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

TL;DR

This paper develops a comprehensive framework for analyzing continuous, simultaneous measurements of noncommuting quantum observables using instrumental Lie groups, providing new insights into instrument evolution and foundational quantum measurement processes.

Contribution

It introduces the Instrument Manifold Program, relating instrument evolution to differential positive transformations and universal Lie groups, advancing understanding of quantum measurement dynamics.

Findings

Instrument evolution described by diffusion on Lie groups.

Universal instrumental Lie group is independent of Hilbert space.

Finite-dimensional universal instruments are fundamental in quantum mechanics.

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.