Simultaneous Measurements of Noncommuting Observables. Positive Transformations and Instrumental Lie Groups

Abstract: We formulate a general program for [...] analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function defined relative to the invariant measure of the instrumental Lie group [...]. This way of considering instrument evolution we call the Instrument Manifold Program. 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[,] the universal instrumental Lie group, which is independent [...] of Hilbert space. The universal instrument is generically infinite dimensional, in which situation the instrument's evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered to be principal and can be analyzed within 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, of position and momentum, and of the three components of angular momentum. These measurements limit to strong simultaneous measurements. For a single observable, this gives the standard decay of coherence between inequivalent irreps; for the latter two, it gives a collapse within each irrep onto the canonical or spherical phase space, locating phase space at the boundary of these instrumental Lie groups.