Abstract We prove that the Hilbert–Schmidt norm of k -particle reduced density matrices of N -body fermionic states is bounded from above by
CkNk/2 C k N k / 2 - matching the scaling behaviour of Slater determinant states. This generalises a recent result of Christiansen [3] on 2-particle reduced density matrices to higher order density matrices. Moreover, our estimate directly yields a lower bound on the von Neumann entropy and the 2-Rényi entropy of reduced density matrices, thereby providing further insight into conjectures of Carlen–Lieb–Reuvers [2, 8].
We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of
N indistinguishable fermions, based on states of
M<N and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced
M- and $(N\ensuremath{-}M)$-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 postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of
M and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of
M-body DMs in some strongly correlated fermionic states is also provided.