That's a cool way to describe it Lawrence...
You might enjoy my essay this time out, because it is more mathematical and focuses on the idea of convergence. I do keep the derivations in the endnotes however. Good to see you in the contest regardless.
All the Best,
Jonathan
I wanted to comment further...
You have achieved a greater level of clarity with this paper than in some of your previous work Lawrence. A lot of prior knowledge was assumed. I know I've picked up a few things, but it seems there is more ease of comprehension and also a greater congruency insofar as a nice conversation between the descriptions and the Math and good agreement between them or applicability throughout. This adds to the paper's impact and clarity.
Of course; there are a few flubs. Right in the abstract; it's a tautology to say obstructions cause obstructions. You might have meant considerations, conditions, limitations, restrictions - or something like that - causes obstructions. And I have no knowledge of Hamitlonians. But overall; it is well-written, where you get your point across, and you have a good point to get across. I was able to understand most of it, and since I didn't come into the process with an understanding of your topic; I have to credit good writing.
Best,
Jonathan
It is a case of editing that resulted in:
Axiomatic incompleteness of any p-adic algorithm illustrates how these obstructions define these obstructions.
Where the first instance of the word obstructions should be incompleteness. Dang! it is something that happens when you write and do all your own proof reading.
LC
It keeps getting better...
And that's good. Even the finest textbooks have lists of errata published later. So there is no shame nor a reason for it.
Best,
Jonathan
Dear Lawrence,
I'm glad you found the time to place an entry in this year's contest. There is, as usual, much to unpack in your contribution, and I'm sure it will amply repay repeated reading.
For now, one thing I'd like to understand better is the relation to set-theoretic forcing you see. Of course, part of this is simply that I don't have a good understanding of forcing itself---as a way of obtaining independence results, it seems that this ought to play right into my own prejudices regarding the connection between quantum mechanics and undecidability. If you can point me to any introductory material that might help me understand, for instance, in what way the move to complex numbers from the reals is related to forcing (I can sort of guess at the extension of the set-theoretic universe, but I'm very hazy on that sort of thing), I'd be thankful.
As for the rest of the paper, if I'm understanding the general gist, you argue that there are topological obstructions inhibiting the SLOCC-conversion of different entanglement classes (such as the W- and GHZ-classes) into one another, and the existence of these obstructions is due to the nonexistence of a general solution algorithm for diophantine equations. This is related to measurement due to the fact that any measurement creates an entangled state between the object system and the probe. Is it then correct to say that the unpredictability of the measurement outcome is thus due the above obstruction, a sort of 'you can't get there from here' type of thing? (Sorry if I'm misunderstanding your arguments, by the way; the paper is highly condensed, and I'm afraid I can't always follow you quickly enough.)
But then, whence entanglement? In my own approach, entanglement itself is due to (basically) the fact that you can only uncover a limited amount of information about any system---so you have two bits to describe a two-qubit system, which might end up describing only the correlations between the spins, without giving definite values to either spin on its own.
I'm also wondering about how, exactly, Gödel numbering comes into play. In a sense, in my approach, measurements code for states, and vice versa, which is how the self-reference comes into play---each state can be written as the set of outcomes for some measurement, while each measurement can be written as the set of states for which it yields a particular outcome. I think there may be some connection here, but I can't quite yet grasp it...
But anyway, there clearly is lots of interest to your essay, I shall be coming back to it. Good luck in the contest!
Cheers
Jochen
Jochen,
The crux centers around the p-adic solutions of Diophantine equations. My paper references Palmer's work, in fact a paper by Hossenfelder and Palmer [ S. Hossenfelder, T. N. Palmer, "Rethinking Superdeterminism," https://arxiv.org/abs/1912.06462v1 ] with the fractal sets in state space. It is a nifty idea in one sense, though I doubt the superdeterminism concept it is meant to uphold. Palmer's notion of incompleteness seems wrong to me, as a fractal as a recursively enumerable set of completely computable. It is the complement of ER sets that are not computable or have some undecidable aspect to them. These sets are forms of Cantor sets, and to define a field or metric geometry one has to use p-adic theory. Incompleteness comes into the picture with the theorem proven by Matiyasevich, which then means the topological obstructions between different entanglement types is a form of G├Âdel incompleteness. In effect the fractal geometry with a p-adic metric has its complement in the remainder of the p-adic set that is undecidable. The other p-adic sets have separate solutions or algorithms not isomorphic.
