子空间的量子递归与算子值Schur函数
Quantum recurrence of a subspace and operator-valued Schur functions
摘要：
最近在[Commun.数学物理，DOI 10.1007/s 00220 -012-1645-2]（另见arXiv：1202.3903），将初始状态视为吸收状态。我们将这一概念的监测复发吸收子空间的任意有限维。生成函数的方法导致了与著名的算子值舒尔函数理论的联系。这是子空间递归的谱特征的基石，概括了上述论文中的一些主要结果。驱动演化的酉步算子的谱分解产生一个谱测度，我们将其投影到子空间上以获得一个新的谱测度，该谱测度是纯奇异的当且仅当子空间是递归的，并且当子空间中的所有状态具有有限的期望返回时间时，该谱测度由具有有限质量的纯点谱组成。子空间递归的概念也将预期返回时间的概念与Aharonov-Anandan相联系起来，与状态递归的情况相反，Aharonov-Anandan相可以是非整数的。更令人惊讶的是，在吸收子空间上平均这种几何相位会产生一个具有拓扑意义的整数，因此平均预期返回时间总是一个有理数。此外，状态递归偶尔可以给出比子空间递归更高的返回概率，这一事实再次揭示了量子系统的反直觉行为。所有这些现象都说明了明确的例子，包括作为一个自然的应用程序的分析网站重现杜撰的散步。
英文摘要：
A notion of monitored recurrence for discrete-time quantum processes was recently introduced in [Commun. Math. Phys., DOI 10.1007/s00220-012-1645-2] (see also arXiv:1202.3903) taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension. The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in the above mentioned paper. The spectral decomposition of the unitary step operator driving the evolution yields a spectral measure, which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent, and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time. This notion of subspace recurrence also links the concept of expected return time to an Aharonov-Anandan phase that, in contrast to the case of state recurrence, can be non-integer. Even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning, so that the averaged expected return time is always a rational number. Moreover, state recurrence can occasionally give higher return probabilities than subspace recurrence, a fact that reveals once more the counterintuitive behavior of quantum systems. All these phenomena are illustrated with explicit examples, including as a natural application the analysis of site recurrence for coined walks.
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