Before revision:  While a variational theorem bounds the GMRQ from above by the sum of the first m eigenvalues of the system's propagator, we show that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty.
After revision:  It is shown that a variational theorem bounds the GMRQ from above by the sum of the first m eigenvalues of the system's propagator, we show that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty.