Looking at turnings overall argument form, I now see that he does address pure states. When he talks about those group element with particular discriminators,these are the eigenkets for his rotational (pre-measurement) operator.
His system is still constrained to be looking at lambda2 (max difference from constants etc).
Interesting also that the inner product norm is to him just a (easy sum of squares) measure of the squiggle path one takes, with respect to another vector (including his eigenket vectors, k).
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