Conditional states and entropy in qudit-qubit systems

We examine the set of conditional qubit states and the ensuing conditional entropy in some relevant correlated qudit-qubit mixed states, arising after a general projective measurement at the qudit. While for a projective measurement at the qubit, the conditional post-measurement qudit states lie on the surface of an ellipsoid, for a measurement at the qudit the set of post-measurement qubit states can form more complex solid regions. In particular, we show the emergence, for some simple classes of mixed states, of sets which are the convex hull of solid ellipsoids, which may lead to cone-like and triangle-like shapes in limit cases. We also provide a full analytic determination of the minimum conditional entropy and the associated minimizing local measurement at the qudit for the previous states. Separable rank 2 mixtures are also discussed.