On Thursday, June 20, 2019 at 8:43:08 AM UTC-5, Bruno Marchal wrote: > > > On 20 Jun 2019, at 00:26, Lawrence Crowell <[email protected] > <javascript:>> wrote: > > On Tuesday, June 18, 2019 at 6:02:54 AM UTC-5, Bruno Marchal wrote: >> >> >> On 18 Jun 2019, at 02:14, Lawrence Crowell <[email protected]> >> wrote: >> >> The stochastic aspects of QM emerge in measurement, where the modulus >> square of amplitudes are probabilities and there are these random outcomes. >> The measurement of a quantum state is not a quantum process, but has >> stochastic outcomes predicted by QM. Based on the Hamkin's work where I >> only looked at the slides and not yet the paper, it seems possible to do >> this with quantum computer. >> >> >> http://jdh.hamkins.org/computational-self-reference-and-the-universal-algorithm-queen-mary-university-of-london-june-2019/ >> >> slides: >> >> >> http://jdh.hamkins.org/wp-content/uploads/Computational-self-reference-and-the-universal-algorithm-QMUL-2019-1.pdf >> >> I wrote a couple of elementary Python codes for the QE machine IBM has to >> prepare states and run then through Hadamard gates. The thought occurred to >> me that this Quining could be done quantum mechanically as a set of >> Hadamard gates that duplicate a qubit or an bipartite entangled qubit. This >> is a part of my ansatz that a measurement is a sort of Gödel numbering of >> quantum states as qubit data in other quantum states. >> >> Quantum computations are mapped into an orthomodular lattice that does >> not obey the distributive property. The distributive law of p and (q or r) >> = (p and q) or (p and r) fails. The reason is due to the Heisenberg >> uncertainty principle. Suppose we let p = momentum in the interval [0, P], >> q = position in the interval [-x, x] and r = particle in interval [x, y]. >> The proposition p and (q or r) is true if this spread in momentum [0, P] is >> equal to the reciprocal of the spread of position [-x, y] with >> >> P = ħ/sqrt(y^2 + x^2). >> >> The distributive law would then mean >> >> P = ħ/|y| or P = ħ/|x| >> >> which is clearly false. This is the major difference with quantum logic >> and Boolean classical logic. These lattices of quantum logic have polytope >> realizations. >> >> This is in fact another way of realizing that QM can't be built up from >> classical physics. If this were the case then quantum orthomodular >> lattices, which act on convex sets on L^p spaces with p = ½ would be >> somehow built from lattices acting on convex sets with p → ∞. This is for >> any deterministic system, whether Newtonian physics or a Turing machine. It >> is this flip between convex sets that is difficult to understand. With p = >> ½ and the duality between two convex sets as 1/p + 1/q = 1 the dual to QM >> also has L^2 measure. This is spacetime with the Gaussian interval. For a p >> → ∞ the dual is q = 1 which is a purely stochastic system, say an idealized >> set of dice or roulette wheel with no deterministic predictability. >> >> The point of Quining statements quantum mechanically is that this might >> be a start for looking at a quantum measurement as a way that quantum >> states encode qubit information of other quantum states. It is a sort of >> Gödel self-reference, and my suspicion is the so called measurement problem >> is not solvable. The decoherence of states is then a case where p = ½ → 1 >> with an outcome. That is pure randomness. >> >> >> With mechanism, that randomness is reduced into the indeterminacy in >> self-multiplication experience. It come from the many-histories internal >> interpretation of arithmetic, in which all sound universal numbers >> converges. The quantum aspect of nature is just how the (sigma_1) >> arithmetical reality looks like from inside. This explains where the >> apparent collapse comes from, in a similar way than Everett, but it >> explains also where the wave comes from. Eventually quantum mechanics is >> just a modal internal view of arithmetic, or anything Turing equivalent. >> The math, and quantum physics confirms computationalism up to now, where >> physicalism and materialism are inconsistent, or consciousness or person >> eliminative. >> >> > Thanks for addressing this. > > I guess in a way I do not entirely understand this. The above illustration > is the main difference between Boolean and quantum logic. > > > OK. I have no problem with this. I agree and understand that quantum logic > cannot be embedded or extended into a