迹族算符的随机微分方程与量子连续测量
Stochastic differential equations for trace-class operators and quantum continual measurements
摘要：
量子力学中时间上连续的测量理论（量子连续测量）是利用工具值和正算子值测度、泛函积分、量子随机微分方程和经典随机微分方程的概念建立起来的。涉及到各种类型的代数方程，以及希尔伯特空间中向量和迹族算子的线性和非线性方程。所有这样的方程要么包含扩散部分，要么包含跳跃部分，或者两者都包含。在本文中，我们介绍了一类线性的迹族运营商，有关的连续测量的理论，我们还记得如何，这样的线的仪器和主方程，因此，量子力学的一般制定。我们不提出的希尔伯特空间制定这样的量子系统的，我们做了一些数学简化：没有时间依赖性引入到系数，只有有界算子的希尔伯特空间的量子系统被认为是。然后引入后验态的概念和后验态所满足的非线性条件，并给出了保证该方程保持纯态和使任意混合态在大时间内变为纯态的条件.最后，我们回顾了关于纯扩散情形下不变测度的存在性和唯一性的已知结果，并给出了一些物理系统的具体例子。
英文摘要：
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.
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