Grothendieck applied to entanglement and optimization

Jop Briët, researcher at the Centrum Wiskunde & Informatica (CWI) in Amsterdam, introduces in his thesis new variations of Grothendieck’s inequality. He applied them to entanglement, an aspect of quantum mechanics, and optimization. On 27 October 2011 Briët defends his dissertation at the University of Amsterdam.

Entanglement means that particles at a large distance may show correlations without any transfer of information. Briët studied entanglement by using so-called nonlocal games, an experimental setting in which two or more players who are not allowed to communicate, need to coordinate their strategies. Based on the winning probability of a nonlocal game, the strength of the correlation can be measured.

In a quantum mechanical world, the players from the experiment can use entangled particles to increase their chance of winning. The stronger the correlation, the better the nonlocal games can be played and the stronger the evidence that the world behaves as in quantum mechanics. Briët's research shows that certain nonlocal games have much lower winning probability for the type of entanglement that can currently be realized experimentally, than for entanglement that should theoretically be possible.

Briët applied Grothendieck’s inequality also to optimization, finding the best solution from a wide range of possibilities. Constructing a railway timetable is an example of a difficult optimization problem. Since determining the optimal choice is time-consuming, optimization often focuses on finding the best possible solution within a reasonable timeframe. Briët analyzed the quality of algorithms that offer precisely such an alternative.

Unintentionally, Briët also solved a 35-year-old problem in Banach algebras. In 2008, mathematicians already translated this problem into a question about nonlocal games. With a modified version of Grothendieck's inequality, Briët found the last piece of the puzzle needed to solve this problem.

The mathematical tools of Grothendieck are very well known. The mathematician is the founder of the modern theory of algebraic geometry and provided significant contributions to number theory. Grothendieck himself disappeared without a trace in 1991.