Entanglement in fermionic Fock space

Abstract

We propose a generalization of the usual SLOCC and LU classification of entangled pure state fermionic systems based on the Spin group. Our generalization uses the fact that there is a representation of this group acting on the fermionic Fock space which when restricted to fixed particle number subspaces recovers naturally the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space changing the particle number. The classification scheme built on the Spin group prohibits naturally entanglement between states containing even and odd number of fermions. In our scheme the problem of classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so called pure spinors. We construct the basic invariants of the Spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states. An intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to configurations of wrapped membranes reinterpreted as qubits.