[quantum-info] Colloquium Institute for Quantum Computing Monday, 25 November

[Matthew Fries]

Colloquium

Institute for Quantum Computing

Monday, 25 November 2013 at 2:30PM

QNC 0101

Quantum mixing time bounds from Poincare and logarithmic Sobolev inequalities

Kristan Temme

Massachusetts Institute of Technology

We will introduce two approaches for estimating the time a finite dimensional quantum Markov process takes to converge to its steady state. These approaches can be seen as the quantum mechanical analogue of logarithmic Sobolov (LS) inequalities and Poincare inequalities. First we discuss bounds on the trace norm mixing in terms of the spectral gap of the generator. The gap can be estimated from a quantum mechanical Poincare inequality. A lower bound on the spectral gap for a particular class of thermalizing generators is proved. Then, we consider a family of logarithmic Sobolev inequalities on finite Lp-spaces. LS bounds can lead to convergence time estimates that are exponentially better than those obtained from the spectral gap alone. We review the framework of non-commutative Lp-spaces and discuss the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups. This relationship is central for the practical derivation of LS inequalities. We will discuss examples, where LS and Poincare bounds can be evaluated explicitly and rapid mixing for quantum mechanical semi-groups can be shown.

Coffee and refreshments served following the talk.