The birth of a new logic
During the 30's, G. Birkhoff and J. von Neumann discovered that quantum theory could be rephrased in terms of a mathematical framework which resembled that of classical probability theory. The main difference with regard to Kolmogorov's approach to probability was that, instead of a Boolean algebra, a non-Boolean lattice was needed in the quantum case. This was based in the algebraic properties of certain subspaces of the Hilbert space. They proposed to use a structure which is known nowadays as a continuous geometry. Since their discovery, many other structures have been studied, among them, von Neumann algebras, effect algebras, and orthomodular posets.
Our research
We study different algebraic structures related to quantum theory. The main goal is to characterize the quantum state space and the properties of quantum systems in different ways. These include the use of projection operators, convex sets, and non-deterministic semantics. Most of these descriptions are based in the use of lattice theory and more general structures, such as orthomodular posets. As a result, one obtains different formal frameworks which allow to uncover the differences between quantum and classical physics, allowing for a better understanding of what quantumness is.
Our goals
One of the main challenges in the foundations of quantum physics is to understand how quantum physics is different from classical physics. The study of quantum structures can help us to understand what is quantumness. In the field of quantum technologies, one of the main goals is to understand why and in which sense quantum computing is different from classical computing. By looking at the logical structures behind quantum physics we aim to gain clues to understand quantum advantage.
