Geometry of quantum inference

The principle of maximum entropy has been a central tool in thermodynamics, data reduction and inverse problems with noisy data. Unfortunately, it has a natural bias towards the uniform distribution, which it picks out in the absence of any other information. For classical systems the generalization to the principle of minimum Kullback information allows for arbitrary prior information. Even though the quantum mechanical version of the Kullback information has no known closed form it is still a useful tool: we use it to define an information theoretic measure of the quantum ''entanglement'' of a pair of systems in a pure state; further, using the geometry of statistical correlations we derive trajectories which closely approximate the optimal quantum inference from prior to posterior, finally, we use this geometry to obtain a near optimal detection scheme in binary communication.