On The Generators of Quantum Dynamical Semigroups
On The Generators of Quantum Dynamical Semigroups
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发表时间：
2019
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通讯作者：
A. Wiedemann
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作者：
A. Wiedemann
近年来，量子动力学半群的有向图诱导生成元被引入和研究，特别是在唯一松弛和不变性的背景下。我们定义了一类对块对角生成器，它允许额外的相互作用系数，但保留了主要的结构特性。也就是说，当基础希尔伯特空间的基由哈密顿算子的本征基给出时（例如一般半群），则半群的作用使对角和非对角矩阵空间保持不变。在这种情况下，我们显式地计算半群的所有不变状态。为了定义这个类，我们提供了一个表征，当允许任意Lindblad算子（特别是，它们不需要像GKSL定理所要求的那样是无迹的）时，Gorini-Kossakowski-Sudarshan-Lindblad（GKSL）方程定义了一个适当的生成元。此外，我们认为匡威结构表明，每个生成器自然产生一个有向图，并在一定的假设下，这个有向图的属性可以利用获得知识的数量和结构的不变状态的相应的半群。我们还考虑更一般的建设冯诺依曼代数的所有有界线性算子的希尔伯特空间，也许是无限维的。特别地，我们证明了对于具有次不变忠实正规态的代数上的每个施瓦茨映射半群，在Hilbert空间的Hilbert Schmidt算子空间上存在一个相伴的压缩半群.此外，我们还证明了如果原半群是弱连续的，则伴随半群是强连续的。引入了Hilbert空间的有界算子半群关于Hilbert空间的标准正交基的生成元的概念。我们描述了这种形式的生成元的量子马尔可夫半群上的冯诺依曼代数的所有有界线性算子的Hilbert空间，其中有一个不变的忠实的正常状态下的假设下，相关的半群的生成元有紧的预解式，或假设下，最小酉膨胀的相关的压缩半群是紧的。
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be traceless as demanded by the GKSL Theorem). Moreover, we consider the converse construction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corresponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In particular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear operators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.