  Amplitude damping changes entangled pure states into usually less-entangled
mixed states. We show, however, that even local amplitude damping of one or two
qubits can result in mixed states more entangled than pure states if one
compares the relative entropy of entanglement (REE) for a given degree of the
Bell-Clauser-Horne-Shimony-Holt inequality violation (referred to as
nonlocality). By applying Monte-Carlo simulations, we find the maximally
entangled mixed states and show that they are likely to be optimal by checking
the Karush-Kuhn-Tucker conditions, which generalize the method of Lagrange
multipliers for this nonlinear optimization problem. We show that the REE for
mixed states can exceed that of pure states if the nonlocality is in the range
(0,0.82) and the maximal difference between these REEs is 0.4. A former
comparison [Phys. Rev. A 78, 052308 (2008)] of the REE for a given negativity
showed analogous property but the corresponding maximal difference in the REEs
is one-order smaller (i.e., 0.039) and the negativity range is (0,0.53) only.
For appropriate comparison, we normalized the nonlocality measure to be equal
to the standard entanglement measures, including the negativity, for arbitrary
two-qubit pure states. We also analyze the influence of the phase-damping
channel on the entanglement of the initially pure states. We show that the
minimum of the REE for a given nonlocality can be achieved by this channel,
contrary to the amplitude damping channel.
