Continuing on below On Thursday, June 20, 2019 at 5:38:53 PM UTC-5, Lawrence Crowell wrote: > > 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]> >> 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 machines are polynomial because one must transmit a classical signal to teleport the outcome. The term is bounded polynomial, because the polynomial time or space is less and depends on the size of the ancillary state space required and not as dependent on the number of qubits processed quantum mechanically. A quantum computer it must be remembered really is a system that establishes constructive and destructive interference of quantum waves so the maximum is the "answer." It is often said quantum processors are running all possible paths at once, which has some truth to it, but with respect to the actual outcome what is measured is the constructive interference.
> >> >> 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. >> >> >> But ... there is no decision procedure to determine whether QM favors collapse, way in the Bohr sense or the GRW sense, or if there is something more in line with many worlds. There are open holes to the problem either way. > >> >> 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 >> >> >> I am not sure about how this relates to a first person structure. That sounds a bit observer dependent. LC > >> >> 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/7c363d11-4686-4185-a16f-f97cdd7365a4%40googlegroups.com.
