Some time ago I asked whether a system entropy below k ln 2 has ever been measured. There were no answers. Also, I found nothing in google scholar.
I found a statement by DeWitt, who wrote that any system is either there or not. Thus, he argues, every system has at least an entropy k ln 2.
On the other hand, many argue that a system that is in its ground state, if the ground state is unique, has an entropy k ln 1 = 0, and thus has vanishing entropy.
So what is the lowest entropy of a system: k ln 2 or zero?
See: B. DeWitt, Quantum field theory in curved space-time, Physics Reports 19, 295 (1975).