The holographic electron density theorem and quantum similarity measures

How much information about the complete molecule is present in a part of th e molecule? Quantum similarity measures provide comparisons between molecul ar electron densities based on integration over the whole space. Such integ ration involves boundaryless electron densities, whereas an early applicati on of the Hohenberg-Kohn theorem to local subsystems of molecules requires these molecules to be confined to bounded, finite regions of the space. How ever, actual molecules have no boundaries, they are not confined to any fin ite region of the space. In order to find deterministic relations between l ocal and global, boundaryless electron densities, and to classify the link between quantum similarity measures involving the full space and local subs ystems, the unique extension property called the holographic property of su bsystems of complete, boundaryless electron densities is established. Any n onzero volume piece of the ground state electron density completely determi nes the electron density of the complete, boundaryless molecule.