Towards the Geometry of Quantum Optical Measurements: Kraus Trajectories in the Instrumental Weyl-Heisenberg Group Manifold

Abstract

Since ancient Greek times, geometry has been a method by which humans can make inferences, by the use of similar triangles, about the world far beyond their immediate senses. Similarity plays an equally important role in quantum measurement, except that instead of triangles, the similarity is between the infinitesimal generators of rotation, canonical transformation, or some other virtual symmetry. For many quantum optical measurements of a single mode, these generators produce Kraus operators that are contained within a twelve-dimensional Lie group: the Instrumental Weyl-Heisenberg group manifold ($\mathrm{IWH}^{\textbf{12}}$). In particular, $\mathrm{IWH}^{\textbf{12}}$ is general enough to include all diffusively generated 'dyne measurements. In this paper, $\mathrm{IWH}^{\textbf{12}}$ will be used to develop two more general classes of 'dyne measurement: one that interpolates between diffusive homodyne and diffusive heterodyne (``slomodyne'') and one that adds to 'dyne measurements adaptive squeezing and displacement. Our analysis uses an approach known as the Instrument Manifold Program (IMP). In particular, a Harish-Chandra-like decomposition of $\mathrm{IWH}^{\textbf{12}}$ is used to calculate the vielbein transformation from a right-invariant basis to the corresponding coordinate basis, allowing us to universally and autonomously navigate $\mathrm{IWH}^{\textbf{12}}$ and locate the Kraus-operator of whatever selective process is being considered as it evolves in time according to the measurement record.