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:1903.10465 · Lecture notes of the Unione Matematica Italiana

Differential Geometry of Quantum States, Observables and Evolution

Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo

Published Jan 1, 2019• 4 citations
arXivHTMLPDF
DOI
10.1007/978-3-030-06122-7_7

Related papers

Differential geometry of bipartite quantum states

Zuhuan Yu, Xianqing Li‐Jost, Qingzhong Li +2 more

arXiv:0711.1211•Reports on Mathematical Physics•Aug 1, 2007•2 citations

THE GEOMETRY OF QUANTUM STATES

Julian Schwinger

World Scientific series in 20th century physics•May 1, 2000•3 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

Geometry of Quantum States

Samuel L. Braunstein, Carlton M. Caves

Springer eBooks•Jan 1, 1995•15 citations

Quantum State Curvature Flow (QSCF)

Jincheng Zhang

Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.

•Aug 17, 2026•0 citations

Geometry of stochastic state vector reduction

Lane P. Hughston

The state space of a quantum mechanical system is a complex projective space, the space of rays in the associated Hilbert space. The state space comes equipped with a natural Riemannian metric (the Fubini-Study metric) and a compatible symplectic structure. The operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry. It is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analysis of quantum dynamics, particularly in connection with state vector reduction. A promising approach to understanding reduction, studied recently by a number of authors, involves the use of nonlinear stochastic dynamics to modify the ordinary linear Schrodinger evolution. Here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse. In this picture, the conventional Schrodinger evolution, which corresponds to the unitary flow associated with a Killing vector of the Fubini-Study metric, is replaced by a more general stochastic flow on the state manifold. In the simplest example of such a flow, the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the Hamiltonian. The conservation of energy is represented by the requirement that the actual process followed by the expectation of the Hamiltonian, as the state evolves, should be a martingale. This requirement implies the existence of a nonlinear term in the drift vector of the state process, which is always oriented opposite to the direction of increasing energy uncertainty. As a consequence, the state vector necessarily collapses to an energy eigenstate, and a martingale argument can be used to show that the probability of collapse to a given eigenstate, from any particular initial state, is, in fact, given by precisely the usual quantum mechanical probability.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 9, 1996•127 citations
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.
Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.
The state space of a quantum mechanical system is a complex projective space, the space of rays in the associated Hilbert space. The state space comes equipped with a natural Riemannian metric (the Fubini-Study metric) and a compatible symplectic structure. The operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry. It is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analysis of quantum dynamics, particularly in connection with state vector reduction. A promising approach to understanding reduction, studied recently by a number of authors, involves the use of nonlinear stochastic dynamics to modify the ordinary linear Schrodinger evolution. Here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse. In this picture, the conventional Schrodinger evolution, which corresponds to the unitary flow associated with a Killing vector of the Fubini-Study metric, is replaced by a more general stochastic flow on the state manifold. In the simplest example of such a flow, the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the Hamiltonian. The conservation of energy is represented by the requirement that the actual process followed by the expectation of the Hamiltonian, as the state evolves, should be a martingale. This requirement implies the existence of a nonlinear term in the drift vector of the state process, which is always oriented opposite to the direction of increasing energy uncertainty. As a consequence, the state vector necessarily collapses to an energy eigenstate, and a martingale argument can be used to show that the probability of collapse to a given eigenstate, from any particular initial state, is, in fact, given by precisely the usual quantum mechanical probability.