Coffman-Kundu-Wootters inequality for fermions

TL;DR

This paper extends the Coffman-Kundu-Wootters inequality to three-fermion systems, identifying fermionic entanglement measures and their relation to known quantum entanglement classes, with implications for understanding fermionic entanglement structure.

Contribution

It derives a new inequality for three fermions, introduces fermionic analogs of entanglement measures, and links these to SLOCC covariants and spinor identities, advancing fermionic entanglement theory.

Findings

Fermionic concurrence implies separability of two-particle reduced states.

Vanishing fermionic concurrence indicates GHZ-class entanglement.

Fierz identities relate fermionic covariants to reduced density matrices.

Abstract

We derive an inequality for three fermions with six single particle states which reduces to the sum of the famous Coffman-Kundu-Wootters inequalities when an embedded three qubit system is considered. We identify the quantities which are playing the role of the concurrence, the three-tangle and the invariant $\det \rho_A+\det \rho_B+\det \rho_C$ for this tripartite system. We show that this latter one is almost interchangeable with the von Neumann entropy and conjecture that it measures the entanglement of one fermion with the rest of the system. We prove that the vanishing of the fermionic "concurrence" implies that the two particle reduced density matrix is a mixture of separable states. Also the vanishing of this quantitiy is only possible in the GHZ class, where some genuie tripartite entanglement is present and in the separable class. Based on this we conjecture that this "concurrence" measures the amount of entanglement between pairs of fermions. We identify the well-known "spin-flipped" density matrix in the fermionic context as the reduced density matrix of a special particle-hole dual state. We show that in general this dual state is always canonically defined by the Hermitian inner product of the fermionic Fock space and that it can be used to calculate SLOCC covariants. We show that Fierz identities known from the theory of spinors relate SLOCC covariants with reduced density matrix elements of the state and its "spin-flipped" dual.