  A strong entanglement monotone, which never increases under local operations
and classical communications (LOCC), restricts quantum entanglement
manipulation more strongly than the usual monotone since the usual one does not
increase on average under LOCC. We propose new strong monotones in mixed-state
entanglement manipulation under LOCC. These are related to the decomposability
and 1-positivity of an operator constructed from a quantum state, and reveal
geometrical characteristics of entangled states. These are lower bounded by the
negativity or generalized robustness of entanglement.
