4 group.

Of course, I have used the idea of quantum teleportation, but we might however drop the idea of this being a quantum teleportation and simply a classical replacement. The result will have a 4-tangle obstruction similar to the 3-tangle obstruction. In that case the cube defines the set of tripartite states and the Hamming 1-distance states removed (110), (101) and (0,1,1) define a double tetrahedron. The 12-cell above is constructed from 4 of these.

That this has some relationship to the 24-cell and the F4 group means this argument is similar to the Kochen-Specker theorem for 4-dimensions. So this insight should work.

Cheers LC

State xA zA xB zB P(λi)

λ1 1 1 1 1 p1

λ2 1 1 1 -1 p2

λ3 1 1 -1 1 p3

λ4 1 1 -1 -1 p4

λ5 1 -1 1 1 p5

λ6 1 -1 1 -1 p6

λ7 1 -1 -1 1 p7

λ8 1 -1 -1 -1 p8

λ9 -1 1 1 1 p9

λ10 -1 1 1 -1 p10

λ11 -1 1 -1 1 p11

λ12 -1 1 -1 -1 p12

λ13 -1 -1 1 1 p13

λ14 -1 -1 1 -1 p14

λ15 -1 -1 -1 1 p15

λ16 -1 -1 -1 -1 p16

9 days later

Hello,

One of my favorites, we recognise a general relevant knowledge about the subjets analysed and extrapolated. I have learnt in the same time several things that I didn t know. Congratulations

ps about the infinity, I beleive that we must rank them and consider this bridge separating this physicality , finite in evolution and this infinity beyond this physicality, a thing that we cannot define and the sciences Community is divided about its philosophical interpretations. I consider personally in my model of spherisation, an infinite eternal consciousness. I beleive returning about this infinity and the infinities and finite systems, that we must rank them, we have a finite universe in logic made of finite systems , coded and we see too this infinity appearing with our numbers and others like pi or the golden number ... and we have this infinity beyond this physicality and this eternity even if we go deeper in philosophy. How must we rank and consider these infinities inside this physicality, it is the real question in fact....

    I have shared it on Facebook with the essay of Tim Palmer too, I beleive that your essays merit it, regards

    Dear Steve,

    Thanks for your kind words. I'm glad you found something useful for you in my essay!

    Infinity is, as many have surmised, a thorny concept, replete with paradoxes. Going back to the Greek philosophers, I believe Anaximander was the first to seriously engage with the notion of 'the infinite' (or apeiron, perhaps more accurately 'the unbounded'). You might think it's a bit of a dodge, but it's not easy at all to get to the notion of infinity from necessarily finite observations.

    Of course, since Cantor, we know that there isn't just one infinity, but ranks of them---his argument, of course, being the original form of the argument I present in my essay. We bump into the limits of 'Undecidability, Uncomputability and Unpredictability' precisely because we can't transgress infinitey---to a machine capable of executing infinitely many steps in finite time, the halting problem would be decidable. We're stuck at the bottom of the arithmetical hierarchy, and thus, all of our reasoning is necessarily bounded.

    Thanks again for your interest!

    Dear Lawrence,

    sorry for taking so long to respond. Unfortunately, my time is limited at the moment, and your posts always take some careful picking apart for me to understand.

    That said, I'm either not quite there yet, or we're not talking about quite the same thing (although perhaps you're suggesting an alternative interpretation of my setting). The CHSH-polytope I reference is a classical entity, living in the probability space spanned by the value-assignments to the observables in the CHSH-experiment (i. e. the hidden variable vectors, and their convex combinations, which just give the probability distributions over experimental outcomes).

    The Kirwan polytope lives in a space of (eigenvalues of) quantum states, and contains information about how a given system is entangled. Are you saying that there exists an entanglement polytope that's the same as the CHSH-polytope? If so, I'm afraid that's not quite clear to me. I mean, I can see that the four-qubit entanglement polytope must be contained in it, but that's trivially the case, because it's just the unit (hyper-)cube. Do you think there's more of a connection than that?

    7 days later

    It has been a while since I checked FQXi. I am a little disappointed in how the essay contest is developing.

    The CHSH polytope is based on the relationship

    I_{chsh} = A_1Г--B_1 + A_1Г--B_2 + A_2Г--B_1 - A_2Г--B_2,

    for Alice and Bob experiments with two outcomes. This curiously is a type of metric that can be interpreted as pseudo-Euclidean. This is also a measure of entropy, for it may be expressed according to conditional probabilities. An arbitrary two-qubit state after Schmidt decomposition can always be written as

    |П€_nвџ© = c_0|n_+, n_+вџ© + c_1|n_в€', n_в€'вџ©.

