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:2601.19111

Introduction to Quantum Entanglement Geometry

Kazuki Ikeda

quant-phhep-thmath-phmath.QAmath.RTSubmitted Jan 27, 2026• Announced Jan 28, 2026• 0 citations
arXivHTMLPDF

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.

Comments
87 pages, 1 figure

Related papers

Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras

Kazuki Ikeda

Quantum entanglement is a basic resource of quantum theory, but its usual definition assumes a fixed decomposition into subsystems. We develop an algebro--geometric framework to address quantum entanglement. For the Severi--Brauer scheme $SB(\mathcal{A})\to X$ of an Azumaya algebra and a factorization type $\mathbf{d}$, we prove that a global locus of product states exists exactly when the associated torsor under $\mathrm{PGL}_n$ reduces to the stabilizer $G_{\mathbf d}$ of the Segre variety. Thus entanglement in families is measured by a geometric obstruction to global subsystem structure. Beyond this reduction criterion, we identify the cohomology that carries computable entangling obstructions. After determinant and incidence data $L\subset R\subset K$ are chosen, the relevant classes lie in the mapping cone relative cohomology $E^\bullet(BL\to BR)$. Their absolute images on $X$, modulo classes induced from $BK$, give determinant entangling obstructions beyond the Brauer class. We construct the moduli of subsystem structures as the quotient $P/G_{\mathbf d}$ and realize it as a locally closed locus in the relative Hilbert scheme, with a compactification by degenerations of product state loci. In the bipartite case, a chosen subsystem structure globalizes the Schmidt rank stratification to a flat filtration with base change compatible resolutions and fiberwise constant numerical invariants. Examples show that reducibility can depend on the underlying torsor, not only on the Brauer class, with an interpretation by entangling monodromy.

math.AGCross-listedarXiv:2601.13764•Jan 21, 2026•0 citations

Geometric Entanglement Entropy on Projective Hilbert Space

Loris Di Cairano

Entanglement for pure bipartite states is most commonly quantified in a state-by-state manner to each pure state of a bipartite system a scalar quantity, such as the von Neumann entropy of a reduced density matrix. This provides a precise local characterization of how entangled a given state is. At the same time, this local description naturally invites a set of complementary, more global questions about the structure of the space of pure states: How abundant are the states with a given amount of entanglement within the full state space? Do the manifolds of constant entanglement exhibit distinct geometric regimes? These questions shift the focus from assigning an entanglement value to a single state to understanding the global organization and geometry of entanglement across the entire manifold of pure states. In this work, we develop a geometric framework in which these questions become natural. We regard the projective Hilbert space of pure states, endowed with the Fubini-Study metric, as a Riemannian manifold and promote bipartite entanglement to a macroscopic functional on this manifold. Its level sets stratify the space of pure states into hypersurfaces of constant entanglement, and we define a geometric entanglement entropy as the log-volume of these hypersurfaces, weighted by the Fubini-Study gradient of entanglement. This quantity plays the role of a microcanonical entropy in entanglement space: it measures the degeneracy of a given entanglement value in the natural quantum geometry. The framework is illustrated first in the simplest case of a single spin-1/2 and then for bipartite entanglement of spin systems, including a two-qubit example where explicit calculations can be carried out, along with a sketch of the extension to spin chains.

quant-pharXiv:2511.21186•Physical Review A•Nov 27, 2025•1 citation

Quantum Entanglement Geometry: A Topological Structure of Quantum Entanglement

Jincheng Zhang

Quantum entanglement, a fundamental phenomenon in quantum mechanics, has captivated scientists and philosophers alike. This paper explores the potential of leveraging quantum entanglement to uncover the underlying geometric structure of quantum entanglement itself, proposing a novel approach to understanding its mathematical foundations. We investigate the creation of quantum geometric models using entanglement, utilizing quantum computation for rapid exploration and simulation, and ultimately, aiming to identify key mathematical properties that could be crucial for advancing quantum computing and information processing.

