  Recently we have presented a hidden variable model of measurements for a
qubit where the hidden variable state space dimension is one-half the quantum
state manifold dimension. The absence of a short memory (Markov) dynamics is
the price paid for this dimensional reduction. The conflict between having the
Markov property and achieving the dimensional reduction was proved in [A.
Montina, Phys. Rev. A, 77, 022104 (2008)] using an additional hypothesis of
trajectory relaxation. Here we analyze in more detail this hypothesis
introducing the concept of invertible process and report a proof that makes
clearer the role played by the topology of the hidden variable space. This is
accomplished by requiring suitable properties of regularity of the conditional
probability governing the dynamics. In the case of minimal dimension the set of
continuous hidden variables is identified with an object living an
N-dimensional Hilbert space, whose dynamics is described by the Schr\"odinger
equation. A method for generating the economical non-Markovian model for the
qubit is also presented.
