Typicality in ensembles of quantum states: Monte Carlo sampling versus analytical approximations

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 parametrization of the wave function in terms of populations and phases, we focus on the probability distribution of the populations in these 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 wave functions describing an ideal system composed of n spins of 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 an 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.