Entanglement in fermion systems

Abstract

We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with fixed number parity yet not necessarily fixed particle number. The "mode entanglement" between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined in [J. Schliemann et al, Phys. Rev. A 64, 022303 (2001)], implying that "particle entanglement" may be seen as minimum "mode entanglement". It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that of the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy are evaluated analytically, extending the results of previous ref. to general states.

Fermionic entanglement in the Lipkin model

“…In [32] an entropic measure of mode entanglement was introduced, which, after minimization over all sp bases, becomes a measure of fermionic entanglement, vanishing iff the state is a SD. The ensuing quantity is based upon the one-body density matrix and represents essentially a minimum distance to a fermionic Gaussian state [35].…”

Spectral properties of reduced fermionic density operators and parity superselection rule

“…For instance, bipartitions with respect to different spin components were considered recently [51]. Mathematically, we deal with the algebraic bipartition in the second quantized description [52][53][54][55][56], which is used in the study of entanglement [45,[57][58][59][60].…”

Multipartite concurrence for identical-fermion systems

“…In the fermionic case, there exists some extended consensus that a pure fermion state is separable if it is a single antisymmetric product state given by a single Slater determinant [5][6][7][8][9][10][11]. In this paper we adopt such a point of view and consider entanglement in systems of identical fermions, meaning entanglement between particles and not entanglement between modes [12].…”