Conservation rules for entanglement transfer between qubits

We consider an entangled but non-interacting qubit pair a1 and b1 that is independently coupled to a set of local qubit systems, aI and bJ, of 0 bit value, respectively. We derive rules for the transfer of entanglement from the pair a1-b1 to an arbitrary pair aI-bJ, for the case of qubit-number conserving local interactions. It is shown that the transfer rule depends strongly on the initial entangled state. If the initial entanglement is in the form of the Bell state corresponding to anti-correlated qubits, the sum of the square of the nonlocal pairwise concurrences is conserved. If the initial state is the Bell state with correlated qubits, this sum can be reduced, even to zero in some cases, to reveal a complete and abrupt loss of all nonlocal pairwise entanglement. We also identify that for the nonlocal bipartitions A-bJ involving all qubits at one location, with one qubit bJ at the other location, the concurrences satisfy a simple addition rule for both cases of the Bell states that the sum of the square of the nonlocal concurrences is conserved.