On entanglement invariant for a double Jaynes-Cummings model

In a recent paper Phys. Rev. A 76 042313 (2007), Sainz and Bjork introduced an entanglement invariant \mathcalE under evolution for a system of four qubits interacting through two isolated Jaynes-Cummings Hamiltonians. This paper proves that this entanglement invariant \mathcalE is closely connected with the linear entropy between two independent subsystems.