I did not go through the "nuts and bolts" work you have done with an explicit Cantor diagonalization of n experiments with k state with f(k,n) as you did. This approach you have taken is to explicitly argue for how this epistemic horizon is a form of mathematical incompleteness. I make the statement this approach is equivalent to the incompleteness of the solution theory for Diophantine equations as p-adic sets. Given the time here of course I have not had the time to seriously work that out. This would of course be a worthy question to pursue. I think though this is correct.
My approach it meant to be more of a practical way this sort of theory could be employed in questions over quantum measurement, decoherence and quantum gravitation. Quantum gravitation or Hawking radiation have much the same implication with quantum decoherence. All processes conserve probability so Tr(¤ü) = sum_i p_i = 1 is conserved. Decoherence changes Tr(¤ü^n) for n > 1, In particular Tr(¤ü^2) is changed. However, this adjustment occurs because this trace with the square of density matrix elements with ¤ü_{ij}^2 for i Ôëá j are removed for this density matrix pertaining to the state. If we consider the total density matrix ¤ü = ¤ü_sÔè--¤ü_r, s = system and r = reservoir, this change in Tr(¤ü^2) occurs for ¤ü_s = Tr_r(¤ü). The entropy measure S = -Tr(¤ü log(¤ü)) by Taylor series involves all powers of the density matrix. So this change reflects a loss of information accounting. I say accounting because what happens is there is a trace of the entire density matrix when the experimenter is concerned entirely with their system
Thanks for the thumb's up on this. This is even though I made an embarrassing tautological language mistake in the abstract due to editing. I have read you essay here, as well as you paper from 2018. I just have not gotten around to commenting yet.
Cheers LC
Hi Lawrence,
Thanks for your post on mine; "There are identical particles in QM"
I respond;
"Yes, only 2 types so 50% must be. But 'conjugate pairs' are opposite! Bohr assumed if one is 'spin up' the other MUST be 'spin down'. THAT was the shockingly simple error leading to all the 'weirdness' nonsense.
In the CORRECT version (DFM); Bob and Alice can reverse their OWN finding, so we don't need 'action at a distance' to violate Bells inequalities and reproduce the data set & Dirac equation.
But doctrine now seems embedded in such a deep hole I doubt we'll find anyone with both the intellect and influence to advance understanding. Do you?"
I hope to read yours soon.
Best
Peter
With spins you can have entanglements that are singlet or triplet for spins anti-aligned or aligned. In fact you can have a superposition of all fur Bell states. With bosons the exchange is positive or with Z cyclicity for n particles. With fermions you have Z_2 for two fermions, indicating a different topology from bosons, and for N fermions this is generalized with Slater determinants.
The connection with gravitation implies there is some generalization of what is meant by entanglement.The equivalency of quantum entanglement with measure of spacetime is some generalization of how entangled and separable states are correlated with each other. This might also be seen in how quantum gravitation as an UV physics is dual to gauge fields at lower energy (smaller mass) or IR scale:
Quantum gravitation at UV = quantum fields at IR,
which is one way of writing the Einstein field equations.
LC
Indeed. But your explanation of 'entanglement' shows you haven't read or understood how the need for that assumption is removed when derived classically. This is big stuff Lawrence! You're locked into Bohr's assumptions leading to tangled solutions & weirdness and ignoring the direct route Bell insisted must exist.
Bob's ability to reverse HIS OWN finding in each case by turning his dial (for ANY particle set) negates the need for ANY 'entanglement' beyond parallel spin axes with opposite orientation. That then reproduces the data and the solution Bell couldn't find.
But you do need to ontologically UNDERSTAND QM's experimental set ups, data set, and the mechanistic sequence which reproduces it. I'm sure you understand the key is in producing Cos^2Theta, the spin stats theorem, and so Dirac equation. Do you have that understanding of 'QM'.
If one works with classical physics of course there is no entanglement. Entanglement is a representation of topological difference between quantum and classical mechanics. The N-tangle that separates entropy configurations with N states in different entanglements is a topological obstruction that does not occur in classical physics.
LC
Lawrence,
Interesting way to describe entanglement, statistical not physical as assumed. Yet if QMs experimental data set can be shown to reproduced, as Bell suggested it should be, with a classical deterministic model, is that not likely a conclusive proof that Bells view was correct? (As Tim Palmers view).
