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Hybrid reconstruction of quantum density matrix: when low-rank meets sparsity

  • Kezhi Li
  • Kai Zheng
  • Jingbei Yang
  • Shuang CongEmail author
  • Xiaomei Liu
  • Zhaokai Li
Article

Abstract

Both the mathematical theory and experiments have verified that the quantum state tomography based on compressive sensing is an efficient framework for the reconstruction of quantum density states. In recent physical experiments, we found that many unknown density matrices in which people are interested in are low-rank as well as sparse. Bearing this information in mind, in this paper we propose a reconstruction algorithm that combines the low-rank and the sparsity property of density matrices and further theoretically prove that the solution of the optimization function can be, and only be, the true density matrix satisfying the model with overwhelming probability, as long as a necessary number of measurements are allowed. The solver leverages the fixed-point equation technique in which a step-by-step strategy is developed by utilizing an extended soft threshold operator that copes with complex values. Numerical experiments of the density matrix estimation for real nuclear magnetic resonance devices reveal that the proposed method achieves a better accuracy compared to some existing methods. We believe that the proposed method could be leveraged as a generalized approach and widely implemented in the quantum state estimation.

Keywords

Quantum state estimation Compressive sensing Low-rank Sparse Reconstruction Fixed-point equation 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61573330).

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of AutomationUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Department of Electrical and Electronic EngineeringImperial College LondonLondonUK
  3. 3.Department of PhysicsUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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