A formulation of density functional theory (DFT) is constructed as an application of the method of maximum entropy for an inhomogeneous fluid in thermal equilibrium. The use of entropy as a systematic method to generate optimal approximations is extended from the classical to the quantum domain. This process introduces a family of trial density operators that are parameterized by the particle density. The optimal density operator is that which maximizes the quantum entropy relative to the exact canonical density operator. This approach reproduces the variational principle of DFT and allows a simple proof of the Hohenberg-Kohn theorem at finite temperature. Finally, as an illustration, we discuss the Kohn-Sham approximation scheme at finite temperature.A formulation of density functional theory (DFT) is constructed as an application of the method of maximum entropy for an inhomogeneous fluid in thermal equilibrium. The use of entropy as a systematic method to generate optimal approximations is extended from the classical to the quantum domain. This process introduces a family of trial density operators that are parameterized by the particle density. The optimal density operator is that which maximizes the quantum entropy relative to the exact canonical density operator. This approach reproduces the variational principle of DFT and allows a simple proof of the Hohenberg-Kohn theorem at finite temperature. Finally, as an illustration, we discuss the Kohn-Sham approximation scheme at finite temperature.

arXiv:2207.10602EntropyDec 21, 20236 citations

Smoothing in Maximum Quantum Entropy

Timothy C. Wallstrom

Springer eBooksJan 1, 19961 citation

Fermion Ensembles of Maximum Entropy

Martin C. Gutzwiller

Ensembles of particles obeying Fermi-Dirac statistics are considered from a point of view which is analogous to some recent work of Mayer on classical particle statistics. The density functions ${\ensuremath{\rho}}_{n}$ are defined for $n$ much smaller than the average number of particles in the ensemble. Since the knowledge of only a few ${\ensuremath{\rho}}_{n}$ is sufficient for the computation of most averages of physical interest, it is important to compare different ensembles with the same density functions, say ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$. The ensemble with the largest entropy is constructed as being the most significant. A number of simple examples are briefly considered.Ensembles of particles obeying Fermi-Dirac statistics are considered from a point of view which is analogous to some recent work of Mayer on classical particle statistics. The density functions ${\ensuremath{\rho}}_{n}$ are defined for $n$ much smaller than the average number of particles in the ensemble. Since the knowledge of only a few ${\ensuremath{\rho}}_{n}$ is sufficient for the computation of most averages of physical interest, it is important to compare different ensembles with the same density functions, say ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$. The ensemble with the largest entropy is constructed as being the most significant. A number of simple examples are briefly considered.

Physical ReviewMar 1, 19623 citations

Classical and quantum entropy

Benjamin W. Schumacher, Michael D. Westmoreland

Cambridge University Press eBooksMar 25, 20100 citations

Efficiently Compressible Density Operators via Entropy Maximization

Serena Di Giorgio, Paulo Mateus

We address the problem of efficiently and effectively compress density operators (DOs), by providing an efficient procedure for learning the most likely DO, given a chosen set of partial information. We explore, in the context of quantum information theory, the generalisation of the maximum entropy estimator for DOs, when the direct dependencies between the subsystems are provided. As a preliminary analysis, we restrict the problem to tripartite systems when two marginals are known. When the marginals are compatible with the existence of a quantum Markov chain (QMC) we show that there exists a recovery procedure for the maximum entropy estimator, and moreover, that for these states many well-known classical results follow. Furthermore, we notice that, contrary to the classical case, two marginals, compatible with some tripartite state, might not be compatible with a QMC. Finally, we provide a new characterisation of quantum conditional independence in light of maximum entropy updating. At this level, all the Hilbert spaces are considered finite dimensional.We address the problem of efficiently and effectively compress density operators (DOs), by providing an efficient procedure for learning the most likely DO, given a chosen set of partial information. We explore, in the context of quantum information theory, the generalisation of the maximum entropy estimator for DOs, when the direct dependencies between the subsystems are provided. As a preliminary analysis, we restrict the problem to tripartite systems when two marginals are known. When the marginals are compatible with the existence of a quantum Markov chain (QMC) we show that there exists a recovery procedure for the maximum entropy estimator, and moreover, that for these states many well-known classical results follow. Furthermore, we notice that, contrary to the classical case, two marginals, compatible with some tripartite state, might not be compatible with a QMC. Finally, we provide a new characterisation of quantum conditional independence in light of maximum entropy updating. At this level, all the Hilbert spaces are considered finite dimensional.

Italian Quantum Information Science Conference (IQIS)Aug 2, 20191 citation

n

much smaller than the average number of particles in the ensemble. Since the knowledge of only a few ${\ensuremath{\rho}}_{n}$ is sufficient for the computation of most averages of physical interest, it is important to compare different ensembles with the same density functions, say ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$. The ensemble with the largest entropy is constructed as being the most significant. A number of simple examples are briefly considered.