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Low rank as a model for quantum and classical estimation problems

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The theory of compressed sensing provides rigorous methods for analyzing the performance of estimators that include a sparsity-enhancing 1-norm regularization term. Since around 2009, a “non-commutative” version of compressed sensing has been developed. Here, the aim is to efficiently recover matrices under a low-rank assumption, most commonly using nuclear-norm regularization. The program was initially motivated by purely classical estimation problems – e.g. the influential “Netflix problem” of predicting user preferences in online shops. However, early on, a fruitful interaction between classical and quantum theory ensued: In one direction, it has been realized that low-rank methods lead to rigorous and very tight performance guarantees for quantum state estimation procedures. In the other direction, mathematical methods originally developed in the context of quantum information theory allowed for a significant generalization and simplification of the rigorous results on low-rank recovery. I will give an introduction to the theory, as well as classical and quantum applications.

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