Jackson Christopher S, Caves Carlton M
Independent Researcher, Gold Beach, OR 97444, USA.
Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA.
Entropy (Basel). 2023 Aug 23;25(9):1254. doi: 10.3390/e25091254.
We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument , without states. The Kraus operators of such measuring processes are time-ordered products of fundamental , which generate nonunitary transformation groups that we call . The temporal evolution of the instrument is equivalent to the diffusion of a , defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the . We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument's evolution is . Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered , and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence inequivalent irreducible representations. For the latter two cases, it leads to a collapse each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.
我们制定了一个通用程序，用于描述和分析对非对易可观测量的连续、微分弱同时测量，该程序侧重于描述测量仪器，而不涉及态。此类测量过程的克劳斯算子是基本算子的时间排序乘积，它们生成我们称为的非酉变换群。仪器的时间演化等同于相对于仪器李群的不变测度定义的一个的扩散。这种扩散可以使用维纳路径积分、随机微分方程或福克 - 普朗克 - 柯尔莫哥洛夫方程进行分析。我们将这种考虑仪器演化的方式称为。我们将仪器流形程序与基于态的随机主方程联系起来。然后我们解释如何使用仪器流形程序，根据我们称为通用仪器李群的通用覆盖来描述仪器演化，该通用覆盖不仅独立于态，还独立于希尔伯特空间。通用仪器一般是无限维的，在这种情况下仪器的演化是。特殊的同时测量有一个有限维的通用仪器，在这种情况下仪器被认为是，并且可以在通用仪器李群的微分几何中进行分析。主仪器是量子力学的基础。我们考虑三个最基本的例子：单个可观测量的测量、位置和动量以及角动量的三个分量。当这些测量连续进行时，它们会收敛到强同时测量。对于单个可观测量，这会导致相干性的标准衰减到不等价不可约表示。对于后两种情况，它会导致每个不可约表示坍缩到经典或球面相空间，相空间位于这些仪器李群的边界。
