Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.16512

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

Tzula B. Propp, Chris S. Jackson

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
arXivHTMLPDF

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.

Related papers

Group Theoretical Methods in Lie Optics

Kurt Bernardo Wolf

Optics has served as the expermental basis and model for much of science since Galielo. The minds at work in mechanics were so too in optics, in its early geometric aspects through Snell up to Newton, and its wave aspects since Huygens through Fourier and including Sommerfeld. Hamilton’s original application of generating functions for canonical transformations was in optics; only later were they recognized as a sharp tool for abstract mechanics through the introduction of the concept of a phase space with a symplectic metric [ 1 ]. This concept in embrionary form, served Heisenberg [ 2 ] and Weyl [ 3 ] for the introduction of quantum mechanics, where the Hamiltonian is quadratic in momentum. The history and scope of Sophus Lie’s work is well known to this audience, as is the fact that its manifold applications have been most successful when applied to systems with quadratic Hamiltonians [ 4 ]. Almost invariably, the kinetic energy term is p 2 /2m. This Hamiltonian is indeed also useful in optics, albeit in its Gaussian approximation, i.e., first-order optics which deals with linear transformations of phase space. The associated methods have been under research since the early seventies from the perspective of quantum mechanics [ 5 ] and groups of integral transforms [ 6 , 7 ].

CRC Press eBooks•Oct 31, 2024•0 citations

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

Christopher S. Jackson, Carlton M. Caves

We formulate a general program for describing and 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 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 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 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 chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, 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 between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

arXiv:2306.06167•Entropy•Aug 23, 2023•5 citations

The canonical group of transformations of a Weyl–Heisenberg frame; applications to Gaussian and Hermitian frames

Maurice A. de Gosson

Journal of Geometry and Physics•Jan 3, 2017•9 citations

Sequential Quantum Measurements and the Instrumental Group Algebra

Christopher S. Jackson

Many of the most fundamental observables—position, momentum, phase point, and spin direction—cannot be measured by an instrument that obeys the orthogonal projection postulate. Continuous-in-time measurements provide the missing theoretical framework to make physical sense of such observables. The elements of the time-dependent instrument define a group called the instrumental group (IG). Relative to the IG, all of the time dependence is contained in a certain function called the Kraus-operator density (KOD), which evolves according to a classical Kolmogorov equation. Unlike the Lindblad master equation, the KOD Kolmogorov equation is a direct expression of how the elements of the instrument (not just the total quantum channel) evolve. Shifting from continuous measurements to sequential measurements more generally, the structure of combining instruments in sequence is shown to correspond to the convolution of their KODs. This convolution promotes the IG to an involutive Banach algebra (a structure that goes all the way back to the origins of POVM and C*-algebra theory), which will be called the instrumental group algebra (IGA). The IGA is the true home of the KOD, similar to how the dual of a von Neumann algebra is the true home of the density operator. Operators on the IGA, which play the analogous role for KODs as superoperators play for density operators, are called ultraoperators and various important examples are discussed. Certain ultraoperator–superoperator intertwining relationships are also considered throughout, including the relationship between the KOD Kolmogorov equation and the Lindblad master equation. The IGA is also shown to have actually two distinct involutions: one respected by the convolution ultraoperators and the other by the quantum channel superoperators. Finally, the KOD Kolmogorov generators are derived for jump processes and more general diffusive processes.

arXiv:2510.13980•Quantum Reports•Nov 30, 2025•1 citation

Quantum Measurements: a modern view for quantum optics experimentalists

Aephraim M. Steinberg

In these notes, based on lectures given as part of the Les Houches summer school on Quantum Optics and Nanophotonics in August, 2013, I have tried to give a brief survey of some important approaches and modern tendencies in quantum measurement. I wish it to be clear from the outset that I shy explicitly away from the "quantum measurement problem," and that the present treatment aims to elucidate the theory and practice of various ways in which measurements can, in light of quantum mechanics, be carried out; and various formalisms for describing them. While the treatment is by necessity largely theoretical, the emphasis is meant to be on an experimental "perspective" on measurement -- that is, to place the priority on the possibility of gaining information through some process, and then attempting to model that process mathematically and consider its ramifications, rather than stressing a particular mathematical definition as the {\it sine qua non} of measurement. The textbook definition of measurement as being a particular set of mathematical operations carried out on particular sorts of operators has been so well drilled into us that many have the unfortunate tendency of saying "that experiment can't be described by projections onto the eigenstates of a Hermitian operator, so it is not really a measurement," when of course any practitioner of an experimental science such as physics should instead say "that experiment allowed us to measure something, and if the standard theory of measurement does not describe it, the standard theory of measurement is incomplete." Idealisations are important, but when the real world breaks the approximations made in the theory, it is the theory which must be fixed, and not the real world.

