Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems
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.
KeywordsLow rank recovery Quantum information Confidence sets Sequential sampling
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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