  We show numerical evidence for a bipartite $d\times d$ pure state
entanglement witness that is readily calculated from the wavefunction
coefficients directly, without the need for the numerical computation of
eigenvalues. This is accomplished by using an approximate analytic
diagonalization of the bipartite state that captures dominant contributions to
the negativity of the partially transposed state. We relate this entanglement
witness to the Log Negativity, and show that it exactly agrees with it for the
class of pure states whose quantum amplitudes form a positive Hermitian matrix.
In this case, the Log Negativity is given by the negative logarithm of the
purity of the amplitudes consider as a density matrix. In other cases, the
witness forms a lower bound to the exact, numerically computed Log Negativity.
The formula for the approximate Log Negativity achieves equality with the exact
Log Negativity for the case of an arbitrary pure state of two qubits, which we
show analytically. We compare these results to a witness of entanglement given
by the linear entropy. Finally, we explore an attempt to extend these pure
state results to mixed states. We show that the Log Negativity for this
approximate formula is exact on the class of pure state decompositions for
which the quantum amplitudes of each pure state form a positive Hermitian
matrix.
