Entanglement versus energy in the entanglement transfer problem

We study the relation between energy and entanglement in an entanglement transfer problem. We first analyze the general setup of two entangled qubits ('a' and 'b') exchanging this entanglement with two other independent qubits ('A' and 'B'). Qubit 'a' ('b') interacts with qubit 'A' ('B') via a spin-exchange-like unitary evolution. A physical realization of this scenario could be the problem of two-level atoms transferring entanglement to resonant cavities via independent Jaynes-Cummings interactions. We study the dynamics of entanglement and energy for the second pair of qubits (tracing out the originally entangled ones) and show that these quantities are closely related. For example, the allowed quantum states occupy a restricted area in a phase diagram entanglement vs energy. Moreover, the curve which bounds this area is exactly the one followed if both interactions are equal and the entire four qubit system is isolated. We also consider the case when the target pair of qubits is subjected to losses and can spontaneously decay