  We show that the basic non-relativistic quantum formulations can be derived
from a least observability principle. The principle extends the least action
principle from classical mechanics by factoring in two assumptions. First, the
Planck constant defines the discrete amount of action a physical object needs
to exhibit during its dynamics in order to be observable. Second, there is
constant vacuum fluctuation along a classical trajectory. A novel method is
introduced to define the information metrics that measures additional
observable information due to vacuum fluctuations, which is then converted to
the additional action through the first assumption. Applying the variation
principle to minimize the total actions allows us to elegantly recover the
basic quantum formulations including the uncertainty relation and the
Schr\"{o}dinger equation in both position and momentum representations. Adding
the no preferred representation assumption, we obtain the transformation
formulation between position and momentum representations. The extended least
action principle shows clearly how classical mechanics becomes quantum
mechanics. Furthermore, it is a mathematical tool that can bring in new
results. By defining the information metrics for vacuum fluctuations using more
general definitions of relative entropy, we obtain a generalized
Schr\"{o}dinger equation that depends on the order of relative entropy. The
principle can be applied to derive more advance quantum formalism such as
quantum scalar field theory.
