  Genuine multimode entanglement in continuous variable systems can be
quantified by exploring the geometry of the state-space, namely via the
generalized geometric measure (GGM) which is defined as the shortest distance
of a given multimode state from a nongenunely multimode entangled state. For
multimode Gaussian states, we derive a closed form expression of GGM in terms
of the symplectic invariants of the reduced states. Following that
prescription, the characteristics of GGM for typical three- and four-mode
Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM
for photon-added as well as -subtracted states having three- and four-modes and
find that both addition and subtraction of photons lead to enhancement of the
genuine multimode entanglement content of the state compared to its Gaussian
counterpart. Our analysis reveals that when an initial three-mode vacuum state
is evolved according to an interacting Hamiltonian, photon addition is more
beneficial in increasing GGM compared to photon subtraction while the picture
is opposite for four-mode case. Specifically, subtracting photons from
four-mode squeezed vacuum states almost always results in higher multimode
entanglement content than that of photon addition with both single as well as
multimode and constrained as well as unconstrained operations. Furthermore, we
observe a novel freezing feature of GGM during some specific cases of photon
subtraction with respect to the number of photons subtracted.
