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Physical Review A

Variational principles for ensembles of quantum states

A. K. Rajagopal, F. A. Buot

Published Mar 1, 1995• 7 citations
Publisher

Abstract

The maximum-entropy principle is used to generalize the variational principle for the ground state (Rayleigh-Ritz minimization principle) to ensembles of unequally weighted states. A hierarchy of nested inequalities for the associated ``free energies'' is deduced. Applications of these inequalities to the estimation of the highest multiplet energy and a generalization of density-functional theory to ensembles of quantum states are discussed. Previous results given in the literature are obtained as special cases of our results under isothermal and adiabatic optimizations.

DOI
10.1103/physreva.51.1770

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The maximum-entropy principle is used to generalize the variational principle for the ground state (Rayleigh-Ritz minimization principle) to ensembles of unequally weighted states. A hierarchy of nested inequalities for the associated ``free energies'' is deduced. Applications of these inequalities to the estimation of the highest multiplet energy and a generalization of density-functional theory to ensembles of quantum states are discussed. Previous results given in the literature are obtained as special cases of our results under isothermal and adiabatic optimizations.
The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues E1{E}_{1}E1​\ensuremath{\le}E2{E}_{2}E2​\ensuremath{\le}...\ensuremath{\le}EM{E}_{M}EM​ of a Hamiltonian H, and given M real numbers w1{w}_{1}w1​\ensuremath{\ge}w2{w}_{2}w2​\ensuremath{\ge}...\ensuremath{\ge}wM{w}_{M}wM​>0, an upper bound for the weighted sum w1{w}_{1}w1​E1{E}_{1}E1​ +w2{w}_{2}w2​E2{E}_{2}E2​+...+wM{w}_{M}wM​EM{E}_{M}EM​ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (w1{w}_{1}w1​=w2{w}_{2}w2​=...=wM{w}_{M}wM​). Applications of the generalized principle are discussed.
The extension of the Rayleigh-Ritz variational principle to ensemble states ρw≡∑kwk|Ψk⟩⟨Ψk| with fixed weights wk lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state ρw as well as the individual eigenstates |Ψk⟩ and eigenenergies Ek . To be more specific, we derive linear bounds d−ΔEw≤ΔQ≤d+ΔEw on the errors ΔQ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of d±≡d±(Q)(w,E) provide valuable insights into the optimal choice of the auxiliary weights wk in practical applications.
A theory of thermodynamics has been recently formulated and derived on the basis of Rényi entropy and its relative versions. In this framework, we define the concepts of partition function, internal energy and free energy, and fundamental quantum thermodynamical inequalities are deduced. In the context of Rényi's thermodynamics, the variational Helmholtz principle is stated and the condition of equilibrium is analyzed. The Rényi maximum entropy principle is formulated and the equality case is discussed. The obtained results reduce to the von Neumann ones when the Rényi entropic parameter ααα approaches 1. The Heisenberg and Schrödinger uncertainty principles on the measurements of quantum observables are revisited. The presentation is self-contained and the proofs only use standard matrix analysis techniques.
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.
The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising, Heisenberg and Hubbard models, superconducting (SC) and superfluid systems, etc. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects and for those who are interested in concrete applications.