Keller Colloquium in Computing and Mathematical Sciences
The exponential function on the space of n by n matrices is not monotone, meaning that if H is a Hermitian matrix, and A is a positive matrix (also Hermitian but with all of its eigenvalues non-negative), then exp(H +A) - exp(H) may have negative eigenvalues. This is the source of many problems. One of the fundamental limit theorems in probability is Cramer's Theorem on Large Deviations, which has been used very successfully in classical statistical mechanics. When one tries to apply it in quantum statistical mechanics, the non-monotonicity of the matrix exponential causes trouble. Though many problems remain open, trace inequalities for the exponential and logarithm function are useful tools that have been used effectively by many researchers. Entropy inequalities in particular are a source of exponential trace inequalities an vice versa. In this talk, I will explain the basic mathematical issues related to this topic. The necessary statistical mechanical aspects are easily explained to a mathematical audience without any assuming any prior knowledge of quantum statistical mechanics; the only tools to be used in the lecture will be linear algebra and some elementary convex analysis. I will present some recent results obtained in collaboration with Elliott Lieb, Jan Maas and Anna Versynina, and will also explain several open problems.