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arXiv · 2606.06676

Unified Framework for Functional Theories of Quantum Systems

Chih-Chun Wang·Julia Liebert·Markus Penz·Christian Schilling
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Abstract

We introduce and study a unified framework for density-functional theory and its variants for quantum systems on finite-dimensional Hilbert spaces. These theories seek to reduce the complexity inherent in the many-body quantum problem by describing ground states through reduced variables. The central ingredients of our unified framework are a generalized choice of basic observables, whose expectation values define precisely those reduced variables, and a fixed part of the Hamiltonian characterizing the class of quantum systems under consideration. It is this minimal structure, which we call the scope of a functional theory, that is necessary and sufficient for the formulation of a functional theory. In particular, it allows one to define the universal functionals, establish their convexity and differentiability properties, address representability questions, and prove a Hohenberg-Kohn-type uniqueness result. A purification construction also relates ensemble and weighted-ensemble functionals to the pure-state variant. Particular emphasis is placed on functional theories with Lie-algebra observable structures, connecting the variational framework to symplectic geometry. The result of this work is a systematic mathematical formulation in which structural results can be proved once and applied across a broad class of finite-dimensional functional theories.

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Chih-Chun Wang, Julia Liebert, Markus Penz, Christian Schilling. 2026-06-04. Unified Framework for Functional Theories of Quantum Systems. https://arxiv.org/abs/2606.06676

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