Generalized Wigner-von Neumann entropy and its typicality.

Area of Science:

• Quantum Information Theory
• Statistical Mechanics
• Mathematical Physics

Background:

• The Wigner-von Neumann entropy is a fundamental concept in quantum mechanics.
• Existing entropy measures have limitations for certain quantum states and systems.

Purpose of the Study:

• To propose a generalized Wigner-von Neumann (GWvN) entropy.
• To extend entropy definitions to both pure and mixed quantum states.
• To explore the properties and applications of this generalized entropy.

Main Methods:

• Developing a generalized mathematical framework for quantum entropy.
• Analyzing the behavior of the GWvN entropy for large Hilbert space dimensions.
• Investigating the dynamic evolution and ensemble properties of the GWvN entropy.

Main Results:

• The GWvN entropy is basis-independent for large N and asymptotically approaches lnN.
• Its dynamic evolution is typical and resembles von Neumann's quantum H theorem.
• The GWvN entropy is additive for composite systems and consistent with Boltzmann and von Neumann entropies.

Conclusions:

• The proposed GWvN entropy offers a unified and typical measure for quantum systems.
• It provides a foundation for deriving statistical ensembles like the Gibbs ensemble.
• This generalization enhances our understanding of entropy in quantum information and statistical mechanics.