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High Dimensional Probability VIII pp 385–430Cite as

Birkhäuser

Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems

Part of the Progress in Probability book series (PRPR,volume 74)

Abstract

We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery of the true matrix in Frobenius norm after a data-driven stopping time \(\hat n\) for the number of measurements that have to be taken. With high probability, this stopping time is minimax optimal. We detail applications to quantum tomography problems where measurements arise from Pauli observables. We also give a theoretical construction of a confidence set for the density matrix of a quantum state that has optimal diameter in nuclear norm. The non-asymptotic properties of our confidence sets are further investigated in a simulation study.

Keywords

  • Low rank recovery
  • Quantum information
  • Confidence sets
  • Sequential sampling

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Fig. 18.1
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Notes

  1. 1.

    The term ‘tomography’ goes back to the use of Radon transforms in early schemes for estimating quantum states of electromagnetic fields [1, 27]. It has become synonymous with ‘quantum density matrix estimation’, even though current methods applied to quantum systems with a finite dimension d have no technical connection to classical tomographic reconstruction algorithms.

  2. 2.

    A more insightful way of proving the first identity is to realise that \(\mathbb E\big (\chi _S(C^x)\big )\) is effectively a Fourier coefficient (over the group \(\mathbb {Z}_2^N\)) of the distribution function of the {−1, 1}N-valued random variable C x (e.g., [11]). Equation (18.21) is then nothing but an inverse Fourier transform.

  3. 3.

    We note that quantum mechanics allows to design measurement devices that directly probe the observable of \(\sigma ^{y_1} \otimes \dots \otimes \sigma ^{y_N}\), without first measuring the spin of every particle and then computing a parity function. In fact, the ability to perform such correlation measurements is crucial for quantum error correction protocols [31]. For practical reasons these setups are used less commonly in tomography experiments, though.

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Acknowledgements

The work of A. Carpentier was done when she was in Cambridge and is partially supported by the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether grant MuSyAD (CA 1488/1-1), by the DFG-314838170, GRK 2297 MathCoRe, by the DFG GRK 2433 DAEDALUS (384950143/GRK2433), and by the DFG CRC 1294 ‘Data Assimilation’, Project A03, and by the UFA-DFH through the French-German Doktorandenkolleg CDFA 01-18. D. Gross acknowledges support by the DFG (SPP1798 CoSIP), Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1—390534769, and the ARO under contract W911NF-14-1-0098 (Quantum Characterization, Verification, and Validation). J. Eisert was supported by the DFG, CRC 183, Project B01, by the DFG GRK DAEDALUS, the DFG SPP1798 CoSIP, the ERC, and the Templeton Foundation.

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Authors and Affiliations

  1. Institut für Mathematische Stochastik, Otto von Guericke Universität Magdeburg, Magdeburg, Germany

    Alexandra Carpentier

  2. Department of Physics, Dahlem Center for Complex Quantum Systems, Freie Universität zu Berlin, Berlin, Germany

    Jens Eisert

  3. Institute of Theoretical Physics, Universität Köln, Köln, Germany

    David Gross

  4. Statistical Laboratory, University of Cambridge, Cambridge, UK

    Richard Nickl

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  1. Alexandra Carpentier
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Correspondence to Richard Nickl .

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Editors and Affiliations

  1. MAP 5, Université Paris Descartes, Paris, France

    Nathael Gozlan

  2. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Prof. Rafał Latała

  3. Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France

    Karim Lounici

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Prof. Mokshay Madiman

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Carpentier, A., Eisert, J., Gross, D., Nickl, R. (2019). Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_18

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