  In this study, we enhance the understanding of entanglement transformations
and their quantification by extending the concept of Schmidt vector from pure
to mixed bipartite states, exploiting the lattice structure of majorization.
The Schmidt vector of a bipartite mixed state is defined using two distinct
methods: as a concave roof extension of Schmidt vectors of pure states, or
equivalently, from the set of pure states that can be transformed into the
mixed state through local operations and classical communication (LOCC). We
demonstrate that the Schmidt vector fully characterizes separable and maximally
entangled states. Furthermore, we prove that the Schmidt vector is monotonic
and strongly monotonic under LOCC, giving necessary conditions for conversions
between mixed states. Additionally, we extend the definition of the Schmidt
rank from pure states to mixed states as the cardinality of the support of the
Schmidt vector and show that it is equal to the Schmidt number introduced in
previous work [Phys. Rev. A 61, 040301 (R), 2000]. Finally, we introduce a
family of entanglement monotones by considering concave and symmetric functions
applied to the Schmidt vector.
