  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 fixed
weights $w_k$ 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 $\rho_{\mathbf{w}}$
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 $ 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 $d_{\pm}\equiv
d_{\pm}^{(Q)}(\mathbf{w},\mathbf{E})$ provide valuable insights into the
optimal choice of the auxiliary weights $w_k$ in practical applications.
