Symplectic Dilations, Gaussian States and Gaussian Channels

2014/05/26 by K. R. Parthasarathy, Parthasarathy, K. R.
Mathematics · #42A82 #60B15 #81R30 #81S25 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Physical sciences #Quantum Physics (quant-ph) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1405.6476

openalex publication_date 2014/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By elementary matrix algebra we show that every real 2n × 2n matrix admits a dilation to an element of the real symplectic group Sp (2(n+m)) for some nonnegative integer m. Our methods do not yield the minimum value of m, for which such a dilation is possible. After listing some of the main properties of Gaussian states in L² (ℝⁿ), we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari, Holevo and Wolf \cite3.

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