Check out exercise 2.56.
For Turing in 1939, this gives him the rationale for taking the log of the bayes factor (and starting the whole decibanning thing). He already knows that evolution in a discrete system can become, for certain systems, a numerical change of phase for a unitary operator (rather than a continuous hamiltonian operator). That one can turn this exponential inside out – and now calculate in terms of a unitary that comes down to adding up logs of values would be quite obvious (and even more an obvious thing to do for folks brought up in the era of slide rule based calculation)l.
For a Turing, rather than think in terms of the Shannon’s noisy channel theorem, he would be thinking more primitively in terms of distinguishability:
This is not exactly the first time we have seen folks focus on that which is not the constant function. But it is the first time we have seen such a coherent, intuitive explanation. It also, by its linkage to POVMs, shows the link to Turing’s argument (which also declared the positive conditions).
We recall how the in the walsh basis, one also has W0.
ok, we can see how in Turings argument, that his one-dimensional representation is unity (once the density converges to the limiting distribution for the particular angular momentum algebra, or fano plane) could be expressed in terms of logs, where the multiplication of H1 terms becomes the addition of the terms’ log values – which should sum to 1 when one considers the “kernel” of U1…Uk as a unitary operator which gives rise to a specifically complete set of orthonormal state descriptors … that includes the M0 term (or constant function, associated with the ground state energy level).
When he states Theorem III, we could be interpreting it as saying that H1 – and its M1 through Mp, addresses each of the 1…p orthonormal wave functions, and excludes M0 (the constant value). Since the Mp sum to zero, all the probability is stored in the amplitude of the remaining wave function (which is the constant function). IN a qudit world, this means that any component of said wave function is constant (though not necessarily the same constant value). The reasoning limitation that only one coset retains positive contribution is based on consequence of there being a quotient group and a limiting condition (that the mean of g has aligned, after all the neighboring averaging effects have reduced the norm by quadratic degree, with that of the wheel,f: i.e. 1). Having made the independent lines in the plane (those cosets now with value 0), all the amplitude in the field has had to flow to the curve that now consolidates the original-plaintext’s dependency (mutual information measure): ie 1 dependent line in the plane, that corresponds to the normalizer. While the information from the plaintext has flowed to the amplitudes of the normalizer elements, one recalls that the normalizer commutes with everything, so the particular log-scores associated with each amplitude doesn’t really matter, there being no uncertainty that the sum of the ratios, in such a normed division algebra, is exactly 1. Those ratios and the product of them are, of course, the bayes factors in the urn model that captures the statistical model of a random repeat rate (finding beetles in depths, that are the “physical” representation of the superposition of the basis operators-kets in his quantum algorithm).
Fascinating. It would be scientifically valuable to study any “secret” historical documents concerning Turing’s interaction with electronics engineers in the period of the design of the spider (pre bombe, that is), when he wanted them (rather than relying on the diagonal boards properties) to be using signals at different phases and thus be performing a quantum algorithm in phase space, as electron flows within a single valve flowed according to the field of the particular particle type. At this point, Turing is thinking in terms of the physics of valves/tubes, rather than as he thought later after having become self-trained in more the electronic engineering side of valves, having learned circuit design.
Don’t suppose GCHQ will; and they certainly wont do it for the likes of me!