Nicolas Berliat

The entropy of a system is a well-known concept. It describes how energy disperses and helps determine the amount of "useful" energy within the system. In the case of quantum systems, another phenomenon comes into play: Heisenberg's uncertainty principle. Intuitively, this principle should imply a lower bound for the entropy. The problem is that mathematically, classical entropy does not exactly describe what we are looking for. Quantum entropy, or Von Neumann entropy, is closer to what we need, but the possibility of states with zero entropy does not align with the uncertainty principle too. So, what solution do we have? In 1979, Wehrl defined an entropy adapted to this problem, based on coherent states. Coherent states are well-known for their simplicity. They take the form of shifted Gaussians (in the case where the phase space is $\mathbb{R}^2$). Furthermore, coherent states are precisely found to minimize Heisenberg's inequality. We will therefore see how to define these coherent states and their connection to this new notion of entropy, as well as the proof provided by Lieb of the Wehrl conjecture.