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

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