Entropy for quantum pure states and quantumHtheorem

Abstract

We construct a complete set of Wannier functions that are localized at both given positions and momenta. This allows us to introduce the quantum phase space, onto which a quantum pure state can be mapped unitarily. Using its probability distribution in quantum phase space, we define an entropy for a quantum pure state. We prove an inequality regarding the long-time behavior of our entropy's fluctuation. For a typical initial state, this inequality indicates that our entropy can relax dynamically to a maximized value and stay there most of time with small fluctuations. This result echoes the quantum H theorem proved by von Neumann [Zeitschrift für Physik 57, 30 (1929)]. Our entropy is different from the standard von Neumann entropy, which is always zero for quantum pure states. According to our definition, a system always has bigger entropy than its subsystem even when the system is described by a pure state. As the construction of the Wannier basis can be implemented numerically, the dynamical evolution of our entropy is illustrated with an example.

Observation of quantum equilibration in dilute Bose gases

“…And ultimately, can we measure the quantum entropies for quantum pure states defined by von Neumann [3,4] or in Ref. [7].…”

“…As a result, the phases e −iEnt/ will quickly be scrambled as t increases. The dephasing occurs, causing the quantum system to equilibrate [3,4,6,7]. In this way, the quantum system enters the second stage, where besides small fluctuations all the observables become constant.…”

“…According to von Neumann [3,4] and others [6,7], a non-integrable quantum many-body system starting from a well-behaved pure state, such as a Gaussian packet and a Bloch state, will eventually evolve dynamically into an equilibrium state which looks intuitively rather random or irregular. As a result, there are two stages of dynamical evolution.…”

“…According to these two theorems show that most of the non-integrable quantum systems, which are ubiquitous in nature [5], will indeed relax dynamically to an equilibrium state, where the macroscopic observables fluctuate only slightly and the entropy is maximized with small fluctuations. These two theorems have now been improved and put in a more transparent framework and on a firmer footing [6,7].…”

Quantum phase space with a basis of Wannier functions

“…Von Neumann's method was developed in a recent work [10] where Kohn's orthogonalization method [11] was employed and a set of Wannier functions localized at Planck cells were found.…”