Turing Head Coherence
Any operator in an n-dimensional Hilbert space can be represented by the generating operators of the special group SU(n), . Specifically, this can be done for the density matrix . Consider the set , , being the SU(n) generators.
Given a density matrix , the associated coherence vector (a.k.a. generalized Bloch vector) is given as the -dimensional vector of expectation values Tr(). In general one has (see Chapter 2 of  for full details):
where , , and is the transition operator between the basis states and in .
The length of is constrained by .
One can classify a density matrix in terms of as follows.
We calculate the coherence vector for the Turing head of our sample machine. (Note: calculating the multi-node coherence vector for the total machine state is beyond the present paper. See [21, 22] for results in that direction.)
Since the Turing head of our sample machine lives in a nine-dimensional Hilbert space , we have to consider 80-dimensional coherence vectors:
Since the and are straightforward, we list the eight only:
The following table summarizes the results for the Turing head coherence vector computation of the above-mentioned QTM for several iterations.
This shows that, outside the excited interaction interval, the Turing head is a pure state and advances to a coherent superposition in the interaction zone before going back to a pure state. Looking to the step operator of the Turing machine, one can see that the head shift operator transforms the pure head state into a coherent superposition at the first marker and the adjoint of collapses the coherent superposition back to a pure basis state.
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