Cina Aghamohammadi, James Crutchfield, John Mahoney, Paul Riechers
Paper #: 15-10-041
Recently it was shown that the causal organization of a classical stochastic process can be substantially compressed using the so-called q-machine representation. The overlaps between quantum signal states play an important role in this compression. At longer length scales, where more of the process’s structure is accounted for, these overlaps become more important and, as it would seem, more difficult to compute. Here we derive useful expressions for these overlaps, including one based on a spectral decomposition, affording us theoretical simplicity as well as greatly improved computational ability. These expressions are based on a new quantum transient structure defined here, the quantum-pairwise-merger-machine (QPMM). Armed with the overlaps, we ultimately proceed to compute the quantum communication cost, defined by the ensemble’s von Neumann entropy. We simplify matters further by making use of a surrogate Gram matrix. We also provide more explicit proofs regarding the connection between the cryptic order of the stochastic process and the optimal coding length of the q-machine.