When performing maximum-likelihood quantum-state tomography, one must find
the quantum state that maximizes the likelihood of the state given observed
measurements on identically prepared systems. The optimization is usually
performed with iterative algorithms. This paper provides a gradient-based upper
bound on the ratio of the true maximum likelihood and the likelihood of the
state of the current iteration, regardless of the particular algorithm used.
This bound is useful for formulating stopping rules for halting iterations of
maximization algorithms. We discuss such stopping rules in the context of
determining confidence regions from log-likelihood differences when the
differences are approximately chi-squared distributed.