  Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are
investigated and related. Based on minimal data provided by the underlying
bialgebras of functions defined on e. g the real line R, the abelian finite
group Z_N, and the canonical Heisenberg-Weyl algebra hw, and by introducing
appropriate functionals on those algebras, examples of ARWs are constructed.
These walks involve short and long range transition probabilities as in the
case of R walk, bistochastic matrices as for the case of Z_N walk, or coherent
state vectors as in the case of hw walk. The increase of classical entropy due
to majorization order of those ARWs is shown, and further their corresponding
evolution equations are obtained. Especially for the case of hw ARW, the
diffusion limit of evolution equation leads to a quantum master equation for
the density matrix of a boson system interacting with a bath of quantum
oscillators prepared in squeezed vacuum state. A number of generalizations to
other types of ARWs and some open problems are also stated. Next, QRWs are
briefly presented together with some of their distinctive properties, such as
their enhanced diffusion rates, and their behavior in respect to the relation
of majorization to quantum entropy. Finally, the relation of ARWs to QRWs is
investigated in terms of the theorem of unitary extension of completely
positive trace preserving (CPTP) evolution maps by means of auxiliary vector
spaces. It is applied to extend the CPTP step evolution map of a ARW for a
quantum walker system into a unitary step evolution map for an associated QRW
of a walker+quantum coin system. Examples and extensions are provided.
