Statistical analysis of compressive low rank tomography with random measurements

Acharya, Anirudh and Guţă, Mădălin (2017) Statistical analysis of compressive low rank tomography with random measurements. Journal of Physics A: Mathematical and Theoretical, 50 (19). p. 195301. ISSN 1751-8121

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Abstract

We consider the statistical problem of 'compressive' estimation of low rank states (r«d ) with random basis measurements, where r, d are the rank and dimension of the state respectively. We investigate whether for a fixed sample size N, the estimation error associated with a 'compressive' measurement setup is 'close' to that of the setting where a large number of bases are measured. We generalise and extend previous results, and show that the mean square error (MSE) associated with the Frobenius norm attains the optimal rate rd/N with only O(rlogd) random basis measurements for all states. An important tool in the analysis is the concentration of the Fisher information matrix (FIM). We demonstrate that although a concentration of the MSE follows from a concentration of the FIM for most states, the FIM fails to concentrate for states with eigenvalues close to zero.

We analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the MSE about its optimal despite a lack of concentration of the FIM for states close to the boundary of the Bloch sphere. We also consider the estimation error in terms of a different metric–the quantum infidelity. We show that a concentration in the mean infidelity (MINF) does not exist uniformly over all states, highlighting the importance of loss function choice. Specifically, we show that for states that are nearly pure, the MINF scales as 1/√N but the constant converges to zero as the number of settings is increased. This demonstrates a lack of 'compressive' recovery for nearly pure states in this metric.

Item Type: Article
Additional Information: This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1751-8121/aa682e/meta
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1088/1751-8121/aa682e
Depositing User: Eprints, Support
Date Deposited: 22 Mar 2017 08:48
Last Modified: 13 Oct 2017 01:24
URI: http://eprints.nottingham.ac.uk/id/eprint/41454

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