Monday, July 16, 2012

1207.3261 (Michael J. Kastoryano et al.)

Quantum logarithmic Sobolev inequalities and rapid mixing    [PDF]

Michael J. Kastoryano, Kristan Temme
A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. These inequalities are shown to lead to very tight bounds on the convergence time of quantum dynamical semigroups to their fixed point. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. The framework of non-commutative Lp-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 Log-Sobolev constants. Essential results for the family of inequalities are proved, and a bound of the generalized Log-Sobolev constant in terms of the spectral gap of the generator of the semigroup is shown. As a main example, illustrating the power of our framework, improved bounds on the mixing time of quantum expanders are obtained.
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