Classical probability, quantum probability

In the first part of the talk, basing on linear algebra, we demonstrate certain parallelism between the models of classical and quantum probability theory, and show that fundamental differences of the second reduce to the two properties: «complementarity» and «nonseparability». The property of nonseparability of composite quantum systems, paradoxical from the classical viewpoint, will be illustrated by consideration of correlation inequalities and the Mermin-Peres magic square game.
In the second part the story moves to separable Hilbert space, where probability operator-valued measures are defined. Their particular cases are well-known overcomplete vector systems of the type of coherent states in quantum optics or «wavelets» in the theory of signals. We will comment on the recent solution of the noncommutative analog of the Gaussian maximizers problem which establishes a new optimal property of the coherent states.