On the Quantum‐Mechanical Kinetic Energy as a Measure of the Information in a Distribution

Abstract

Standard concepts of information theory, including Shannon's entropy, Fisher's information, Jaynes' principle of entropy maximization, Fisher's locality information matrix, and Kullback and Leibler's information measure, are described and extended to many dimensions as appropriate, to establish precise connections between the many body quantum‐mechanical kinetic energy functional T[Ψ] and information measures. Implications for density functional theory of electronic structure are discussed, and elementary examples are displayed to illustrate the argument. Among the several exact relations obtained, one of special interest is the identity where the first term is the intrinsic accuracy or Fisher's information for locality of the one‐particle density (normalized to 1), \documentclass{article}\pagestyle{empty}\begin{document}$ \rho \left( 1 \right) = \int { \cdot \cdot \cdot \int {|{\rm \psi |}^{\rm 2} d_{{\rm T}_2 } \cdot \cdot \cdot } } \,d_{{\rm T}_{\rm N} ,} $\end{document}, and the second term is the average over the one‐particle density of Fisher's information associated with the conditional density \documentclass{article}\pagestyle{empty}\begin{document}$ f\left( {2,\;3,\; \cdot \cdot \cdot ,\;N|1} \right) \equiv \,|\psi |^2 /\rho \left( 1 \right) $\end{document} that is, the second term is the average over the marginal distribution ρ(1) of the trace of Fisher's information matrix for the distribution f(2,3, …, N|1). Because of this formula, the quantum mechanical variation principle may be precisely stated as a principle of minimal information.