Anna Vershynina: Geometric inequalities and contractivity of bosonic semigroups

Опубликовано 1 февраля 2017, 1:36

"We consider a quantum generalization of the classical heat equation,
whose associated semigroup consists of additive Gaussian classical noise
channels. For this quantum diffusion semigroup, we establish various
contractivity properties: we prove a Nash inequality and a logarithmic
Sobolev inequality, which lead to an ultracontractivity result. This
implies that the purity of any state decreases inverse polynomially in
time under the evolution, while its entropy grows logarithmically.
Additionally, we establish several functional inequalities related to
the semigroup, which constitute quantum versions of celebrated classical
statements: these include concavity of the entropy along the flow, as
well as isoperimetric inequalities for Fisher information and entropy.
The additive Gaussian classical noise channel is a special instance of a
convolution operation defined for a classical probability distribution
on phase space and a quantum state of a bosonic system. We find a new
entropy power inequality for this convolution operation. These results
can be understood as quantum information-theoretic analogues of
inequalities from geometric analysis."