Investigating Bias in Maximum Likelihood Quantum State Tomography

Maximum likelihood quantum state tomography yields estimators that are consistent, provided that the likelihood model is correct, but the maximum likelihood estimators may have bias for any finite data set. The bias of an estimator is the difference between the expected value of the estimate and the true value of the parameter being estimated. This paper investigates bias in the widely used maximum likelihood quantum state tomography. Our goal is to understand how the amount of bias depends on factors such as the purity of the true state, the number of measurements performed, and the number of different bases in which the system is measured. For that, we perform numerical experiments that simulate optical homodyne tomography under various conditions, perform tomography, and estimate bias in the purity of the estimated state. We find that estimates of higher purity states exhibit considerable bias, such that the estimates have lower purity than the true states.
Comment: New version where we fixed error in evenly spaced phases. Expanded analysis of evenly spaced phases. 8 figures. Submitted to Phys. Rev. A