量子对数Sobolev不等式与快速混合
Quantum logarithmic Sobolev inequalities and rapid mixing
摘要：
在有限维量子态空间上引入了一族对数Sobolev不等式.回顾了非交换$\bL_p$ -空间的框架，讨论了量子对数Sobolev不等式与量子半群超压缩性之间的关系.这种关系是中央的对数Sobolev（LS）常数的下限的推导。证明了这类不等式的基本结果，并给出了广义LS常数关于半群生成元谱隙的一个上界。这些不等式提供了一个框架，当LS常数和谱隙是相同的顺序时，量子动力学半群的收敛时间的改进的界限的推导。有限维状态空间上的收敛界与量子信息理论领域特别相关。我们提供了一些例子，其中改进的边界上的混合时间的几个半群，包括去极化半群和量子扩展。
英文摘要：
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. 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; including the depolarizing semigroup and quantum expanders.
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