Here is the latest version of “Noncommutative analogs of random-real forcing”, the paper that introduces generic quantum states as noncommutative analogs of random real numbers.

Abstract: The measure algebra used in random-real forcing is isomorphic (as is well known) to the projection lattice of a commutative von Neumann algebra $\mathcal{R}$, and it is not difficult to show that a generic filter $G$ on this lattice induces a normal state $\omega_G$ on $\mathcal{R}$, with which it is interdefinable. We show that this construction also works when carried out with noncommutative von Neumann algebras, inducing generic states on them that can be seen as noncommutative analogs of random reals. We consider whether these forcing notions can or must collapse $2^{\aleph_0}$.