arXiv:1406.5535•Jun 20, 2014•0 citations

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group

Christopher S. Jackson, Carlton M. Caves

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

arXiv:2306.01045•Entropy•Aug 16, 2023•3 citations
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 (IWH12\mathrm{IWH}^{\textbf{12}}IWH12). In particular, IWH12\mathrm{IWH}^{\textbf{12}}IWH12 is general enough to include all diffusively generated 'dyne measurements. In this paper, IWH12\mathrm{IWH}^{\textbf{12}}IWH12 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 IWH12\mathrm{IWH}^{\textbf{12}}IWH12 is used to calculate the vielbein transformation from a right-invariant basis to the corresponding coordinate basis, allowing us to universally and autonomously navigate IWH12\mathrm{IWH}^{\textbf{12}}IWH12 and locate the Kraus-operator of whatever selective process is being considered as it evolves in time according to the measurement record.
Optics has served as the expermental basis and model for much of science since Galielo. The minds at work in mechanics were so too in optics, in its early geometric aspects through Snell up to Newton, and its wave aspects since Huygens through Fourier and including Sommerfeld. Hamilton’s original application of generating functions for canonical transformations was in optics; only later were they recognized as a sharp tool for abstract mechanics through the introduction of the concept of a phase space with a symplectic metric [ 1 ]. This concept in embrionary form, served Heisenberg [ 2 ] and Weyl [ 3 ] for the introduction of quantum mechanics, where the Hamiltonian is quadratic in momentum. The history and scope of Sophus Lie’s work is well known to this audience, as is the fact that its manifold applications have been most successful when applied to systems with quadratic Hamiltonians [ 4 ]. Almost invariably, the kinetic energy term is p 2 /2m. This Hamiltonian is indeed also useful in optics, albeit in its Gaussian approximation, i.e., first-order optics which deals with linear transformations of phase space. The associated methods have been under research since the early seventies from the perspective of quantum mechanics [ 5 ] and groups of integral transforms [ 6 , 7 ].
We formulate a general program for describing and 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 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 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 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 chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, 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 between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.
Many of the most fundamental observables—position, momentum, phase point, and spin direction—cannot be measured by an instrument that obeys the orthogonal projection postulate. Continuous-in-time measurements provide the missing theoretical framework to make physical sense of such observables. The elements of the time-dependent instrument define a group called the instrumental group (IG). Relative to the IG, all of the time dependence is contained in a certain function called the Kraus-operator density (KOD), which evolves according to a classical Kolmogorov equation. Unlike the Lindblad master equation, the KOD Kolmogorov equation is a direct expression of how the elements of the instrument (not just the total quantum channel) evolve. Shifting from continuous measurements to sequential measurements more generally, the structure of combining instruments in sequence is shown to correspond to the convolution of their KODs. This convolution promotes the IG to an involutive Banach algebra (a structure that goes all the way back to the origins of POVM and C*-algebra theory), which will be called the instrumental group algebra (IGA). The IGA is the true home of the KOD, similar to how the dual of a von Neumann algebra is the true home of the density operator. Operators on the IGA, which play the analogous role for KODs as superoperators play for density operators, are called ultraoperators and various important examples are discussed. Certain ultraoperator–superoperator intertwining relationships are also considered throughout, including the relationship between the KOD Kolmogorov equation and the Lindblad master equation. The IGA is also shown to have actually two distinct involutions: one respected by the convolution ultraoperators and the other by the quantum channel superoperators. Finally, the KOD Kolmogorov generators are derived for jump processes and more general diffusive processes.
In these notes, based on lectures given as part of the Les Houches summer school on Quantum Optics and Nanophotonics in August, 2013, I have tried to give a brief survey of some important approaches and modern tendencies in quantum measurement. I wish it to be clear from the outset that I shy explicitly away from the "quantum measurement problem," and that the present treatment aims to elucidate the theory and practice of various ways in which measurements can, in light of quantum mechanics, be carried out; and various formalisms for describing them. While the treatment is by necessity largely theoretical, the emphasis is meant to be on an experimental "perspective" on measurement -- that is, to place the priority on the possibility of gaining information through some process, and then attempting to model that process mathematically and consider its ramifications, rather than stressing a particular mathematical definition as the {\it sine qua non} of measurement. The textbook definition of measurement as being a particular set of mathematical operations carried out on particular sorts of operators has been so well drilled into us that many have the unfortunate tendency of saying "that experiment can't be described by projections onto the eigenstates of a Hermitian operator, so it is not really a measurement," when of course any practitioner of an experimental science such as physics should instead say "that experiment allowed us to measure something, and if the standard theory of measurement does not describe it, the standard theory of measurement is incomplete." Idealisations are important, but when the real world breaks the approximations made in the theory, it is the theory which must be fixed, and not the real world.
The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.