Contractivity properties of a quantum diffusion semigroup

Abstract

We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

The results of this paper were presented at the "48th Symposium on Mathematical Physics; Gorini-Kossakowski-Lindblad-Sudarshan Master Equation - 40 Years After", held in Torun, Poland (June 10-12, 2016). v2: minor typos and proof of Theorem 7 corrected. v3, v4: minor typos corrected and overall presentation improved. v5: Published version. v6: metadata modified

References in corpus (2)

• Normal form decomposition for Gaussian-to-Gaussian superoperators
• Geometric inequalities from phase space translations

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• The conditional entropy power inequality for quantum additive noise channels
• Geometric inequalities from phase space translations
• Conditional quantum entropy power inequality for $d$-level quantum systems
• Coherent state coding approaches the capacity of non-Gaussian bosonic channels