  We derive a general approximate solution to the problem of minimizing the
conditional entropy of a qudit-qubit system resulting from a local projective
measurement on the qubit, which is valid for general entropic forms and becomes
exact in the limit of weak correlations. This entropy measures the average
conditional mixedness of the post-measurement state of the qudit, and its
minimum among all local measurements represents a generalized entanglement of
formation. In the case of the von Neumann entropy, it is directly related to
the quantum discord. It is shown that at the lowest non-trivial order, the
problem reduces to the minimization of a quadratic form determined by the
correlation tensor of the system, the Bloch vector of the qubit and the local
concavity of the entropy, requiring just the diagonalization of a $3\times 3$
matrix. A simple geometrical picture in terms of an associated correlation
ellipsoid is also derived, which illustrates the link between entropy
optimization and correlation access and which is exact for a quadratic entropy.
The approach enables a simple estimation of the quantum discord. Illustrative
results for two-qubit states are discussed.
