Quantum tomography with small number of copies: a simple estimator for qubit states

In quantum tomography, one performs repeated measurements on $N$ copies of a given but unknown state and constructs an estimator for the state from the gathered data. A common way of converting the data into an estimator is the maximum-likelihood (ML) method, where the estimator is the state with the largest probability of giving rise to the observed data. ML methods methods work well for large $N$, since the likelihood function for large $N$ is sharply peaked around its maximum. For small $N$, however, there is a significant neighborhood of states around the maximum with nearly equal probability of giving rise to the data. One can then imagine using the likelihood function as a weight to construct an estimator as an average over states. This motivates the introduction of the ``mean estimator,'' also previously discussed for quantum tomography in the spirit of Bayesian estimation by Blume-Kohout [NJP 12, 043034(2010)]. Here, we extend the mean estimator for a classical die problem to an estimator for qubit states, and demonstrate its advantage over ML estimators. We also discuss a way of overcoming the common complaint of rank-deficiency in ML estimators for our estimator. This simple estimator should be useful as a convenient first estimate for any qubit tomography experiment.