Many-body entanglement in fermion systems

We introduce a general bipartite-like representation and Schmidt decomposition of an arbitrary pure state of N indistinguishable fermions, based on states of M<N and (N-M) fermions. It is directly connected with the reduced M- and (N-M)-body density matrices (DMs), which have the same spectrum in such states. The concept of M-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized M-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as post-measurement states. Moreover, such entropy is an upper bound to the average bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of M and N-M effectively distinguishable fermions. Analytic evaluation of the spectrum of M-body DMs in some strongly correlated fermionic states is also provided.