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:quant-ph/9806028 · Journal of Mathematical Physics

Connections and metrics respecting purification of quantum states

Jochen Dittmann, Armin Uhlmann

Published Jul 1, 1999• 27 citations
arXivHTMLPDF

Abstract

Standard purification interlaces Hermitian and Riemannian metrics on the space of density operators with metrics and connections on the purifying Hilbert–Schmidt space. We discuss connections and metrics which are well adopted to purification, and present a selected set of relations between them. A connection, as well as a metric on state space, can be obtained from a metric on the purification space. We include a condition, with which this correspondence becomes one to one. Our methods are borrowed from elementary *-representation and fiber space theory. We lift, as an example, solutions of a von Neumann equation, write down holonomy invariants for cyclic ones, and “add noise” to a curve of pure states.

DOI
10.1063/1.532884

Related papers

Geometry of quantum evolution for mixed quantum states

Ole Andersson, Hoshang Heydari

The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert–Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.

arXiv:1312.3360•Physica Scripta•Apr 1, 2014•6 citations

Geometry of quantum evolution for mixed quantum states

Ole Andersson, Hoshang Heydari

Abstract. The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert-Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework. 1.

•Aug 16, 2016•12 citations

Quantum information geometry and standard purification

Anna Jenčová

We investigate relations between Uhlmann’s parallelism, monotone Riemannian metrics and dual affine connections on the space of density matrices.

Journal of Mathematical Physics•May 1, 2002•13 citations

Purification complexity without purifications

Shan-Ming Ruan

Abstract We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.

arXiv:2006.01088•Journal of High Energy Physics•Jan 18, 2021•19 citations

The Stratified Information Geometry of Mixed Quantum States: Uhlmann Purification Bundles and Caustic Control Singularities in Open Systems

Ahmed Hala

This paper develops a rigorous, self-contained differential-geometric framework for tracking mixed quantum states in open physical systems, providing an analytical structural extension to the pure-state Kâhler geometry of Paper 1. Because open quantum dynamics induced by environment-driven dissipation and decoherence break the global manifold homogeneity, we model the total space of density operators S(H) as a stratified space. We prove that the individual rank-k subsets constitute embedded smooth real manifolds of dimension 2k(N + 1) − k2 − 1, satisfying the algebraic conditions of Whitney Regularity Condition B to guarantee continuous tangent space alignment across interfaces. To assign coordinate-independent connection and metric tensors over this stratified landscape, we formalize the global principal U (n) fiber bundle structure using Uhlmann’s purification theorem. By lifting mixed density operators to pure-state amplitudes W ∈ GL(n, C) on a bipartite workspace, we show that minimizing the ambient flat Hilbert–Schmidt metric along the vertical fibers induces the non-degenerate Bures metric on the base manifold. We derive the exact matrix elements of the corresponding Symmetric Logarithmic Derivative (SLD) fields in an arbitrary non-degenerate eigenbasis, establishing the mixed-state information scaling boundary gQF IM = 4gB. Finally, we characterize the non-Abelian Yang–Mills curvature FU = dAU + AU ∧ AU of the Uhlmann connection, demonstrating how non-commutativity emerges naturally under environmental coupling. By isolating the measure-zero boundary configurations where eigenvalues or matrix rank drop, we generalize our control immersion criterion to identify both rank deficient and eigenvalue-vanishing caustic singularities. This metric-determinant verification pipeline allows researchers to systematically diagnose, map, and structurally bypass trajectory bottlenecks during Lindbladian control path optimization on real-world noisy quantum backends

•Jun 30, 2026•0 citations

The geometry of entanglement: metrics, connections and the geometric phase

Péter Lévay

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.

arXiv:quant-ph/0306115•Journal of Physics A Mathematical and General•Jan 19, 2004•83 citations
Standard purification interlaces Hermitian and Riemannian metrics on the space of density operators with metrics and connections on the purifying Hilbert–Schmidt space. We discuss connections and metrics which are well adopted to purification, and present a selected set of relations between them. A connection, as well as a metric on state space, can be obtained from a metric on the purification space. We include a condition, with which this correspondence becomes one to one. Our methods are borrowed from elementary *-representation and fiber space theory. We lift, as an example, solutions of a von Neumann equation, write down holonomy invariants for cyclic ones, and “add noise” to a curve of pure states.
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert–Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.
Abstract. The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert-Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework. 1.
Abstract We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.
This paper develops a rigorous, self-contained differential-geometric framework for tracking mixed quantum states in open physical systems, providing an analytical structural extension to the pure-state Kâhler geometry of Paper 1. Because open quantum dynamics induced by environment-driven dissipation and decoherence break the global manifold homogeneity, we model the total space of density operators S(H) as a stratified space. We prove that the individual rank-k subsets constitute embedded smooth real manifolds of dimension 2k(N + 1) − k2 − 1, satisfying the algebraic conditions of Whitney Regularity Condition B to guarantee continuous tangent space alignment across interfaces. To assign coordinate-independent connection and metric tensors over this stratified landscape, we formalize the global principal U (n) fiber bundle structure using Uhlmann’s purification theorem. By lifting mixed density operators to pure-state amplitudes W ∈ GL(n, C) on a bipartite workspace, we show that minimizing the ambient flat Hilbert–Schmidt metric along the vertical fibers induces the non-degenerate Bures metric on the base manifold. We derive the exact matrix elements of the corresponding Symmetric Logarithmic Derivative (SLD) fields in an arbitrary non-degenerate eigenbasis, establishing the mixed-state information scaling boundary gQF IM = 4gB. Finally, we characterize the non-Abelian Yang–Mills curvature FU = dAU + AU ∧ AU of the Uhlmann connection, demonstrating how non-commutativity emerges naturally under environmental coupling. By isolating the measure-zero boundary configurations where eigenvalues or matrix rank drop, we generalize our control immersion criterion to identify both rank deficient and eigenvalue-vanishing caustic singularities. This metric-determinant verification pipeline allows researchers to systematically diagnose, map, and structurally bypass trajectory bottlenecks during Lindbladian control path optimization on real-world noisy quantum backends
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of S7S^7S7 over the quaternionic projective space HP1≃S4{\bf HP}^1\simeq S^4HP1≃S4 with an SU(2)≃S3SU(2)\simeq S^3SU(2)≃S3 fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on HP1{\bf HP}^1HP1 between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.