  Trends of genuine entanglement in Haar uniformly generated multimode pure
Gaussian states with fixed average energy per mode are explored. A
distance-based metric known as the generalized geometric measure (GGM) is used
to quantify genuine entanglement. The GGM of a state is defined as its minimum
distance from the set of all non-genuinely entangled states. To begin with, we
derive an expression for the Haar averaged value of any function defined on the
set of energy-constrained states. Subsequently, we investigate states with a
large number of modes and provide a closed-form expression for the Haar
averaged GGM in terms of the average energy per mode. Furthermore, we
demonstrate that typical states closely approximate their Haar averaged GGM
value, with deviation probabilities bounded by an exponentially suppressed
limit. We then analyze the GGM content of typical states with a finite number
of modes and present the distribution of GGM. Our findings indicate that as the
number of modes increases, the distribution shifts towards higher entanglement
values and becomes more concentrated. We quantify these features by computing
the Haar averaged GGM and the standard deviation of the GGM distribution,
revealing that the former increases while the latter decreases with the number
of modes.
