  In static classical statistical systems the problem of information transport
from a boundary to the bulk finds a simple description in terms of wave
functions or density matrices. While the transfer matrix formalism is a type of
Heisenberg picture for this problem, we develop here the associated
Schr\"odinger picture that keeps track of the local probabilistic information.
The transport of the probabilistic information between neighboring
hypersurfaces obeys a linear evolution equation, and therefore the
superposition principle for the possible solutions. Operators are associated to
local observables, with rules for the computation of expectation values similar
to quantum mechanics. We discuss how non-commutativity naturally arises in this
setting. Also other features characteristic of quantum mechanics, such as
complex structure, change of basis or symmetry transformations, can be found in
classical statistics once formulated in terms of wave functions or density
matrices. We construct for every quantum system an equivalent classical
statistical system, such that time in quantum mechanics corresponds to the
location of hypersurfaces in the classical probabilistic ensemble. For suitable
choices of local observables in the classical statistical system one can, in
principle, compute all expectation values and correlations of observables in
the quantum system from the local probabilistic information of the associated
classical statistical system. Realizing a static memory material as a quantum
simulator for a given quantum system is not a matter of principle, but rather
of practical simplicity.
