The team defines thermal equilibrium and macroscopic operations using quantities such as energy and magnetization. It introduces a quantum entropy compatible with thermodynamic entropy within that framework, including for some pure quantum states.
The authors derive two formulations of the second law, including entropy non-decrease under the studied conditions. This is a mathematical proof, with no new experimental device announced.
RIKEN specifies a limitation: the proof concerns operations whose duration does not increase with system size. Extending it to longer operations requires further conditions; the result is not an assumption-free proof covering every physical situation.