Peter
If you read my paper, you can see I discuss a fair amount of material on the geometric aspects of quantum mechanics and entanglement. The references I include are from authors who have contributed to these developments. The most elementary form of this is the Fubini-Study metric.
Quantum mechanics is curious in that it is completely deterministic as a wave dynamics, but the amplitudes of the wave give probabilities that occur in measurement or decoherence spontaneously. There is no established idea of a causal process that gives any outcome of a quantum measurement. Hence quantum mechanics has this strange sort of duality between what is deterministic and what is stochastic. QM as a wave dynamics is deterministic, and yet unobserved as such. However, this determines probability distributions and the actual observed outcome is purely stochastic.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ВЅ as an L^2 measure theory. It then has a corresponding q = ВЅ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q в†' в€ћ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's Game of life or classical mechanics. A quantum measurement is a transition between p = ВЅ for QM and в€ћ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel's theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
I am aware of your stance on these things. I personally think this is something that Cervantes wrote about. However, if you are determined to stay on your steed Rosinante and pursue your quest then you have the freedom to do so.
Cheers LC
Hi LC,
As usual, you made an excellent work. I see that your Essay has connections with the Essay of Szangolies, that I have found interesting too. In a certain sense, you extends Szangolies' approach to the search of quantum gravity. I like a lot your sentence that "Spacetime built from entanglements or QM equivalent to GR means conservation of quantum information and the equivalence principle are either equivalent themselves or are in some duality with each other". I really hope that you are correct on this. But, till now, it seems that Nature needs to generate a breakdown of one of them if it wants to save the other. This is the big problem in order to realize a theory of quantum gravity.
I wish you good luck in the contest.
Cheers, Ch.
This implies some relationship between CHSH polytopes and Kirwan polytopes. I am not exactly sure how that will work. To be honest I seems to at least tangentially have something to do with the Born rule. The CHSH polytope pertains to conditional probabilities with entanglements and the Kirwan polytope with eigenvalues of entanglements.
Cheers LC
Hi Lawrence, a very good essay I must say. I am curious so I am going to ask you several simple questionS. What are for you the causes of our geometries, topologies, matters and properties ? and philosophically speaking also , what is the cause of all this ? Regards
The word cause is probably not quite appropriate. The word source might be better. I would say a source for the topological obstruction may be this epistemic horizon or a fundamental undecidability of states. The entanglement symmetries of GHZ and W states are separated by this 3-tangle for 3 states. This is a topological obstruction that has a measure based on the degree of uncomputability of states of one entanglement by another. As a result the basis for these topological obstructions should be a measure of unobservability or of deciding one set of measured states based on another set of measured states.
Cheers LC
Updated comment: The idea of superdeterminism is really a statement of nonlocality. The idea that no two points or events in the universe are completely independent is a nonlocality expected of quantum gravity. Anything approaching a black hole is never seen to cross the event horizon, but at the same time Hawking radiation occurs. This means that quantum states do not have a unique location in space, but rather have a nonlocality that is probably a salient feature of quantum gravitation.
Palmer and Hossenfelder place this in the context of hidden variables, where an average over these gives standard QM result. This has a Gaussian or standard statistical distribution that on average removes this dependency between regions of spacetime. This means the statistical independence of establishing initial states and the subsequent measurements are not entirely secure. Palmer places this in the setting of undecidability, but on a version of undecidability that is somewhat controversial. This is the Blum, Shub, and Smale (BSS) concept of undecidability of fractals. I work within a more standard idea, where the complement of fractal sets are undecidable. This leads into the undecidable nature of Diophantine sets and p-adic numbers.
Cheers LC
Hi , it is well generalised. What is this source, it is what I try to encircle. What is the cause of our reality and its geometries, topologies, matters and properties? and why all this is undecidable and uncomputable, is it due to philosphical limitations or errors or is it because we know so few about our main physics? If I can Lawrence, I d klike to have your general philosophy about these mathematical and physical objects and the philosophy correlated , why we have these geonetriesm topologies, matters and properties? and how this universe transforms and codes this energy to create this physicality for you, That will permit us to go deeper about the generality . I liked your essay, it is one of my favorites, friendly, regards.
