  In this paper we study the entanglement in symmetric $N$-quDit systems. In
particular we use generalizations to $U(D)$ of spin $U(2)$ coherent states and
their projections on definite parity $\mathbb{C}\in\mathbb{Z}_2^{D-1}$
(multicomponent Schr\"odinger cat) states and we analyse their reduced density
matrices when tracing out $M<N$ quDits. The eigenvalues (or Schmidt
coefficients) of these reduced density matrices are completely characterized,
allowing to proof a theorem for the decomposition of a $N$-quDit Schr\"odinger
cat state with a given parity $\mathbb{C}$ into a sum over all possible
parities of tensor products of Schr\"odinger cat states of $N-M$ and $M$
particles. Diverse asymptotic properties of the Schmidt eigenvalues are studied
and, in particular, for the (rescaled) double thermodynamic limit
($N,M\rightarrow\infty,\,M/N$ fixed), we reproduce and generalize to quDits
known results for photon loss of parity adapted coherent states of the harmonic
oscillator, thus providing an unified Schmidt decomposition for both
multi-quDits and (multi-mode) photons. These results allow to determine the
entanglement properties of these states and also their decoherence properties
under quDit loss, where we demonstrate the robustness of these states.
