  Entanglement monotone is defined as a convex measure of entanglement that
does not increase on average under local operations and classical communication
(LOCC). Here we call an entanglement monotone a strict entanglement monotone
(SEM) if it decreases strictly on average under LOCC. We show that, for any
convex roof extended entanglement monotone that on pure states is given by a
function of the reduced states, if the function is strictly concave, then it is
a SEM. Moreover, we prove that the negativity and the relative entropy of
entanglement, which are not defined by the convex roof structure, are also
SEMs. In addition, if the squashed entanglement could be obtained by some
optimal extension, then it is a SEM as well. Our results imply that
entanglement is strictly decreasing on average under LOCC.
