  Mode entanglement in many-body quantum systems is an active area of research.
It provides crucial insight into the suitability of many-body systems for
quantum information processing tasks. Local super-selection rules must be taken
into account when assessing the amount of physically accessible entanglement.
This requires amending well-established entanglement measures by incorporating
local parity and local particle number constraints. In this paper, we report on
mode entanglement present in the analytically solvable system of N-Harmonium.
To the knowledge of the authors, this is the first analytic study of the
physically accessible mode and mode-mode entanglement of an interacting
many-body system in a continuous state space. We find that super-selection
rules dramatically reduce the amount of physically accessible entanglement,
which vanishes entirely in some cases. Our results strongly suggest the need to
re-evaluate intra and inter-mode entanglement in other fermionic and bosonic
systems.
