The Annals of Statistics

Optimal large-scale quantum state tomography with Pauli measurements

Tony Cai, Donggyu Kim, Yazhen Wang, Ming Yuan, and Harrison H. Zhou

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Quantum state tomography aims to determine the state of a quantum system as represented by a density matrix. It is a fundamental task in modern scientific studies involving quantum systems. In this paper, we study estimation of high-dimensional density matrices based on Pauli measurements. In particular, under appropriate notion of sparsity, we establish the minimax optimal rates of convergence for estimation of the density matrix under both the spectral and Frobenius norm losses; and show how these rates can be achieved by a common thresholding approach. Numerical performance of the proposed estimator is also investigated.

Article information

Ann. Statist., Volume 44, Number 2 (2016), 682-712.

Received: January 2015
Revised: August 2015
First available in Project Euclid: 17 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 81P50: Quantum state estimation, approximate cloning
Secondary: 62C20: Minimax procedures 62P35: Applications to physics 81P45: Quantum information, communication, networks [See also 94A15, 94A17] 81P68: Quantum computation [See also 68Q05, 68Q12]

Compressed sensing density matrix Pauli matrices quantum measurement quantum probability quantum statistics sparse representation spectral norm minimax estimation


Cai, Tony; Kim, Donggyu; Wang, Yazhen; Yuan, Ming; Zhou, Harrison H. Optimal large-scale quantum state tomography with Pauli measurements. Ann. Statist. 44 (2016), no. 2, 682--712. doi:10.1214/15-AOS1382.

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