## Statistically efficient tomography of low rank states with incomplete measurements

Acharya, Anirudh and Kypraios, Theodore and Guţă, Mădălin (2016) Statistically efficient tomography of low rank states with incomplete measurements. New Journal of Physics, 18 (4). 043018. ISSN 1367-2630

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## Abstract

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 $O(r\mathrm{log}d)$ bases to achieve the average Fisher information over all bases. We present numerical evidence for random states of up to eight atoms, which suggests that a similar behaviour holds in the case of Pauli bases measurements, for randomly chosen states. The relation to similar problems in compressed sensing is also discussed.

Item Type: Article https://nottingham-repository.worktribe.com/output/785323 University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences https://doi.org/10.1088/1367-2630/18/4/043018 Eprints, Support 02 Oct 2017 11:18 04 May 2020 17:46 http://eprints.nottingham.ac.uk/id/eprint/46912

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