Maximum entropy principle and Landau free energy inequality

ABSTRACT

In this paper, the following problem is considered: given two Hermitian matrices H and K and two real numbers x and y, determine a positive semidefinite matrix ρ such that the von Neumann entropy $-\mathrm{T}\mathrm{r}\rho \log \rho$ is maximum, subject to the condition that $\mathrm{T}\mathrm{r}\rho H=x$ and $\mathrm{T}\mathrm{r}\rho K=y$. This question arises in information theory and in statistical mechanics in connection with the maximum-entropy inference principle. To answer it, we use this principle and numerical range methods.

Keywords:

• Maximum-entropy principle
• Landau free energy inequality
• Von Neumann entropy
• numerical range

2000 AMS Subject Classifications:

• 47A12
• 62F30
• 54C10

Acknowledgments

The authors are grateful to the Referee for bringing ref. [19] to their attention and for constructive criticisms.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.