  The paper investigates the non-local property of quantum mechanics in the
quantum hydrodynamic analogy (QHA) given by Madelung. The role of the quantum
potential in generating the non-local dynamics of quantum mechanics is
analyzed. The work shows how in presence of noise the non-local properties as
well as the quantization of the action are perturbed. The resulting stochastic
QHA dynamics much depend by the strength of the interaction: Strongly bounded
systems (such as linear ones) lead to quantum entangled stochastic behavior,
while weakly bounded ones may be not able to maintain the quantum superposition
of states on large distances and may loose their macro-scale quantum coherence
acquiring the classical stochastic evolution . The work shows that in the frame
of the stochastic approach it is possible to have freedom between quantum
weakly bounded systems. The stochastic QHA model shows that the wave-function
collapse to an eigenstates (deriving by interaction of a quantum microscopic
system with a classical (macroscopic) one) can be described by the model itself
as a kinetic quantum (relaxation) process to a stationary state. Since the time
of the wave function decay to the eigenstate represents the minimum duration
time of a measurement, the minimum uncertainty principle is shown to be
compatible with the relativistic postulate about the light speed as the maximum
velocity of transmission of interaction. About this topic, the paper shows that
the Lorenz invariance of the relativistic quantum potential (coming from the
Dirac equation) enforces the hypothesis that the superluminal transmission of
information are not present in measurements on quantum entangled state.
