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Spectral thresholding in quantum state estimation for low rank states

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Quantum Information and Technology is a young research area at the overlap between quantum physics and “classical” fields such as computation theory, information theory, statistics and probability and control theory. The paradigm is that quantum systems such as atoms and photons, are carriers of a new type of information, whose processing is governed by the formalism of quantum mechanics. This has found numerous applications in computation, cryptography, precision metrology, and significant experimental efforts are dedicated towards the practical implementation of such technologies.

One of the key component of many quantum engineering experiments is the statistical analysis of measurement data. In particular, in ion trap experiments one deals with the problem of reconstructing large density matrices (positive, complex matrices of trace one) representing the joint state of several atoms, from i.i.d. counts of collected from measurements on identical prepared atoms. Since the matrix dimension scales exponentially with the number of atoms, current techniques can cope with at most 10 atoms, and one of the key questions is how statistically reconstruct large dimensional states.

In this talk I will discuss two new estimation methods for quantum tomography in ion experiments, their theoretical properties and simulations results. Both methods consist in computing the least squares estimator as first step, followed by setting certain “statistically insignificant” eigenvalues to zero. Since in many experiments the goal is to produce a pure (rank one) density matrix, low rank density matrices provide a natural lower dimensional model for experiments. For such states, the thresholding methods provide a significant improvement compared with the least squares estimator; in fact, our upper and lowe bounds show that up to logarithmic factors, the mean square error has the optimal scaling in terms of dimension and sample size.

This talk is part of the Statistics series.

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