Quantum state tomography: Mean squared error matters, bias does not

Abstract

Because of the constraint that the estimators be bona fide physical states, any quantum state tomography scheme - including the widely used maximum likelihood estimation - yields estimators that may have a bias, although they are consistent estimators. Schwemmer et al. (arXiv:1310.8465 [quant-ph]) illustrate this by observing a systematic underestimation of the fidelity and an overestimation of entanglement in estimators obtained from simulated data. Further, these authors argue that the simple method of linear inversion overcomes this (perceived) problem of bias, and there is the suggestion to abandon time-tested estimation procedures in favor of linear inversion. Here, we discuss the pros and cons of using biased and unbiased estimators for quantum state tomography. We conclude that the little occasional benefit from the unbiased linear-inversion estimation does not justify the high price of using unphysical estimators, which are typically the case in that scheme.