Dear David,

thank you for your interest in my article. Regarding the topic of infinity, I must confess that it does make me somewhat uneasy---but in the end, our most successful current theories of physics (not just quantum field theory and general relativity, but also plain old Maxwellian electrodynamics) depend on the continuum of the real numbers, generally including an uncountable number of degrees of freedom. As these do yield spectacularly accurate predictions in many cases, one might well surmise that they do at least get something right.

That said, there are intriguing arguments that quantum gravity, due to the Bekenstein bound, only includes a finite number of degrees of freedom within a finite volume. But I'm afraid these considerations are rather beyond the scope of the current topic.

Dear Joseph,

thanks for taking the time and commenting on my essay. In some sense, you can indeed view things like entanglement as information that's present in the system as a whole, but not reducible to any of its parts: while a classical system, such as the two differently colored cards in the envelopes I briefly consider, contains all of its information in the states of each individual subsystem, that's no longer the case in quantum mechanics---there, the state of each subsystem (of a maximally entangled system) will be maximally mixed, but the total system is not a combination of maximally mixed states.

Hence, there is information within the total state that's not reducible to its components---the basis of what's sometimes called 'quantum holism'.

Whether, of course, that has anything to do with intuition, or other macroscopic phenomena, seems rather doubtful to me. However, you might find this recent paper of renowned physicist Don Page interesting: https://arxiv.org/abs/2001.11331

Dear Jonathan,

thanks for your kind comment. I can't claim myself to be an expert in category theory, and really, the introduction I give to Lawvere's result is a kind of bastardized set-theoretical version, which one can get away with because Lawvere's theorem applies to categories that are cartesian closed, of which Set (the category with sets as objects and total functions as morphisms) is an example.

For a much better introduction into the subject, I can only highly recommend Noson Yanofsky's paper: https://arxiv.org/abs/math/0305282

For the core idea, I think the important part is to grasp the connection between the category-theoretic argument and the diagonalization; everything else flows from there. If you've got any questions, I'd be happy to try and address them!

Dear Boris,

thank you for your comment and for taking an interest in my essay, but you are much too kind.

I will have a look at your essay, and see whether I have anything useful to say on the subject.

If an entangled pair consists of an apple and an orange, and instead of measuring their extrinsic properties (like position and momentum in the original EPR), one instead decides (as Bell subsequently did in his theorem) to substitute a measurement of an intrinsic property (like skin-texture or polarization), then there is going to be a big problem, when you have thus unwittingly assumed that the measurement of the skin-texture of the orange can be substituted for (or compared to) a measurement of the skin-texture of the apple, in the same manner in which measurements of their position and momentum can be substituted. Consequently, unlike the original EPR thought experiment, Bell's theorem is only valid for perfectly identical pairs of entangled particles.

By exploiting this rarely discussed "loophole", it is easy to demonstrate that a peculiar set of non-identical pairs of entangled particles (those manifesting only a single-bit-of-information), will perfectly reproduce the peculiar correlations observed in Bell-tests, in a purely classical manner; due to frequent "false positives", caused by mistaking the normal behavior of entangled "fraternal twins", for an abnormal behavior of entangled "identical twins". From this perspective, quantum theory should be interpreted as merely describing behavior analogous to a poorly-designed "drug test", rather than describing any behavior of the drugs (substances) themselves; tests in which "up" states (the drug is present) are frequently being mistaken for "down" states (the drug is not present) and vice-versa.

Rob McEachern

    Dear Robert,

    thanks for highlighting your model. I see you can reproduce the singlet-correlation with a detection efficiency of 72%; out of curiosity, what is the maximal detection efficiency you can allow and still observe a violation of the CHSH inequality?

    Furthermore, the identical nature of particles is, of course, a central tenet of quantum mechanics---quantum particle statistics are derived assuming that exchanging any two particles does not lead to a new configuration of the system. This has many observable consequences that seem to be hard to explain otherwise. How do you propose to account for this?

    Jochen,

    The detection efficiency given in the paper, is not the standard conditional efficiency (single detector) usually reported; it is the product of both detectors. So the corresponding, conditional efficiency is the sqrt(0.72)=0.85, which is already above the theoretical limit for a classical process. With optimization, of the matched-filtering, it ought to be possible to perfectly reproduce the entire correlation curve (not just the few points evaluated in most Bell tests) at even higher efficiencies.

