  Simple examples are constructed that show the entanglement of two qubits
being both increased and decreased by interactions on just one of them. One of
the two qubits interacts with a third qubit, a control, that is never entangled
or correlated with either of the two entangled qubits and is never entangled,
but becomes correlated, with the system of those two qubits. The two entangled
qubits do not interact, but their state can change from maximally entangled to
separable or from separable to maximally entangled. Similar changes for the two
qubits are made with a swap operation between one of the qubits and a control;
then there are compensating changes of entanglement that involve the control.
When the entanglement increases, the map that describes the change of the state
of the two entangled qubits is not completely positive. Combination of two
independent interactions that individually give exponential decay of the
entanglement can cause the entanglement to not decay exponentially but,
instead, go to zero at a finite time.
