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. 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. 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.