    The "fraternal twins" are statistically identical. So they obey the same probability of detection statistics predicted by quantum theory. But there are two detection distributions that are important, in any detection theory; unfortunately quantum theory only computes one of those two. It only computes the probability that something will be detected, but it never even attempts to predict the probability that that "something" is actually the thing that the system was supposed to detect (Probability of False Alarm). This would not be an issue, if all the things being detected were in fact "identical" particles, as has been assumed. But when the particles are only very similar (statistically identical), rather than perfectly identical, it does matter. This is exactly the problem in a Bell test; the number of detections agrees with the theory, but only because the actual state of the detection is frequently incorrect - enough to change the computed correlation statistics, as the result of non-random, systematic errors in the process. In effect, lopsided, "edge-on", "up" polarized coins are frequently being mistaken for perfect "face-on", "down" polarized coins and vice-versa.

    The point is, there is a very important distinction between "assuming that exchanging any two particles does not lead to a new configuration of the system", and assuming that exchanging any two particles does not lead to a new detectable configuration of the system, when the detection process (matched filtering) is not perfect enough to distinguish between "fraternal twins" and "identical twins", in every instance.

    Rob McEachern

    9 days later

    Regarding horizons...

    I have found another useful analogy for Schwarzschild event horizons to be the virtual ground or amplitude null at the summing junction of an inverting op-amp circuit. This too was suggested by my study of the Misiurewicz point M3,1, the 'edge of chaos' point. The suggestion here is that other types of black hole horizons could be studied with the toolkit of category theory, by finding the correct circuit diagram analogy, and that the Mandelbrot Set provides somewhat of a map.

    I have attached a diagram. Let me know if this makes sense. It would make a black hole horizon like a phase-reversed mirror that appears black or as a black body because whatever strikes it is (energetically or temporally?) inverted.

    All the Best,

    JonathanAttachment #1: MandelAmp.jpg

      Apologies if this is too far off-topic...

      I am following up on a comment you made in my essay forum. Still interested in your thoughts.

      JJD

      I am starting to get a handle on your paper Jochen...

      It appears Lawvere's theorem is a template for quite an array of meaningful conjectures, which makes it very powerful. Not so easy to grasp for those unfamiliar with the language of category theory however. I will stay the course and try to grasp what you are saying.

      In fact; I think grasping is a great metaphor here, because the story is about what gets caught before it slips behind the epistemic horizon, and only new information is available. The derivation of the word 'think' is about grasping coming from the word 'tong' a device that lets one pick things up and examine them.

      The essence of Lawvere would be caught up in the same idea. If the tongs are used to pick up hot items coming from a forge or kiln, one can examine only one item at a time and the rest are either heating or cooling, so one loses information about or control over the items not in your tongs. So the metaphor of grasping and holding vs. what slips away applies to Lawvere's theorem.

      Best,

      Jonathan

        Yes, Lawvere's theorem is certainly a very deep result, if anything underutilised in my essay---as I noted, the simple set-theoretical framework in which I present it doesn't really do justice to the full category-theoretic treatment.

        There have been attempts of bringing black hole type losses of predictability within the framework of formal incompleteness, such as this one: https://link.springer.com/article/10.1007/s11128-008-0089-2.

        I am not sure I see how to connect this to the qualitative similarity you point out in regards to the Misiurewicz point; I'm largely ignorant on that topic, I'm afraid. However, you might be interested in some of the works of Louis H. Kauffmann (http://homepages.math.uic.edu/~kauffman/), particularly on what he calls 'reentrant forms'---for instance, in 'Knot Logic' (http://homepages.math.uic.edu/~kauffman/KnotLogic.pdf), he considers the Koch snowflake as an example, and I have a hunch something similar could be applied to the Mandelbrot set and its recursive dynamics.

        Thank you Jochen,

        The link to the paper by Srikanth and Hebri is appreciated, and it looks very interesting. I don't know enough about how Godelian incompleteness relates to the BH information paradox. I will check that out when I can, and perhaps discuss here further if timely.

        I think the work of Louis Kauffman is pretty amazing. I have had some communication with him, but not in a while. My research has progressed much further since then and perhaps its time for an outreach, and I will check out the linked material which I appreciate your citing.

        All the Best,

        Jonathan

        I finally read through your paper. It is very interesting that you do make some connection with Gödel incompleteness. I do though appear there is a need to "wring out" violations of Bell's inequalities --- pun intended. Maybe a form of PR box or Tsirelson bound argument with a diagonalization of possible measurements will work. This is important, for to use the language of my paper this is where the topological obstruction is manifested.

        In effect the CHSH or bell inequality may be thought of as a sort of metric. As with geometry non-Euclidean spaces have different metrics, and this is particularly the case for spaces with different topologies. As with Nagel and Newman in their book I see this issue as similar to the incompleteness of geometry as an axiomatic system to determine the truth of the 5th axiom.

        I have no clear stance on the matter of counterfactual definiteness. That also seems to be something dependent on various quantum interpretations. I question whether this is something that is simply undecidable. As Palmer puts it this means the statistical independence of state preparation and measurement is not something provable or derivable from QM.

        Cheers LC

          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?