Complex entropy and resultant information measures

Abstract

Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$ , due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, $$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$ are shown to determine the real and imaginary parts of the state complex Shannon entropy, $$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$ a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, $$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$ and the gradient entropy: $$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$