The Annals of Statistics

Asymptotic equivalence of quantum state tomography and noisy matrix completion

Yazhen Wang

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Matrix completion and quantum tomography are two unrelated research areas with great current interest in many modern scientific studies. This paper investigates the statistical relationship between trace regression in matrix completion and quantum state tomography in quantum physics and quantum information science. As quantum state tomography and trace regression share the common goal of recovering an unknown matrix, it is nature to put them in the Le Cam paradigm for statistical comparison. Regarding the two types of matrix inference problems as two statistical experiments, we establish their asymptotic equivalence in terms of deficiency distance. The equivalence study motivates us to introduce a new trace regression model. The asymptotic equivalence provides a sound statistical foundation for applying matrix completion methods to quantum state tomography. We investigate the asymptotic equivalence for sparse density matrices and low rank density matrices and demonstrate that sparsity and low rank are not necessarily helpful for achieving the asymptotic equivalence of quantum state tomography and trace regression. In particular, we show that popular Pauli measurements are bad for establishing the asymptotic equivalence for sparse density matrices and low rank density matrices.

Article information

Ann. Statist., Volume 41, Number 5 (2013), 2462-2504.

First available in Project Euclid: 5 November 2013

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Zentralblatt MATH identifier

Primary: 62B15: Theory of statistical experiments
Secondary: 62P35: Applications to physics 62J99: None of the above, but in this section 65F10: Iterative methods for linear systems [See also 65N22] 65J20: Improperly posed problems; regularization 81P45: Quantum information, communication, networks [See also 94A15, 94A17] 81P50: Quantum state estimation, approximate cloning

Compressed sensing deficiency distance density matrix observable Pauli matrices quantum measurement quantum probability quantum statistics trace regression fine scale trace regression low rank matrix sparse matrix


Wang, Yazhen. Asymptotic equivalence of quantum state tomography and noisy matrix completion. Ann. Statist. 41 (2013), no. 5, 2462--2504. doi:10.1214/13-AOS1156.

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