Publication: Least-bias state estimation with incomplete unbiased measurements
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2015-11
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American Physical Society
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Abstract
Measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high-dimensional quantum systems. The unbiased nature of these bases optimizes the uncertainty hypervolume. However, imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von Neumann entropy, a popular figure of merit in this context. Furthermore, this imposition typically leads to mock density matrices that are not even positive definite. This provides a strong argument against perfunctory applications of linear estimation strategies. We propose to use instead the physical state estimators that maximize the Shannon entropy of the unmeasured outcomes, which quantifies our lack of knowledge fittingly and gives physically meaningful statistical predictions.
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©2015 American Physical Society. Many of the ideas in this paper originated at the Workshop on Mathematical Methods of Quantum Tomography at Fields Institute (Toronto). Z.H., J.R., and Y.S.T. are grateful for the support of the European Social Fund and the state budget of the Czech Republic [Project No. CZ.1.07/2.3.00/30.0004 (POST-UP)], the Grant Agency of the Czech Republic (Grant No. 15-031945), the IGA Project of the Palacky University (Grant No. IGA PrF 2015-002), and the sustainability of postdoc positions at Palacky University. L.L. S.S. acknowledges the support from UCM-Banco Santander Program (Grant No. GR3/14) and helpful discussions with M. Grassl. H.K.N.'s, J.H.C.'s, and B.-G.E.'s work was funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and the National Research Foundation of Singapore. H.K.N. was also funded by a Yale-NUS College start-up grant.
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