•Aug 31, 2026•0 citations

On Multiquantum Bits, Segre Embeddings and Coxeter Chambers

Noémie C. Combe

This work explores the interplay between quantum information theory, algebraic geometry, and number theory, with a particular focus on multiqubit systems, their entanglement structure, and their classification via geometric embeddings. The Segre embedding, a fundamental construction in algebraic geometry, provides an algebraic framework to distinguish separable and entangled states, encoding quantum correlations in projective geometry. We develop a systematic study of qubit moduli spaces, illustrating the geometric structure of entanglement through hypercube constructions and Coxeter chamber decompositions. We establish a bijection between the Segre embeddings of tensor products of projective spaces and binary words of length $n-1$, structured as an $(n-1)$-dimensional hypercube, where adjacency corresponds to a single Segre operation. This reveals a combinatorial structure underlying the hierarchy of embeddings, with direct implications for quantum error correction schemes. The symmetry of the Segre variety under the Coxeter group of type $A$ allows us to analyze quantum states and errors through the lens of reflection groups, viewing separable states as lying in distinct Coxeter chambers on a Segre variety. The transitive action of the permutation group on these chambers provides a natural method for tracking errors in quantum states and potentially reversing them. Beyond foundational aspects, we highlight relations between Segre varieties and Dixon elliptic curves, drawing connections between entanglement and number theory.

arXiv:2502.00461•Feb 1, 2025•0 citations

Quantum Entanglement Geometry

Jincheng Zhang

Quantum entanglement, a phenomenon where two or more particles become linked regardless of distance, has captivated physicists and philosophers alike. This research explores the potential of utilizing geometric principles to understand and manipulate entanglement, proposing a novel model to represent quantum entanglement as a geometric structure. We aim to investigate how the inherent geometry of entangled states can be leveraged to optimize quantum computation and communication protocols. The core claim is that establishing a geometric model of quantum entanglement will unlock new avenues for enhancing the performance of these technologies. This paper delves into the foundational concepts and proposes a mechanism for translating entanglement into a geometric framework, ultimately seeking to reveal hidden properties and potential optimizations within the space of entangled quantum states.

•Aug 31, 2026•0 citations

Geometry of Quantum Logic Gates

M. W. AlMasri

In this work, we investigate the geometry of quantum logic gates within the holomorphic representation of quantum mechanics. We begin by embedding the physical qubit subspace into the space of holomorphic functions that are homogeneous of degree one in each Schwinger boson pair $(z_{a_j}, z_{b_j})$. Within this framework, we derive explicit closed-form differential operator representations for a universal set of quantum gates, including the Pauli operators, Hadamard, CNOT, CZ, and SWAP, and demonstrate that they preserve the physical subspace exactly. Restricting to unit-magnitude variables ($|z| = 1$) reveals a toroidal space $\mathbb{T}^{2N}$, on which quantum gates act as canonical transformations: Pauli operators generate Hamiltonian flows, the Hadamard gate induces a nonlinear automorphism, and entangling gates produce correlated diffeomorphisms that couple distinct toroidal factors. Beyond the torus, the full Segal--Bargmann space carries a natural Kähler geometry that governs amplitude dynamics. Entanglement is geometrically characterized via the Segre embedding into complex projective space, while topological protection arises from the $U(1)^N$ fiber bundle structure associated with the Jordan--Schwinger constraint.

