  We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz
quantization, of certain constraints on Poisson brackets coming from "hard"
symplectic geometry. It turns out that they can be interpreted in terms of the
quantum noise of observables and their joint measurements in operational
quantum mechanics. Our findings include various geometric mechanisms of quantum
noise production and a noise-localization uncertainty relation. The methods
involve Floer theory and Poisson bracket invariants originated in function
theory on symplectic manifolds.
