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:2403.17435

Restrictions of holomorphic sections to products

Manimugdha Saikia

Published Mar 26, 2024• 0 citations
arXivHTMLPDF

Abstract

We associate quantum states with subsets of a product of two compact connected Kähler manifolds $M_1$ and $M_2$. To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of $M_1\times M_2$ such that the states associated with these subsets are the original states, to begin with.

DOI
10.48550/arxiv.2403.17435

Related papers

Entanglement and products

Tatyana Barron, Noah Wheatley

We address a general question whether geometry of submanifolds of an integral compact Kähler manifold is characterized by invariants that come from analysis. In geometric quantization, we have an integral compact Kähler manifold M and a holomorphic line bundle L on this manifold. There is a known procedure how to associate a sequence of mixed states (ρN), N=1,2,3,…, to a submanifold Λ of M. Do analytic properties of this sequence reflect the geometry of Λ ? In this paper, we consider the case when M is a product of two integral compact Kähler manifolds. We show that, when Λ is a product submanifold of M, then the entanglement of formation of ρN is zero for all sufficiently large N.

arXiv:2104.10597•Linear and Multilinear Algebra•Mar 5, 2022•0 citations

Quantum Mechanics and Operator algebras on the Hilbert ball

Katsunori Kawamura

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

arXiv:funct-an/9710002•Oct 21, 1997•1 citation

Quantum States from Tangent Vectors

J. M. Isidro

We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite–dimensional, compact, classical phase spacesC, by explicitly constructing Hilbert–space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum states (except the vacuum) appear as tangent vectors toC. The vacuum state appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are classified by the elements of Pic (C), the Picard group of C. In this way Pic (C) appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case whenC is complex projective space CP n.

•Jan 1, 2004•2 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

QUANTUM STATES FROM TANGENT VECTORS

JOSÉ M. ISIDRO

We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces [Formula: see text], by explicitly constructing Hilbert-space vector bundles over [Formula: see text]. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle [Formula: see text], plus a complex line bundle [Formula: see text]. Quantum states (except the vacuum) appear as tangent vectors to [Formula: see text]. The vacuum state appears as the fibrewise generator of [Formula: see text]. Holomorphic line bundles [Formula: see text] are classified by the elements of [Formula: see text], the Picard group of [Formula: see text]. In this way [Formula: see text] appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when [Formula: see text] is complex projective space CPn.

arXiv:quant-ph/0407159•Modern Physics Letters A•Oct 10, 2004•3 citations

Coherent states associated to the real Jacobi group

Stefan Berceanu, Piotr Kielanowski, Anatol Odzijewicz +2 more

Using the coherent states attached to the complex Jacobi group—the semi‐direct product of the Heisenberg‐Weyl group with the real symplectic group—we obtain the resolution of unity of coherent states defined on the Kähler manifold which is the product of the n‐dimensional complex plane with the Siegel upper half‐plane.

AIP conference proceedings•Jan 1, 2007•12 citations
We associate quantum states with subsets of a product of two compact connected Kähler manifolds M1M_1M1​ and M2M_2M2​. To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of M1×M2M_1\times M_2M1​×M2​ such that the states associated with these subsets are the original states, to begin with.
We address a general question whether geometry of submanifolds of an integral compact Kähler manifold is characterized by invariants that come from analysis. In geometric quantization, we have an integral compact Kähler manifold M and a holomorphic line bundle L on this manifold. There is a known procedure how to associate a sequence of mixed states (ρN), N=1,2,3,…, to a submanifold Λ of M. Do analytic properties of this sequence reflect the geometry of Λ ? In this paper, we consider the case when M is a product of two integral compact Kähler manifolds. We show that, when Λ is a product submanifold of M, then the entanglement of formation of ρN is zero for all sufficiently large N.
Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C∗^{*}∗-algebra is represented as a function algebra on the set of pure states with a noncommutative ∗*∗-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space P(H){\cal P}({\cal H})P(H) of a Hilbert space H{\cal H}H. The space P(H){\cal P}({\cal H})P(H) is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature ccc. The Hilbert ball BHB_{\cal H}BH​ is defined by the open unit ball in H{\cal H}H and it is a Kähler manifold with c<0c<0c<0. We introduce the quantum mechanics on BHB_{\cal H}BH​. As an application, we show the structure of the noncommutative function algebra on BHB_{\cal H}BH​.
We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite–dimensional, compact, classical phase spacesC, by explicitly constructing Hilbert–space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum states (except the vacuum) appear as tangent vectors toC. The vacuum state appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are classified by the elements of Pic (C), the Picard group of C. In this way Pic (C) appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case whenC is complex projective space CP n.
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.
We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces [Formula: see text], by explicitly constructing Hilbert-space vector bundles over [Formula: see text]. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle [Formula: see text], plus a complex line bundle [Formula: see text]. Quantum states (except the vacuum) appear as tangent vectors to [Formula: see text]. The vacuum state appears as the fibrewise generator of [Formula: see text]. Holomorphic line bundles [Formula: see text] are classified by the elements of [Formula: see text], the Picard group of [Formula: see text]. In this way [Formula: see text] appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when [Formula: see text] is complex projective space CPn.