quant-pharXiv:2602.15080•Modern Physics Letters A•Feb 17, 2026•0 citations
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.
Quantum entanglement is a basic resource of quantum theory, but its usual definition assumes a fixed decomposition into subsystems. We develop an algebro--geometric framework to address quantum entanglement. For the Severi--Brauer scheme SB(A)→XSB(\mathcal{A})\to XSB(A)→X of an Azumaya algebra and a factorization type d\mathbf{d}d, we prove that a global locus of product states exists exactly when the associated torsor under PGLn\mathrm{PGL}_nPGLn​ reduces to the stabilizer GdG_{\mathbf d}Gd​ of the Segre variety. Thus entanglement in families is measured by a geometric obstruction to global subsystem structure. Beyond this reduction criterion, we identify the cohomology that carries computable entangling obstructions. After determinant and incidence data L⊂R⊂KL\subset R\subset KL⊂R⊂K are chosen, the relevant classes lie in the mapping cone relative cohomology E∙(BL→BR)E^\bullet(BL\to BR)E∙(BL→BR). Their absolute images on XXX, modulo classes induced from BKBKBK, give determinant entangling obstructions beyond the Brauer class. We construct the moduli of subsystem structures as the quotient P/GdP/G_{\mathbf d}P/Gd​ and realize it as a locally closed locus in the relative Hilbert scheme, with a compactification by degenerations of product state loci. In the bipartite case, a chosen subsystem structure globalizes the Schmidt rank stratification to a flat filtration with base change compatible resolutions and fiberwise constant numerical invariants. Examples show that reducibility can depend on the underlying torsor, not only on the Brauer class, with an interpretation by entangling monodromy.
Entanglement for pure bipartite states is most commonly quantified in a state-by-state manner to each pure state of a bipartite system a scalar quantity, such as the von Neumann entropy of a reduced density matrix. This provides a precise local characterization of how entangled a given state is. At the same time, this local description naturally invites a set of complementary, more global questions about the structure of the space of pure states: How abundant are the states with a given amount of entanglement within the full state space? Do the manifolds of constant entanglement exhibit distinct geometric regimes? These questions shift the focus from assigning an entanglement value to a single state to understanding the global organization and geometry of entanglement across the entire manifold of pure states. In this work, we develop a geometric framework in which these questions become natural. We regard the projective Hilbert space of pure states, endowed with the Fubini-Study metric, as a Riemannian manifold and promote bipartite entanglement to a macroscopic functional on this manifold. Its level sets stratify the space of pure states into hypersurfaces of constant entanglement, and we define a geometric entanglement entropy as the log-volume of these hypersurfaces, weighted by the Fubini-Study gradient of entanglement. This quantity plays the role of a microcanonical entropy in entanglement space: it measures the degeneracy of a given entanglement value in the natural quantum geometry. The framework is illustrated first in the simplest case of a single spin-1/2 and then for bipartite entanglement of spin systems, including a two-qubit example where explicit calculations can be carried out, along with a sketch of the extension to spin chains.
Quantum entanglement, a fundamental phenomenon in quantum mechanics, has captivated scientists and philosophers alike. This paper explores the potential of leveraging quantum entanglement to uncover the underlying geometric structure of quantum entanglement itself, proposing a novel approach to understanding its mathematical foundations. We investigate the creation of quantum geometric models using entanglement, utilizing quantum computation for rapid exploration and simulation, and ultimately, aiming to identify key mathematical properties that could be crucial for advancing quantum computing and information processing.
This work explores the interplay between quantum information theory, algebraic geometry, and number theory, with a particular focus on multiqubit systems, their entanglement structure, and their classification via geometric embeddings. The Segre embedding, a fundamental construction in algebraic geometry, provides an algebraic framework to distinguish separable and entangled states, encoding quantum correlations in projective geometry. We develop a systematic study of qubit moduli spaces, illustrating the geometric structure of entanglement through hypercube constructions and Coxeter chamber decompositions. We establish a bijection between the Segre embeddings of tensor products of projective spaces and binary words of length n−1n-1n−1, structured as an (n−1)(n-1)(n−1)-dimensional hypercube, where adjacency corresponds to a single Segre operation. This reveals a combinatorial structure underlying the hierarchy of embeddings, with direct implications for quantum error correction schemes. The symmetry of the Segre variety under the Coxeter group of type AAA allows us to analyze quantum states and errors through the lens of reflection groups, viewing separable states as lying in distinct Coxeter chambers on a Segre variety. The transitive action of the permutation group on these chambers provides a natural method for tracking errors in quantum states and potentially reversing them. Beyond foundational aspects, we highlight relations between Segre varieties and Dixon elliptic curves, drawing connections between entanglement and number theory.
Quantum entanglement, a phenomenon where two or more particles become linked regardless of distance, has captivated physicists and philosophers alike. This research explores the potential of utilizing geometric principles to understand and manipulate entanglement, proposing a novel model to represent quantum entanglement as a geometric structure. We aim to investigate how the inherent geometry of entangled states can be leveraged to optimize quantum computation and communication protocols. The core claim is that establishing a geometric model of quantum entanglement will unlock new avenues for enhancing the performance of these technologies. This paper delves into the foundational concepts and proposes a mechanism for translating entanglement into a geometric framework, ultimately seeking to reveal hidden properties and potential optimizations within the space of entangled quantum states.
In this work, we investigate the geometry of quantum logic gates within the holomorphic representation of quantum mechanics. We begin by embedding the physical qubit subspace into the space of holomorphic functions that are homogeneous of degree one in each Schwinger boson pair (zaj,zbj)(z_{a_j}, z_{b_j})(zaj​​,zbj​​). Within this framework, we derive explicit closed-form differential operator representations for a universal set of quantum gates, including the Pauli operators, Hadamard, CNOT, CZ, and SWAP, and demonstrate that they preserve the physical subspace exactly. Restricting to unit-magnitude variables (∣z∣=1|z| = 1∣z∣=1) reveals a toroidal space T2N\mathbb{T}^{2N}T2N, on which quantum gates act as canonical transformations: Pauli operators generate Hamiltonian flows, the Hadamard gate induces a nonlinear automorphism, and entangling gates produce correlated diffeomorphisms that couple distinct toroidal factors. Beyond the torus, the full Segal--Bargmann space carries a natural Kähler geometry that governs amplitude dynamics. Entanglement is geometrically characterized via the Segre embedding into complex projective space, while topological protection arises from the U(1)NU(1)^NU(1)N fiber bundle structure associated with the Jordan--Schwinger constraint.