Introduction to Quantum Entanglement Geometry

• URL: http://arxiv.org/abs/2601.19111v1
• Date: Tue, 27 Jan 2026 02:30:23 GMT
• Title: Introduction to Quantum Entanglement Geometry
• Authors: Kazuki Ikeda,
• Abstract summary: This article is aimed at viewing entanglement in finite-dimensional quantum many-body systems as a phenomenon of global geometry.<n>We show that the holonomy of the gluing can produce an entangling quantum gate, and can appear as an obstruction class distinct from the usual Berry numbers or Chern numbers.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This article is an expository account aimed at viewing entanglement in finite-dimensional quantum many-body systems as a phenomenon of global geometry. While the mathematics of general quantum states has been studied extensively, this article focuses specifically on their entanglement. When a quantum system varies over a classical parameter space, each fiber may look like the same Hilbert space, yet there may be no global identification because of twisting in the gluing data. Describing this situation by an Azumaya algebra, one always obtains the family of pure-state spaces as a Severi-Brauer scheme. The main focus is to characterize the condition under which the subsystem decomposition required to define entanglement exists globally and compatibly, by a reduction to the stabilizer subgroup of the Segre variety, and to explain that the obstruction appears in the Brauer class. As a consequence, quantum states yield a natural filtration dictated by entanglement on the Severi-Brauer scheme. Using a spin system on a torus as an example, we show concretely that the holonomy of the gluing can produce an entangling quantum gate, and can appear as an obstruction class distinct from the usual Berry numbers or Chern numbers. For instance, even for quantum systems that have traditionally been regarded as having no topological band structure, the entanglement of their eigenstates can be related to global geometric universal quantities, reflecting the background geometry.

Related papers

• Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras [0.0]
  We study pure-state entanglement in families of projective state spaces that are locally trivial but globally twisted.<n>For a given factorization type, we show that the existence of a global locus of product states is equivalent to a reduction of the underlying projective linear torsor to the stabilizer of the corresponding Segre variety.<n>We construct the moduli space of subsystem structures, identify it with a natural torsor quotient, and realize it as a locally closed locus in the relative Hilbert scheme.
  arXiv  Detail & Related papers  (2026-01-20T09:16:42Z)
• Supernematic [0.0]
  Quantum theory of geometrically frustrated systems is usually approached as a gauge theory where the local conservation law becomes the Gauss law.<n>We show that it can do something fundamentally different: enforce a global conserved quantity via a non-perturbative tiling invariant.<n>This establishes a route to a novel quantum many-body state driven by constraints.
  arXiv  Detail & Related papers  (2025-11-13T18:58:17Z)
• Topological Mixed States: Phases of Matter from Axiomatic Approaches [15.433292838001103]
  In closed quantum systems, topological orders are understood through the equivalence classes of ground states of gapped local Hamiltonians.<n>Here, we fill this gap by proposing an approach based on three axioms: ($i$) local recoverability, ($ii$) absence of long-range correlations, and ($iii$) spatial uniformity.<n>From these axioms, a rich set of topological data naturally emerges; importantly, these data are robust under relaxation of axioms.
  arXiv  Detail & Related papers  (2025-06-04T17:58:45Z)
• Observation of quantum superposition of topological defects in a trapped ion quantum simulator [10.307677845109378]
  We report the observation of quantum superposition of topological defects in a trapped-ion quantum simulator. Our work provides useful tools for non-equilibrium dynamics in quantum Kibble-Zurek physics.
  arXiv  Detail & Related papers  (2024-10-20T13:27:13Z)
• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution [21.39913099096503]
  We introduce the concept of imaginary-time Lindbladian evolution as an alternative framework.<n>This new approach defines gapped quantum phases in open systems through the spectrum properties of the imaginary-Liouville superoperator.<n>Our findings demonstrate universal properties at quantum criticality, such as nonanalytic behaviors of steady-state observables, divergence of correlation lengths, and closing of the imaginary-Liouville gap.
  arXiv  Detail & Related papers  (2024-08-06T14:53:40Z)
• Entanglement signatures of a percolating quantum system [0.0]
  Entanglement measures have emerged as one of the versatile probes to diagnose quantum phases and their transitions. We show that when the underlying lattice has percolation disorder, free fermions at a finite density show interesting entanglement properties due to massively degenerate ground states.
  arXiv  Detail & Related papers  (2024-03-22T18:00:07Z)
• Quantum Chaos on Edge [36.136619420474766]
  We identify two different classes: the near edge physics of sparse'' and the near edge of dense'' chaotic systems. The distinction lies in the ratio between the number of a system's random parameters and its Hilbert space dimension. While the two families share identical spectral correlations at energy scales comparable to the level spacing, the density of states and its fluctuations near the edge are different.
  arXiv  Detail & Related papers  (2024-03-20T11:31:51Z)
• On the tensorial structure of general covariant quantum systems [0.0]
  The structure of the observable algebra is related to a tensor product decomposition of the Hilbert space. It has been remarked that the Hamiltonian may determine this tensor product structure.
  arXiv  Detail & Related papers  (2023-12-20T19:03:23Z)
• Distribution of quantum gravity induced entanglement in many-body systems [14.37149160708975]
  We show that two distant test masses, each in a spatially superposed quantum state, become entangled due to their mutual gravitational interaction. We extend this treatment to a many-body system in a general setup and study the entanglement properties of the time-evolved state.
  arXiv  Detail & Related papers  (2023-11-14T16:37:32Z)
• Numerical investigations of the extensive entanglement Hamiltonian in quantum spin ladders [9.617349193925188]
  Entanglement constitutes one of the key concepts in quantum mechanics and serves as an indispensable tool in the understanding of quantum many-body systems. We perform extensive numerical investigations of extensive entanglement properties of coupled quantum spin chains.
  arXiv  Detail & Related papers  (2023-11-03T04:06:20Z)
• Realizing the entanglement Hamiltonian of a topological quantum Hall system [10.092164351939825]
  Topological quantum many-body systems, such as Hall insulators, are characterized by a hidden order encoded in the entanglement between their constituents. Entanglement entropy, an experimentally accessible single number that globally quantifies entanglement, has been proposed as a first signature of topological order. We use a synthetic dimension, encoded in the electronic spin of dysprosium atoms, to implement spatially deformed Hall systems.
  arXiv  Detail & Related papers  (2023-07-12T15:40:06Z)
• Discord and Decoherence [0.0]
  We investigate how quantum discord is modified by a quantum-to-classical transition. We find that the evolution of quantum discord in presence of an environment is a competition between the growth of the squeezing amplitude and the decrease of the state purity.
  arXiv  Detail & Related papers  (2021-12-09T17:01:54Z)
• Geometrical Rabi oscillations and Landau-Zener transitions in non-Abelian systems [0.0]
  We propose universal protocols to determine quantum geometric properties in non-Abelian systems. Our schemes suggest a way to prepare eigenstates of the quantum metric.
  arXiv  Detail & Related papers  (2021-05-06T14:09:52Z)
• Quantum particle across Grushin singularity [77.34726150561087]
  We study the phenomenon of transmission across the singularity that separates the two half-cylinders. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection. This allows to comprehend the distinguished status of the so-called bridging' transmission protocol previously identified in the literature.
  arXiv  Detail & Related papers  (2020-11-27T12:53:23Z)
• Experimental measurement of the divergent quantum metric of an exceptional point [10.73176455098217]
  We report the first experimental measurement of the quantum metric in a non-Hermitian system. The specific platform under study is an organic microcavity with exciton-polariton eigenstates, which demonstrate exceptional points.
  arXiv  Detail & Related papers  (2020-11-24T11:31:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.