Asymptotic equivalence of quantum state tomography and noisy matrix completion
The Annals of Statistics · 2013 · Volume 41 · Issue 5
Abstract
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.
Keywords
62B15, 62J99, 62P35, 65F10, 65J20, 81P45, 81P50, compressed sensing, deficiency distance, density matrix, fine scale trace regression, low rank matrix, observable, Pauli matrices, quantum measurement, Quantum probability, quantum statistics, sparse matrix, Trace regression
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