classical logic. This is related to > the fact that there is no local hidden variable theory compatible with the > quantum experiments. > > But this does not mean that quantum logic cannot have a classical > explanation. In fact the quantum formalism is by itself a classical > description, even local and deterministic, but hard to interpret in any > local realistic way. > > Assuming the mechanist hypothesis, we have a similar (to QM) form of > indeterminacy, due to the fact that we can be duplicated, and in that case > the person who is duplicated cannot predict with certainty which of the > copies she will feel to be, as both will be right to say that they have > survived in the place where they are reconstituted. We can come back on > this if you want to know more. That leads to the problem that no machine > can know which computations (which exists in arithmetic as we know since > Gödel-Turing 1930s papers) support her, and we know that there is an > infinity of such computations in arithmetic: this eventually rediuce > physics (the art of predicting the observable) into a relative statistics > on all computations in arithmetic. > > In fact with mechanism, we have a canonical “many-world” interpretation of > elementary arithmetic. And with mechanism, it should explain the existence > and persistence of the physical laws (and indeed up to now this is > confirmed, notably by the Everett formulation of QM). > > It requires a little more than elementary arithmetic. Graph theory maybe. A coloring scheme for graphs with Borel groups of upper right triangular matrices would work. The Heisenberg group is a form of a Borel group. The arithmetic you refer to appears to be the additivity of the probabilities, which is the same thing as Tr(ρ) for ρ the density matrix. I can go into greater detail on this. There are maps to the quotient space of the AdS spacetime as well.
I am not terribly worried about interpretations of QM. These are auxiliary postulates or physical axioms. I do think these are some aspect of the decoherence of quantum states or measurement being a sort of self-reference. > > > It is not clear to me in what way quantum mechanics is σ_1 arithmetic > viewed from the "inside." I guess I am not sure what is meant by σ_1 > arithmetic. > > > The sigma_1 arithmetical sentences are the sentences provably equivalent > (in PA, say) with sentences having the shape “ExP(x), with P a decidable or > recursive (sigma_0) predicate. > So is σ_0 the same thing as primitive recursive? There is a bit of symbolic representation that I am not familiar with. > > Turing-completeness or Turing-universality is equivalent sigma_1 > completeness, i.e. the ability to prove all true sigma_ sentences. > > Intuitively it is obvious that you and me, all humans, and in fact all > computers, are sigma_1 complete. If is true that ExP(x), and if P is > decidable, then by testing 0, 1, 2, … we will eventually find that x, and > be able to verify it satisfies p. The reverse is true also: if something > can prove all true sigma_1 sentences, then it can emulate all computations, > and it provides “one more” formal definition of computation, and one more > universal machine. > > A normal form theorem by Kleene makes it possible to identify halting > computations and true sigma_1 sentence. The set of all true sigma_1 > sentences is more or less equivalent with the universal dovetailing (a > procedure which generate all programs and execute them all). > > It has been shown that RA, or SK are Turing-complete theories, and thus > constitute universal machine or machinery. > > RA is classical logic + the seven axioms: > > 1) 0 ≠ s(x) > 2) x ≠ y -> s(x) ≠ s(y) > 3) x ≠ 0 -> Ey(x = s(y)) > 4) x+0 = x > 5) x+s(y) = s(x+y) > 6) x*0=0 > 7) x*s(y)=(x*y)+x > > > SK is theory (without logic!): > > Rules: > > 1) If A = B and A = C, then B = C > 2) If A = B then AC = BC > 3) If A = B then CA = CB > > Axioms: > > 4) KAB = A > 5) SABC = AC(BC) > > > This looks pretty elementary, though 4 and 5 look a bit odd.. I am not sure how useful it is with quantum computation. With my idea about Gödel in the quantum it is where a set of ancillary states are set to become copies of other states, or they in effect emulate them through entanglement. This will requires a Hadamard gate process, which is needed to duplicate states or just to set up a prepared state. LC > > > The space of computation for quantum computers is not clear. Aaronson > showed the space