A characterization of the Helmholtz free energy

Abstract

Let \documentclass[12pt]{minimal}\begin{document}$\mathcal {{{H}}}$\end{document}H be a finite dimensional complex Hilbert space, \documentclass[12pt]{minimal}\begin{document}${\mathbb {B}}(\mathcal {{{H}}})_{+}$\end{document}B(H)+ be the set of all positive semi-definite operators (matrices) on \documentclass[12pt]{minimal}\begin{document}$\mathcal {{{H}}}$\end{document}H and φ is a (not necessarily linear) map of \documentclass[12pt]{minimal}\begin{document}${\mathbb {B}}(\mathcal {{{H}}})_{+}$\end{document}B(H)+ preserving the generalized Helmholtz free energy. In this paper, under suitable conditions we prove that there exists either a unitary or an anti-unitary operator U on \documentclass[12pt]{minimal}\begin{document}$\mathcal {{{H}}}$\end{document}H such that φ(A) = UAU* for any \documentclass[12pt]{minimal}\begin{document}$A \in {\mathbb {B}}(\mathcal {{{H}}})_{+}$\end{document}A∈B(H)+.