I got it mostly right in https://yorkporc.wordpress.com/2014/01/22/modeling-turings-quantum-machine-in-terms-of-the-quantum-harmonic-oscillator/. But, we can do better.
Turing was not modeling the use of angular momentum to generate wave functions in qu-bits but in qu-dits.
Thus, when he made his argument about a+m.b.a-m he was reallyt talking about a series of a- and a+ operators acting on the eigenstates . Whereas in a qubit world typically taught when introducing the quantum mechanics spin operators and the spin/orbital angular momentum algebra we have just the a+ (and a-) raising (lowering) operator, in a qudit world we get variants of the a+ operator: namely each m of am+ . And similarly for a-.
We know from the angular momentum algebra, which is really all about wave function generator polynomials, that [Lx, Ly] = … Lz. And, similar relations exist for the spin version of the same algebra.
And we know a+ is (Lx + Ly) and a- is (Lx – Ly).
Turing writes Lx as I, Ly as J and Lz as K. Now, a+m requires the modeling of the sum (I + J) of I and J; and a-m the modeling of a difference (I – J) between I and J. One needs to model 1) the sign, and 2) the m.
1) the sign can be modeled by considering a series of non-stepping enigma drums U, with U1, U2, U3, … being the first 3 of several drums. If U2 is rotated at indicator-time at an offset of –1, relative to U1 we get the usual conjugation of labels (and rod identification in the Friedman table of the wheel). This conjugation relation between the first couplet of wheels can be better expressed, per Turing, as U1.R-1.U2 – where U1 and U2 could be written both as U, since both actually have the same wirings and their relative offset at indicator-time is now captured by the R term.
2) we now consider m, by generalizing the above to qudits rather than qubits. In so doing, we can now rotate the second wheel at indicator-time using R+m or R-m. That is, m offset steps.
Remember that in phase space, we are nominally rotating the entire 7-point geometry m times in the direction of m’s sign.
In the case that a negative R to some m’th power conjugates the current input character, we see loss of a m quanta of photons as the system changes angular momentum level by m levels – which accounts for how the “information” (i.e entropy) in the plaintext can be lost. In the physics case, the energy is given up to the bosionic environment – whereas in the purer information theory sense, we loose entropy given out as electromagnetic/quantum noise (suitable for NSA to capture). What is left is, as in physics, a lower state of energy at equilibrium in the associated wave function represented by number (and position) of the zero crossings in what is a function that might be approximated using the relevant “walsh function”.
We should also interpret his macro argument. When he noted that cauchy-schwartz for a mean = 1 wheel induces bounded averaging (and reduction of complexity), we should also note his argument that the limiting final state in phase space is continuous. That is, he is really modeling the poincare sphere – which has continuous range of state – though the function may collapse to only 2 certain values (in qubit land) or m (in qudit land). The way in which the collapse happens is based on how energy at each of the n’th pole of the m orbit revolutions flows to one of the smaller number of poles in the now m-1 orbit levels. Presumably, there is a nearest neighbor argument to be found, here (or one thinks of Gallagher’s quantization rules in A2D).
The appropriate metaphor for all this REALLY is the cyclotron, rather than an enigma wheel cage, which does similar concentration of entropy when refining uranium oxide into nuclear fuel. And perhaps, with that association, one sees why such early quantum computing models were kept so secret, for so long.
Let’s remember, now in modern crypto, how quantum ciphers rely on changing the series of bases, using unitary operators; and changing from m operations on a bit …to 1 operation on an ‘m-bit’ having changed perspective of m ‘1-bit’ fields to a single ‘m-bit’ field. Of course, there are different ways to represent a 1-bit field, as for m-bit fields. Between changing how the field itself is represented, how how such a field changes from being a number of singularities to a singularity of an index-n polynomial, and how then unitaries built from the non-Hermition S+ and S- operators are applied to change basis, one gets a space for fashioning suitable complexity that defeats classical computation power – in both the cipher making and cryptanalysis worlds. Now, one asks, is a ‘brutish’ quantum “computing” machine such as cyclotron suitable in any way for helping the cryptanalysis or “search” side of the process when applied to classical bits?