Hannover 2016 – wissenschaftliches Programm
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Q: Fachverband Quantenoptik und Photonik
Q 18: Quantum Information: Concepts and Methods III
Q 18.8: Vortrag
Dienstag, 1. März 2016, 12:45–13:00, e214
Improving compressed sensing with the diamond norm — •Martin Kliesch1, Richard Kueng2, Jens Eisert1, and David Gross2 — 1Freie Universität Berlin — 2Universität zu Köln
In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank.
In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that –for a class of matrices saturating a certain norm inequality– the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices, which also contains quantum channels. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include not only the task of quantum process tomography but also signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes.
The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.