Abstract: In this talk, I will introduce the basic setup of quantum measurement theory, and explain how it can be realized over projective manifolds via the use of Berezin-Toeplitz quantization. In particular, I will introduce various ways of describing the associated quantum noise, and show how it arises naturally from the Riemannian geometry of the manifold. I will then present some applications to Donaldson's program towards the search for canonical metrics in Kähler geometry. This project is part of a collaboration with Leonid Polterovich, and includes joint works with Victoria Kaminker, David Kazhdan and Dor Shmoish.