  We discuss the cases where local decoherence selectively degrades one type of
entanglement more than other types. A typical case is called state ordering
change, in which two input states with different amounts of entanglement
undergoes a local decoherence and the state with the larger entanglement
results in an output state with less entanglement than the other output state.
We are also interested in a special case where the state with the larger
entanglement evolves to a separable state while the other output state is still
entangled, which we call selective entanglement breaking. For three-level or
larger systems, it is easy to find examples of the state ordering change and
the selective entanglement breaking, but for two-level systems it is not
trivial whether such situations exist. We present a new strategy to construct
examples of two-qubit states exhibiting the selective entanglement breaking
regardless of entanglement measure. We also give a more striking example of the
selective entanglement breaking in which the less entangled input state has
only an infinitesimal amount of entanglement.
