Contractivity properties of a quantum diffusion semigroup
  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. 
Nilanjana Datta, Yan Pautrat and Cambyse Rouze
quant-ph math-ph math.MP