  From the consideration of measuring bipartite mixed states by separable pure
states, we introduce algebraic sets in complex projective spaces for bipartite
mixed states as the degenerating locus of the measurement. These algebraic sets
are independent of the eigenvalues and only measure the "position" of
eigenvectors of bipartite mixed states. They are nonlocal invariants ie.,
remaining invariant after local unitary transformations. The algebraic sets
have to be the sum of the linear subspaces if the mixed states are separable,
and thus we give a "eigenvalue-free" criterion of separability.Based on our
criterion, examples are given to illustrate that entangled mixed states which
are invariant under partial transposition or fufill entropy and disorder
criterion of separability can be constructed systematically.We reveal that a
large part of quantum entanglement is independent of eigenvalue spectra and
develop a method to measure this part of quantum enatnglement.The results are
extended to multipartite case.
