Bipartite representations and many-body entanglement of pure states of $N$ indistinguishable particles

We analyze a general bipartite-like representation of arbitrary pure states of $N$ indistinguishable particles, valid for both bosons and fermions, based on $M$- and $(N-M)$-particle states. It leads to exact $(M,N-M)$ Schmidt-like expansions of the state for any $M<N$ and is directly related to the isospectral reduced $M$- and $(N-M)$-body density matrices ${\rho }^{(M)}$ and ${\rho }^{(N-M)}$. The formalism also allows for reduced yet still exact Schmidt-like decompositions associated with blocks of these densities, in systems having a fixed fraction of the particles in some single particle subspace. Monotonicity of the ensuing $M$-body entanglement under a certain set of quantum operations is also discussed. Illustrative examples in fermionic and bosonic systems with pairing correlations are provided, which show that in the presence of dominant eigenvalues in ${\rho }^{(M)}$, approximations based on a few terms of the pertinent Schmidt expansion can provide a reliable description of the state. The associated one- and two-body entanglement spectrum and entropies are also analyzed.