On the second law of thermodynamics in isolated quantum systems

The second law of thermodynamics states that over time, the entropy of an isolated system increases to an equilibrium. But a truly isolated system, quantum or otherwise, must have recurrences – it must eventually return to its initial state. Classically, resolutions of this issue can be found in arguments about likelihood of states and initial conditions. But in the quantum regime there is a further issue: the most widely-used definition of quantum entropy, the von Neumann entropy, is constant for isolated quantum systems. This raises the question: in what sense does the entropy of an isolated quantum system actually satisfy the second law? In this talk I will present a solution to this based on the entropy of observables – their Shannon entropy. In doing so I will show that a variant of the second law is recovered: the entropy relative to a given observable for isolated quantum systems tends towards its equilibrium value (with fluctuations occurring after equilibrium is achieved). Analytically-derived bounds on entropy equilibration will be presented, alongside numerical illustrations of these arguments via a one-dimensional quantum Ising model chain of spins.