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] > <javascript:>> 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. 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 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. 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. How QM flips from a p = ½ system to a p = 1 system is unknown. 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. 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] <javascript:>. > 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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/8819a3ce-6e7d-443c-ba3a-2555fccac0d1%40googlegroups.com.
