Quantum logarithmic Sobolev inequalities and rapid mixing

Understand

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced.

• The framework of non-commutative $\bL_p$-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed.
• This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants.
• Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup.