Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates

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Bardet, Ivan 
Rouzé, Cambyse 

jats:titleAbstract</jats:title>jats:pWe generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called jats:italicamalgamated</jats:italic>jats:inline-formulajats:alternativesjats:tex-math$$\mathbb {L}_p$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miL</mml:mi> mml:mip</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the jats:italicdecoherence-free</jats:italic> algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated jats:inline-formulajats:alternativesjats:tex-math$$\mathbb {L}_p$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miL</mml:mi> mml:mip</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> norms are not uniformly convex for jats:inline-formulajats:alternativesjats:tex-math$$1\le p \le 2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mn1</mml:mn> mml:mo≤</mml:mo> mml:mip</mml:mi> mml:mo≤</mml:mo> mml:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.</jats:p>


Funder: MCQST

Funder: Technische Universität München (1025)

4901 Applied Mathematics, 4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences
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Annales Henri Poincaré
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Springer Science and Business Media LLC