I beleive that in fact Lawrence the generality of this philosophy about this source is important and the link with the foundamental objects. The fact to consider that all is Waves , strings and fields instead of particles or the opposite becone a main general key to really understand why we have this physicality. If we consider that this universe is just an energy oscillating tranformaing the enery in matters and if we consider that we have just photons like main essence, it is a kind of prison of beleifs for me. I see that these strings or geometrodynamics are a real fashion inside the theoretical sciences Community, maybe the main cause is due to Witten and Einstein, many maybe have counfound the field medal of Witten for a relevant work in maths about the fields and his theoriy of strings. It is two things totally different in fact, we cannot affirm that these strings are foundamental objects at this planck scale and the same for the Cosmic fields linked with thes quantum strings. We cannot affirm simply, I consider personally in my model coded 3D particles and they can explain also all our geometries, topologies, matters and properties. That is why I d like to know your general philosophy about the source like you told me.
Todd Brun found [ https://arxiv.org/pdf/gr-qc/0209061v1.pdf ] that P = NP is true for closed timelike curves. This is a short, readable and decent paper. The extension to all PSPACE and undecidable propositions is of course difficult to prove explicitly. However, a spacetime that permits CTCs will present Cauchy horizons, and in principle an observer can in a finite time verify whether a Turing machine halts or does not halt, even if the proper time of that TM is infinite. This is of course an in principle argument.
It is potentially interesting in the context of P = NP vs P в‰ NP whether this result really does mean this is undecidable proposition. P = NP appears true in a spacetime with CTCs, such as AdS or wormholes and so forth. We have no knowledge whether P = NP can hold in our more normal dS-like spacetime with positive vacuum energy.
Aaronson and Watrous found [ https://arxiv.org/pdf/0808.2669.pdf ] that classical systems on closed timelike curves can perform some BQP algorithms quantum computers execute. This emboldens my hypothesis that quantum physics and spacetime physics are categorically equivalent, or that spacetime is an epiphenomenon of large N-entanglements. On the top of page 5 of this paper is an interesting diagram. This illustrates a register or memory system with two parts, one part for a spacetime such as what we observe with open timelike curves and another with part with closed timelike curves. Aaronson and Watrous argue the Deutsche self-consistency condition on CTCs should hold and that a quantum wave corresponding to the causality respecting must also constructively interfere with the CTC wave function. The argument then is it is possible to emulate all PSPACE this way.
Spacetimes such as Gödel's universe, the global metric on AdS or the timelike interior of a Kerr black hole have CTCs. BTW, it appears that Gödel had some mental obsession with closed loopy systems! The question might be raised, what is the separation or distinction between causal respecting CR and closed timelike curve CTC spaces? The diagram on the AW paper suggests there is some quantum wave interference between a wave function associated with the CR and CTC spacetimes. The de Sitter and anti-de Sitter spaces respectively fit as a single sheet hyperboloid surrounding a light cone and two hyperboloids bounded within the conical openings. These meet at I^{±∞}, which means they share the same quantum information as defined by the AdS/CFT correspondence of Maldacena et al. . In the setting of holography we have something similar, and there are arguments of AdS black hole correspondence as well.
This according to A. Almheiri1, R. Mahajan, J. Maldacena, and Y. Zhao (AMMZ) [ arXiv:1908.10996v1 [hep-th] ] also has some correspondence with the interior state of a black hole. This paper is rather qualitative and speculative. The idea is the interior of a black hole has "islands" of states defined by a dimension difference of one. We might compare this to how the Reisner-Nordstrom metric, and by extension the Kerr metric, has a near horizon condition for an accelerated observer equivalent to AdS_2Г--S^2. The AdS globally has CTCs, and locally we consider conformal patches that restrict away from CTCs and respects CR. What I am working on now is to illustrate how the AMMZ islands correspond to local AdS regions or conformal patches. This would imply event horizons or boundaries imposes restrictions away from a complete correspondence. This is in line with my FQXi paper on topological restrictions between entanglement types and their correspondence with Szangolies' concept of the epistemic horizon.
The issue with P = NP vs P ≠NP is then still open. As I approach this with p-adic numbers and the Gödel undecidability of these sets, a complex number version corresponds to problems in algebraic geometry. Mulmuley has devoted much work on the algebraic geometry of computation. This leads to interesting issues with the Riemann ζ-function.
LC
