Generalized Wigner-von Neumann entropy and its typicality

Abstract

We propose a generalization of the quantum entropy introduced by Wigner and von Neumann in 1929 [Zeitschrift f\"ur Physik 57, 30 (1929)]. Our generalization is applicable to both quantum pure states and mixed states. When the dimension $N$ of the Hilbert space is large, as a result of typicality, this generalized Wigner-von Neumann (GWvN) entropy becomes independent of choices of basis and is asymptotically equal to $\ln N$. The dynamic evolution of our entropy is also typical, reminiscent of quantum H theorem proved by von Neumann. For the microcanonical ensemble, the GWvN entropy is equivalent to the Boltzmann entropy; for a subsystem in a mixed state, the GWvN entropy is equivalent to the familiar von Neumann entropy, which is zero for pure states. The GWvN entropy can be used to derive the Gibbs ensemble.