  Random Quantum States are presently of interest in the fields of quantum
information theory and quantum chaos. Moreover, a detailed study of their
properties can shed light on some foundational issues of the quantum
statistical mechanics such as the emergence of well defined thermal properties
from the pure quantum mechanical description of large many body systems. When
dealing with an ensemble of pure quantum states, two questions naturally arise:
what is the probability density function on the parameters which specify the
state of the system in a given ensemble? And, does there exist a most typical
value of a function of interest in the considered ensemble? Here two different
ensembles are considered: the Random Pure State Ensemble (RPSE) and the Fixed
Expectation Energy Ensemble (FEEE). By means of a suitable parameterization of
the wave function in terms of populations and phases, we focus on the
probability distribution of the populations in such ensembles. A comparison is
made between the distribution induced by the inherent geometry of the Hilbert
Space and an approximate distribution derived by means of the minimization of
the informational functional. While the latter can be analytically handled, the
exact geometrical distribution is sampled by a Metropolis-Hastings algorithm.
The analysis is made for an ensemble of wavefunctions describing an ideal
system composed of n spins 1/2 and reveals the salient differences between the
geometrical and the approximate distributions. The analytical approximations
are proven to be useful tools in order to obtain ensemble averaged quantity. In
particular we focus on the distribution of the Shannon entropy by providing an
explanation of the emergence of a typical value of this quantity in the
ensembles.
