Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems

The Hohenberg–Kohn (HK) theorem for interacting electrons is a cornerstone of modern electronic structure calculations. For a general quantum system, a HK-type Hamiltonian in the form of (Formula presented.) can always be defined, by grouping those terms with fixed or preknown coefficients into an internal part of the Hamiltonian (Formula presented.), and factorizing the remaining as the superposition of a set of Hermitian operators (Formula presented.). It is asked whether the HK theorem can be extended to such a general setting, so that the ground-state expectation values of (Formula presented.) as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. It is shown that the question can be addressed by the invertibility of generalized density correlation matrix (GDCM) defined with respect to the (Formula presented.) operators. This criterion is applied to several representative examples, including the quantum Ising dimer, frustration-free systems, N-level quantum systems and a fermionic Hubbard chain. It is suggested that for a finite-size system, finding an invertible GDCM under one single (Formula presented.) configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space.