Mathematical aspects of continuous variable quantum information

• Applicant: Professor Dr. Robert König
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 318917567

• Final Report Year: 2019

Final Report Abstract

We have studied a number of problems in continuous-variable quantum information theory and quantum communication. We have established a range of new entropic inequalities for bosonic systems, which govern the behavior of entropic quantities for Gaussian quantum systems. This contributes towards closing the gap between classical and quantum information theory. As one application, we have found the first bounds on the additivity violation of non-Gaussian bosonic channels for a general class of noisy channels, modeled by channels made up of beamsplitters and classical noise. While the question of whether additivity holds for these channels remains open, we could establish that potential additivity violations must necessarily be small. As a consequence, well-established coding techniques for classical communication, which use classical modulation of coherent states and which were known to be optimal in certain special cases, also perform well for these non-Gaussian channels. These results widen our understanding of continuous-variable channels beyond the case of channels with symmetries. Further results include a characterization of Gaussian additive noise channels, and a measure concentration result for the symplectic spectrum of the covariance matrix associated with reduced density operators of bipartite random pure Gaussian states. Finally, we studied information degradation under repeated action of a noise channel: here we constructed an algorithm for numerically computing the so-called Dobrushin curve in the finite-dimensional setting.

Publications

• Generalized log-majorization and multivariate trace inequalities. Annales Henri Poincaré, 18(7):2499–2521, March 2017
  Hiai, Fumio; König, Robert & Tomamichel, Marco
  
• Geometric inequalities from phase space translations. Journal of Mathematical Physics, 58(1):012206, January 2017
  Huber, Stefan; König, Robert & Vershynina, Anna
  
• On quantum additive Gaussian noise channels. Quantum Information and Computation, 17(3–4):0283–0302, 2017
  Martin Idel and Robert König
  
• Uncertainty relations: An operational approach to the error-disturbance tradeoff. Quantum, 1:20, July 2017
  Renes, Joseph M.; Scholz, Volkher B. & Huber, Stefan
  
• Coherent state coding approaches the capacity of non-gaussian bosonic channels. Journal of Physics A: Mathematical and Theoretical, 51(18):184001, April 2018
  Stefan Huber and Robert König
  
• The conditional entropy power inequality for quantum additive noise channels. Journal of Mathematical Physics, 59(12):122201, December 2018
  De Palma, Giacomo & Huber, Stefan
  
• Jointly constrained bilinear semidefinite programming with an application to Dobrushin curves. IEEE Transactions on Information on Aug 21, 2019
  Stefan Huber, Robert K¨nig, and Marco Tomamichel