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The extension of the Rayleigh-Ritz variational principle to ensemble states$\rho_{\mathbf{w}}\equiv\sum_k w_k |\Psi_k\rangle \langle\Psi_k|$ with fixedweights $w_k$ lies ultimately at the heart of several recent methodologicaldevelopments for targeting excitation energies by variational means. Prominentexamples are density and density matrix functional theory, Monte Carlosampling, state-average complete active space self-consistent field methods andvariational quantum eigensolvers. In order to provide a sound basis for allthese methods and to improve their current implementations, we prove thevalidity of the underlying critical hypothesis: Whenever the ensemble energy iswell-converged, the same holds true for the ensemble state $\rho_{\mathbf{w}}$as well as the individual eigenstates $|\Psi_k\rangle$ and eigenenergies $E_k$.To be more specific, we derive linear bounds $d_-\Delta{E}_{\mathbf{w}} \leq\Delta Q \leq d_+ \Delta\Delta{E}_{\mathbf{w}}$ on the errors $\Delta Q $ ofthese sought-after quantities. A subsequent analytical analysis and numericalillustration proves the tightness of our universal inequalities. Our resultsand particularly the explicit form of $d_{\pm}\equivd_{\pm}^{(Q)}(\mathbf{w},\mathbf{E})$ provide valuable insights into theoptimal choice of the auxiliary weights $w_k$ in practical applications.

Ground and Excited States from Ensemble Variational Principles