is a bounded quantum polynomial space, which contains P > and now appears to extend into NP. The measure of quantum computing is > PSPACE is as yet not known. > > > For my “mind-body” interest, we need only to know that quantum digital > machines do not violate the Church-Turing thesis. > It seems to me that David Deutsch has already shown that the universal > quantum Turing machine emulates all machines polynomially, so Aaronson is > correct. But of course, we can expect this is false if we put a rounded > polynomial measure on the computations. Typically, we can expect an > exponential slow-down when a classical machine emulates a quantum > algorithm, although this has not been yet proved. Most people believe in > this conjecture, and that motivates the research in quantum computation. > > > > > Quantum logic are in nondistributive orthomodular lattices of p = ½ convex > functions, classical probability systems p = 1 and deterministic systems > without a definable measure. We do not think of deterministic classical > systems, or for that matter Turing machines as having a measure over which > one integrates a density. The classical probability system and > deterministic system are in a dual relationship, as are quantum mechanics > and spacetime physics with L^2 measure. > > > OK. > > > > How QM flips from a p = ½ system to a p = 1 system is unknown. > > > Indeed. It is the problem. > > Now, this is less mysterious when we abandon the collapse, as this makes > the quantum indeterminacy a particular case of the first person > indeterminacy, and the math confirms that we do find a quantum logic there. > > I do not claim that this solves all interpretation problem; but with > Mechanism, we have no choice: we must reduce physics into a statistics on > the first person view distributed on all computations. If I did not get a > non boolean quantum logic there, I would probably believe that Mechanism > (as an hypothesis in cognitive science) is refuted, or made implausible. > > > > > There was a recent paper that demonstrated how a quantum system about to > enter decoherence exhibited some behavior, which means there may be some > process involved whereby a quantum deterministic system transforms into a > set of classical probabilities. This process may have some analogues I > think with singular perturbation theory. > > > I would need more on this to evaluate if this is consistent with digital > mechanism or not. Then, I might need to progress more on the “arithmetical > quantum logic” related to that first person statistics calculus. > > Bruno > > > > > LC > > >> >> Now of course we can ask what we mean by random, and that is undefinable. >> Given any set of binary strings of length n there are N = 2^n of these, and >> in general for n → ∞ there is no universal Turing machine which can >> compress these into any general algorithm, or equivalently the Halting >> problem can't be solved. A glance at this should indicate that N is the >> power set of n and this is not Cantor diagonalizable. Chaitin found there >> is an uncomputable Halting probability for any subset of these strings. >> Randomness is then something that can't be encoded in an algorithm, only >> pseudo-randomness. >> >> The situation is then similar to the fifth axiom of geometry. In geometry >> one may consider the 5th axiom as true and remain within a consistent >> geometry. One may similarly stay within the confines of QM, but there is >> this nagging issue of decoherence or measurement. One may conversely assume >> the 5th axiom is false, but now one has a huge set of geometries that >> are not consistent with each other. Similarly in QM one may adopt a >> particular quantum interpretation. >> >> >> >> QM cannot be invoked except as a toll to test Mechanism >> (computationalism). >> >> Bruno >> >> >> LC >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/everything-list/504fa0ed-686e-4e17-bbdc-68dfa609008f%40googlegroups.com >> >> <https://groups.google.com/d/msgid/everything-list/504fa0ed-686e-4e17-bbdc-68dfa609008f%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> >> >> > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] <javascript:>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/8819a3ce-6e7d-443c-ba3a-2555fccac0d1%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/8819a3ce-6e7d-443c-ba3a-2555fccac0d1%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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