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Sunday, February 26, 2012

Turing and wavefunction collapse

Some interesting discussion by Turing biographer and mathematical physicist Andrew Hodges of Turing's early thoughts about the brain as a quantum computer and the possible connection to quantum measurement. I doubt the brain makes use of quantum coherence (i.e., it can probably be efficiently simulated by a Turing machine), but nevertheless these thoughts led Turing to the fundamental problems of quantum mechanics. He came close to noticing that a quantum computer might be outside the class of machines that a Universal Turing Machine could efficiently simulate.

Hodges' Enigma (biography of Turing) is an incredible triumph. Turing's life was tragic, but at least he was granted a biographer worthy of his contributions to mankind.

A shorter precis of Turing's life and thought, also by Hodges, can be found here.

Hodges: ... Turing described the universal machine property, applying it to the brain, but said that its applicability required that the machine whose behaviour is to be imitated

…should be of the sort whose behaviour is in principle predictable by calculation. We certainly do not know how any such calculation should be done, and it was even argued by Sir Arthur Eddington that on account of the indeterminacy principle in quantum mechanics no such prediction is even theoretically possible.

... Turing here is discussing the possibility that, when seen as as a quantum-mechanical machine rather than a classical machine, the Turing machine model is inadequate. The correct connection to draw is not with Turing's 1938 work on ordinal logics, but with his knowledge of quantum mechanics from Eddington and von Neumann in his youth. Indeed, in an early speculation, influenced by Eddington, Turing had suggested that quantum mechanical physics could yield the basis of free-will (Hodges 1983, p. 63). Von Neumann's axioms of quantum mechanics involve two processes: unitary evolution of the wave function, which is predictable, and the measurement or reduction operation, which introduces unpredictability. Turing's reference to unpredictability must therefore refer to the reduction process. The essential difficulty is that still to this day there is no agreed or compelling theory of when or how reduction actually occurs. (It should be noted that ‘quantum computing,’ in the standard modern sense, is based on the predictability of the unitary evolution, and does not, as yet, go into the question of how reduction occurs.) It seems that this single sentence indicates the beginning of a new field of investigation for Turing, this time into the foundations of quantum mechanics. In 1953 Turing wrote to his friend and student Robin Gandy that he was ‘trying to invent a new Quantum Mechanics but it won't really work.’

[ Advances in the theory of decoherence and in experimental abilities to precisely control quantum systems have led to a much better understanding of quantum measurement. The unanswered question is, of course, whether wavefunctions actually collapse or whether they merely appear to do so. ]

At Turing's death in June 1954, Gandy reported in a letter to Newman on what he knew of Turing's current work (Gandy 1954). He wrote of Turing having discussed a problem in understanding the reduction process, in the form of

…‘the Turing Paradox’; it is easy to show using standard theory that if a system start in an eigenstate of some observable, and measurements are made of that observable N times a second, then, even if the state is not a stationary one, the probability that the system will be in the same state after, say, 1 second, tends to one as N tends to infinity; i.e. that continual observation will prevent motion. Alan and I tackled one or two theoretical physicists with this, and they rather pooh-poohed it by saying that continual observation is not possible. But there is nothing in the standard books (e.g., Dirac's) to this effect, so that at least the paradox shows up an inadequacy of Quantum Theory as usually presented. ...

[ This is sometimes referred to as the Quantum Zeno Effect. A modern understanding of measurement incorporating decoherence shows that this is not really a paradox. ]

Turing as polymath:

In a similar way Turing found a home in Cambridge mathematical culture, yet did not belong entirely to it. The division between 'pure' and 'applied' mathematics was at Cambridge then as now very strong, but Turing ignored it, and he never showed mathematical parochialism. If anything, it was the attitude of a Russell that he acquired, assuming that mastery of so difficult a subject granted the right to invade others. Turing showed little intellectual diffidence once in his stride: in March 1933 he acquired Russell's Introduction to Mathematical Philosophy, and on 1 December 1933, the philosopher R. B. Braithwaite minuted in the Moral Science Club records: 'A. M. Turing read a paper on 'Mathematics and logic.' He suggested that a purely logistic view of mathematics was inadequate; and that mathematical propositions possessed a variety of interpretations, of which the logistic was merely one.' At the same time he was studying von Neumann's 1932 Grundlagen den Quantenmechanik. Thus, it may be that Eddington's claims for quantum mechanics had encouraged the shift of Turing's interest towards logical foundations. And it was logic that made Alan Turing's name.

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