    We choose the measurement settings in the following way

    A_1 = m_1В·Пѓ, A_2 = m_2В·Пѓ,

    B_1 = (1/в€љ2)(m_1В·Пѓ + m_2В·Пѓ), B_2 = (1/в€љ2)(m_1В·Пѓ в€' m_2В·Пѓ).

    Here n, m_1 and m_2 are the unit vectors perpendicular to each other. Now find the expectation value of the CHSH operator in the state |П€_nвџ©. We get

    вџЁП€_n|I_{chsh}|П€_nвџ© = 2в€љ2C.

    The expectation of I then has this bound.

    This CHSH polytope is I think related to the Kirwan polytope. The CSHS comes from the relationship with different basis measurements, while the Kirwan polytope is based on eigenstates. For x and z measurements we can think of there being two copies of the Kirwan polytope. The CHSH is then a discrete lattice for some form of covering space.

    Cheers LC

    Dear Jochen, Very interesting paper. To understand Assumption 1 I had to get rid of the idea that measurements are usually repeated under identical conditions; in this case the measurement may change with n and this is necessary to derive the paradox. It may be helpful to think in terms of settings (as in the EPR-Bell-Bohm situation), so that the choice of a measurement is a choice of the settings. This introduces a tacit Free Choice assumption into the argument. The contradiction is reminiscent of, perhaps even equivalent to, the so-called paradox of predictability, see e.g. the review by Rummens and Cuypers, Determinism and the Paradox of Predictability, Erkenntnis 72, 233-249 (2010), https://link.springer.com/article/10.1007/s10670-009-9199-1. I must admit that I find inferences or suggestions of the kind that an undecidable proposition can be modeled by a quantum superposition suspicious - the former are very general, the latter arise is a very specific mathematical context (Hilbert space) and one needs additional arguments to really make the inference. As far as I know, no one has managed to do this convincingly.

    Having said this, I will continue follow your work with great interest. Best wishes, Klaas

      Dear Klaas,

      thank you for your comment! I'm glad you found something of interest in my essay.

      Regarding the importance of free choice, indeed you can make an argument that one might be constrained to make only measurements for which f(n,k) yields a definite value, thus never running into the sorts of phenomena following from its indefiniteness. That we do, apparently, run into them would then be evidence that there is no such constraint (which of course doesn't entail that we have free choice).

      However, one does not need to think about changing measurement conditions to derive the paradox, merely about the totality of all possible measurements on the system---whether they are ever performed or not. The index n is then essentially just an attempt to enumerate these measurements, with the argument then showing that no such enumeration can be complete.

      As for superposition, you're right to point out that the structure of linear operators on Hilbert space is a quite specific one; but in the end, the project of deriving a theory from underlying principles is one to derive the specific from the general---for comparison, the structure of Lorentz transformations on Minkowski space is also quite specific, while following from the very general principle of relativity, together with the constancy of the speed of light.

      Indeed, if one views Hilbert space as a concrete realization of an abstract propositional structure---the orthomodular lattice of its subspaces---then one can show that this essentially follows from the notion that there exists a maximum amount of information that can be extracted from any system (https://link.springer.com/article/10.1007/s10702-005-1129-0). This is both connected to undecidability (as in Chaitin's principle, you can't derive---under a suitable measure of complexity---a theorem more complex than the set of axioms), and superposition, with the failure of the distributive law in quantum logic.

      For simple (that is, not subject to Gödelian phenomena) axiom systems, this correspondence was demonstrated by Brukner (https://link.springer.com/article/10.1007/s11047-009-9118-z) and Paterek et al (https://iopscience.iop.org/article/10.1088/1367-2630/12/1/013019/meta), who show an explicit way to encode axioms in a quantum system and demonstrate that a given measurement will produce random outcomes whenever the corresponding proposition is not derivable from the axioms. In a sense, my work is simply an extension of this to cases where undecidability is not due to the limitations of the axiom system, but to the inherent limitations imposed by the limitative theorems of metamathematics (although in the treatment using Lawvere's theorem, one can pass over first establishing a correspondence with formal axiomatic systems).

