On entropy and complexity of coherent states

TL;DR

This paper explores the relationship between entropy and complexity in coherent states of the group SL(d+1,C), revealing their geometric origins and demonstrating their connection through Kähler potentials and Calabi's diastasis function.

Contribution

It establishes a geometric framework linking entropy and complexity of coherent states via Kähler and symplectic geometry, highlighting the Fubini-Study metric's optimality.

Findings

Entropy equals the Kähler potential in dual symplectic variables.

Complexity logarithm equals Calabi's diastasis function.

Fubini-Study metric's optimality is analyzed through deformation.

Abstract

Consanguinity of entropy and complexity is pointed out through the example of coherent states of the group $SL(d+1,\C)$. Both are obtained from the K\"ahler potential of the underlying geometry of the sphere corresponding to the Fubini-Study metric. Entropy is shown to be equal to the K\"ahler potential written in terms of dual symplectic variables as the Guillemin potential for toric manifolds. The logarithm of complexity relating two states is shown to be equal to Calabi's diastasis function. Optimality of the Fubini-Study metric is indicated by considering its deformation.