Mathematics of quantum entropy

15/11/2019, 14:00 - 15:00 in MAP/0G/017

Berta, Mario (Imperial College London)


Entropy is a fundamental, multidisciplinary concept linking a priori unconnected areas of science such as statistical mechanics, thermodynamics, information theory, and theoretical computer science – well-understood in the case of classical systems. In contrast, when it comes to systems described by quantum mechanics, our knowledge about entropy is still rather limited. The reason is that quantum states are entangled and the resulting non-commutativity poses a big challenge for lifting results from the classical to the quantum setting. Entropy inequalities that are known to hold in the non-commutative case, such as the strong sub-additivity, give crucial insights into the entanglement structure of quantum states. In my talk, I will prove refined quantum entropy inequalities based on multivariate trace inequalities that extend the Golden-Thompson and Araki-Lieb-Thirring inequalities to arbitrarily many matrices. I will present applications in physics and computer science: First, I will give rigorous approximation guarantees for thermal states of strongly interacting quantum spin chains, and second, I will discuss the security analysis of quantum cryptographic protocols.

Mathematical Sciences Research Centre