      Having said that, I of course don't claim to have a complete reconstruction of quantum theory in hand. There are still different options possible---for instance, it's not easy to see why one should use Hilbert spaces over the complex field, and not over the reals or quaternions. In that sense, perhaps one should think of the connections I point out, as of yet, as 'family resemblances', rather than strict formal equivalences. I view them as enticing prospects that seem sufficiently promising for me to carry on exploring this point of view; but I would not be the first wanderer to be deceived by tantalizing lights in the dark forest.

      Cheers

      Jochen

      Dear Jochen,

      You wrote a really excellent Essay, have my sincere congrats. You chose a very hot topic by discussing it from an original point of view. Your approach of reconstructing quantum mechanics is in a certain sense similar to my attempt of reconstructing quantum gravity through its fundamental bricks, that are black holes. In addition, from the philosophical point of view my position is near local realism too. You deserves my highest score, I wish you good luck in the contest.

      Cheers, Ch.

        You are welcome, I agree fully , it is difficult to understand these infinities inside this physicality that we observe and try to understand. We search a kind of universal partition with these numbers and these foundamental mathematical and physical objects. I consider the 3d coded spheres and a gravitational coded aether sent from this central cosmological sphere, it is there that these finite series of spheres are coded by a kind a infinite eternal consciousness that we cannot define, we can just understand this physicality and its laws. The reals, irrationals, rationals, imaginaries, primes, p adics analyses , harmonics of fourier and this and that seem under a specific universal partition but we know so few still, I consider that these 3D quantum spheres of this aether play between the zero absolute and the planck temperature and they have codes permitting the geonmetries, topologies and properties of matters in this space time, I have considered the Ricci flow, the lie derivatives, the poincare conjecture, the topological and euclidian spaces, the lie groups, the heat equations and other mathematical Tools , and I have invented with a person the assymetric Ricci flow, that permits to create the unique things and all Shapes , I try to find the good mathematics for this formalisation and the good partition, but it is not easy.I have quantize with this logic the quantum gravitation, I have just considered different distances like if our actual standard model was just emergent due to codes farer .

        Best Regards

        Dear Jochen,

        I really enjoyed reading your essay. I particularly liked your clean-cut presentation of the principles of finiteness and extensibility. You might like to have a look at my essay wherein I outline finiteness as a program to (re)construct an alternative, indeterministic classical physics (a program that we are developing with Nicolas Gisin). It would be nice to find an analogous (but of course not completely identical) feature of extensibility in indeterministic classical physics. We can maybe discuss this.

        Meanwhile, congratulations again, top rate so far!

        Flavio

          Dear Christian,

          thanks for your kind words. I agree that out approaches seem to share the same spirit, of trying to find the 'lynchpin' issues from which everything else may (hopefully) unfold, and come to be explicable. Sometimes, a complicated problem may resist forceful attempts to crack it, while crumbling under a gentle tap at the right place. Let's hope we've found the right place to tap!

          Cheers

          Jochen

          Dear Flavio,

          thanks for your kind words! I'm aware of your work with Nicolas Gisin, although I haven't yet had the time to study it thoroughly. I'll take your essay as an opportunity to rectify this; if there is a connection between our approaches, maybe we can narrow in on what, precisely, it is that separates the classical and the quantum. I'll get back to you after I've had a look at your essay.

          Cheers

          Jochen

          You can see easily that I consider coded particles like causes of our reality. I don t consider strings and 1D main Cosmic filds creating these geonetries, topologies, matters and properties, nor a geometrodynamics. I beleive that the strings are a fashion philosophically speaking like if all was fields, Waves, oscillations, I prefer to consider particles coded in a gravitational aether where the space dispappears and is coded if I can say. I doubt that this universe is an enormous heat and that we have only photons like primordial essence and after vibrations, oscillations creating this physicality. I beleive that the strings and thsi GR have created a kind of prison for the thinkers, but it is just my opinion of course, we have for me a deeper logic to all this puzzle.

