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

Abstract

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 ρ^(M) and ρ^(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 ρ^(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.