Quantum logarithmic Sobolev inequalities and rapid mixing

By Michael J. Kastoryano, Kristan Temme. 2012

176 citations recorded by the source corpus.

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. 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 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. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained, incl...

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/36e0f5eda75d8a10bfbbc3a1eb08373a61389fdb

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Citations in the corpus

Local random quantum circuits are approximate polynomial-designs: numerical results (2012, background): “At least two other conditions that would be interesting to investigate are the rapid-scrambling criterion [57, 72] and the log-Sobolev condition [53].”
Quantum error mitigation (2022, background): “|| It has been shown that the relative entropy of between two output states from noisy circuits under local depolarizing noise decreases exponentially with the circuit depth (Kastoryano and Temme, 2013; M¨uller-Hermes et al. , 2016), which implies that the sampling cost in Eq.”