          Hi, All this is very interesting about the polytopes, and the plays of maths. We search after all what are the foundamentals of this universe. The polytops can converge, but for this we must be sure about their properties and if they are foundamental, we know that we have many different polytopes , like the Lie Groups also and this E8 for example,or the infinite polytopes, the abstracts ones or the complex polytopes also. And dualities appear also. Now the real question is , must we consider these polytops really considering the QFT ? is it just a tool to rank and study better the fields in our standard model ? the real question is there, and we can extrapolate philosophically deeper, are we sure that all is made of Waves and fields ? like in the strings theory , or in the geometrodynamics, because if we have coded particles instead of fields creating our physicality , so we must consider particles and not fields implyinmg these geometries, topologies,properties of matters and so the effects possible in extrapolating the maths. The maths are Always interesting but they must be utilised with the biggest wisdom considering the interpretations and assumptions, we cannot extrapolate and conclude all what we want.The problem foundamental for me is that we consider still these geometrisations due to fields , like if we had a 1D main field from this Cosmic scale and permitting with the oscillations to create the reality with these 1D strings at this planck scale, all is false if the particles are coded and in 3D, don t forget that we can create all SHAPEs, geonetries, topologies with coded 3D particles, 3D spheres for example, now imagine this, imagine that the codes of geometrisations and properties are inside these particles , imagine a Ricci flow, the Hamilton Ricci flow, a kind of assymetric Ricci flow to create the unique things, imagine too this poincare conjecture and the heat equation and imagine the plays of maths with the topological and euclidian spaces, and the lie derivatives and lie groups, we can create all geometries and topologies also, so we arrive at big philosophical questions about these foundamental objects and the main cause of these objects and their properties. You can tell all what you want with polytopes, we cannot affirm that it is foundamental simply. The same for my reasoning considering these 3D spheres coded at this planck scale considering a gravitational coded aether sent from the central cosmological sphere. I beleive that we must prove what we extrapolate simply and at this moment we are limited simply. The aim is not to create mathematical partitions but to find the real universal partition, it is totally different at my humble opinion. The convex polytopes and the linear transformations must be sure after all, and the vectors and scalars also, the problem is that we cannot affirm in fact, so the same for the extrapolations and assumptions. The secret maybe if I can is to superimpose a deeper logic to this universe , the fields, strings, geometrodynamics and the fact to consider only photons like main essence imply a prison for the majority of thinkers, that is why we cannot explain our unknowns mainly for me.Think beyond the box and maybe consider coded particles, the Waves particles duality is respected because they are inm motions and in contact in a superfluid these particles....Regards

          Jochen. I enjoyed your paper. You may find my essay. "Clarification of Physics--" interesting. I introduce a self creating system, a new basic level to the current epistemically horizon and show how it fits into the creation of a multiverse that includes "our" physical universe. I would appreciate your comments on my essay. John D Crowell

            here is a general post about the entropical spherical informations

            Entropical spherical informations and general universal communications , the sortings, superimposings, synchronisations and the link with quantum 3D spheres and the general spherisation of the universe .Why and how ? sources, signals and encodings .....

            The complexity appears with the quantities of informations and can be ranked between the minimal and maximal of informations . For this let s consider a main universal emission from the central cosmological sphere, it is there that this infinite energy codes and transform thsi energy in matters, 3D finite series of spheres for me in a gravitational coded aether where this space disappears playing between the cold and heat generally.The source is from there and the aether is the source but it encodes also and recepts in function of evolutive codes and properties disered to create the diversity and communications of evolution in logic.

            The works of Shannon can converge and the uncertainty can be better understood at my humble opinion seeing the complexity and number of these finite series having probably the same number than our cosmological finite series of spheres, there is like an universal link between this finite number,

            the redondance and the equiprobability can be better understood if we know the real universal meaning of this general thought

            The thermodynamics can converge considering two main constants for this gravitational aether, like codes playing between this zero absolute and this planck temperature, it is an assumption but when we consider all the properties of these series, we can understand better the synchronisations, the sortings, the superimposings with all the motions, rotations , oscillations of these 3D spheres.

            The second principle in thermodynamics become relevant , Q/T correlated with this entropy and we can converge with the entropy of Shannon and the topological entropy in considering several mathematical Tools of ranking, like the lie derivatives, the topological and euclidian spaces, the Ricci flow and an assymetric Ricci flow, the poincare conjecture , the lie groups and others mathematical Tools. See that the motions, rotations , oscillations, volumes, densities, mass, angles, senses of rotations, moments, and other physical properties can help for the rankings and for a better understanding of communications ,uncertainties and probabilities.

            The potential of these series so become the key and the distribution also of informations in function of codes of evolution and properties of matters. It is a question of internal energy and distribution of this energy in function of internal codes and informations. The relevance becomes the infinity of combinations.

            Regards

            6 days later

            Dear John,

            thank you for your comment. I'm glad you enjoyed my thoughts. The notion of self-creating systems is a very interesting one, and close to my thoughts in some regards, so I'll definitely have a look at your essay.

            Good luck in the contest!

            Cheers

            Jochen