Math Physics Seminar
One of the founding ideas of quantum information science is the observation that some experimentally verifiable correlations cannot be reproduced in a classical ("local realistic") theory, but require quantum mechanics. In this talk I will tell the story, based on the smallest scenario: just two parties, each of which can choose two different yes/no measurements. Correlations are then represented by points in four dimensions, and geometric features of the relevant sets of correlations can be well visualized in three dimensional sections.
I will give a very detailed description of this geometry, taking the opportunity to introduce some of the concepts and methods that proved to be useful in this and sometimes in more complex scenarios. The minimal case has some beautiful but not generalizable features: Applying the sine function to each coordinate precisely maps the classical correlations to the quantum correlations. Moreover, the quantum set is self-dual.
Characterizing higher correlation sets is in general intractable. I will briefly discuss recent developments on "Tsirelson's problem" that even bring in undecidability issues. I may also adress the question, whether Hilbert spaces over the reals can be proved, by way of correlation inequalities, to be insufficient for quantum mechanics.