Pure states, quantum filters and ultrafilters

We will describe how the usual notion of an ultrafilter on N extends to the notion of a maximal quantum filter. Such objects correspond to pure states of quantum systems the same way that ultrafilters correspond to points of  the Cech-Stone compactification of the integers linking set-theory with the C*-algebraic formalism of quantum physics.

In the second part of the talk we will describe the main idea of our negative solution to Anderson's conjecture concerning the possibility of describing every maximal quantum filter by an ultrafilter on N and an orthonormal basis of the Hilbert space. The proof uses an old idea of Fichtenholz and Kantorovich of a construction of an independent family of subsets of N of size continuum. The second part is based on the paper
"P. Koszmider;  A non-diagonalizable pure state; Proc. Natl. Acad. Sci. USA, 2020, 117 (52) 33084-33089".