Entropy for Quantum Pure States and Its Dynamical Relaxation

TL;DR

This paper introduces a new entropy measure for quantum pure states based on a quantum phase space representation, demonstrating its relaxation to a maximum value over time and its potential for numerical analysis.

Contribution

It constructs a complete set of Wannier functions to define a quantum phase space entropy for pure states, extending the understanding of quantum entropy and its dynamical behavior.

Findings

Entropy relaxes to a maximum value over time.

The entropy fluctuation is bounded and diminishes with time.

Numerical examples illustrate the entropy's dynamical evolution.

Abstract

We construct a complete set of Wannier functions which 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 in [Zeitschrift f\"ur Physik {\bf 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.