Many-body entanglement in fermion systems

TL;DR

This paper develops a framework for understanding many-body entanglement in fermion systems using a generalized Schmidt decomposition and analyzes the properties of reduced density matrices, revealing new insights into fermionic entanglement and entropy behavior.

Contribution

It introduces a bipartite-like representation and Schmidt decomposition for pure fermion states, connecting to reduced density matrices and defining a natural concept of M-body entanglement.

Findings

Derived majorization relations for M-body density matrices.

Showed entropy bounds for fermionic operations.

Analytic spectra for strongly correlated states.

Abstract

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.