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Statistically efficient tomography of low rank states with incomplete measurements

The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random selection of settings, and that the number of settings can be significantly reduced with only a negligible increase in error. We present an argument to explain these findings based on a concentration inequality for the Fisher information matrix. In the more general setup of random basis measurements we use this argument to show that for certain rank $r$ states it suffices to measure in $O(r\log d)$ bases to achieve the average Fisher information over all bases. We present numerical evidence for states upto 8 atoms, supporting a conjecture on a lower bound for the Fisher information which, if true, would imply a similar behaviour in the case of Pauli bases. The relation to similar problems in compressed sensing is also discussed.