Geometry of Quantum Logic Gates

M. W. AlMasri

arxiv: 2602.15080 · v2 · submitted 2026-02-16 · 🪐 quant-ph

Geometry of Quantum Logic Gates

M. W. AlMasri This is my paper

Pith reviewed 2026-05-15 22:14 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum gatesholomorphic representationSchwinger bosonstoroidal spacecanonical transformationsentanglementKähler geometrydifferential operators

0 comments

The pith

Quantum gates receive closed-form differential operator forms in the holomorphic representation while exactly preserving the qubit subspace.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper embeds the physical qubit subspace into holomorphic functions homogeneous of degree one in Schwinger boson pairs. It derives explicit differential operators for Pauli operators, Hadamard, CNOT, CZ and SWAP gates. These operators are shown to preserve the physical subspace exactly. On the unit-magnitude torus, the gates act as canonical transformations including Hamiltonian flows and diffeomorphisms. The approach also uses Kähler geometry and Segre embedding to describe amplitude dynamics and entanglement.

Core claim

We embed the physical qubit subspace into the space of holomorphic functions homogeneous of degree one in each Schwinger boson pair. 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 reveals a toroidal space on which quantum gates act as canonical transformations, with Pauli operators generating Hamiltonian flows, the Hadamard gate inducing a nonlinear automorphism, and entangling gates producing correlated diffeomorphisms. The full space carries a Kähler geometry that g

What carries the argument

The embedding into holomorphic functions homogeneous of degree one in each Schwinger boson pair, on which gates are realized as differential operators.

Load-bearing premise

The physical qubit subspace embeds exactly into the space of holomorphic functions homogeneous of degree one in each Schwinger boson pair without losing information or adding unphysical states.

What would settle it

Demonstrating that applying one of the derived differential operators to a physical state produces a non-zero component outside the homogeneous degree-one subspace would disprove the exact preservation.

Figures

Figures reproduced from arXiv: 2602.15080 by M. W. AlMasri.

**Figure 2.** Figure 2: FIG. 2. Schematic illustration of entanglement geometry in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

read the original abstract

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 K\"ahler 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper spells out differential operator forms for standard gates in the holomorphic Schwinger-boson picture and ties them to toroidal geometry and Segre embeddings, but the constructions follow directly from prior literature without new results.

read the letter

The main point is that the author derives explicit closed-form differential operators for the Pauli gates, Hadamard, CNOT, CZ, and SWAP inside the space of holomorphic functions that are homogeneous of degree one in each Schwinger boson pair, then shows these operators map the physical subspace to itself. On the unit circle this becomes action on a torus, with Paulis as Hamiltonian flows and entangling gates as correlated diffeomorphisms. The Segre embedding is used to describe entanglement geometrically and the U(1) fibers are linked to topological aspects. These pieces are laid out clearly enough that a reader already comfortable with the Bargmann representation can follow the geometric gloss without much trouble. The derivations appear to be straightforward applications of the standard Schwinger operators once the homogeneity constraint is fixed, so the preservation property holds by construction rather than by new insight. The Kähler structure on the full Segal-Bargmann space is invoked to govern amplitude dynamics, but again this is standard background. The soft spot is that nothing here resolves an open question or produces a computation that could not be done with existing tools; the explicit forms are useful for visualization but do not change the underlying algebra. The topological-protection claim stays at the level of bundle structure without showing concrete protection against errors or decoherence. This work is aimed at people who already like geometric quantum mechanics and want the gate operators written out in holomorphic language for intuition or teaching. It is not aimed at readers looking for new algorithms or falsifiable predictions. The manuscript deserves a serious referee if the full derivations are complete and the geometric claims are backed by explicit checks, because the presentation is coherent even if the advance is incremental.

Referee Report

1 major / 2 minor

Summary. The manuscript embeds the physical qubit subspace into holomorphic functions homogeneous of degree one in each Schwinger boson pair (z_{a_j}, z_{b_j}). It derives explicit closed-form differential operator representations for a universal gate set (Pauli operators, Hadamard, CNOT, CZ, SWAP) and shows these operators map the physical subspace exactly onto itself. Restricting to |z|=1 yields a toroidal geometry T^{2N} on which the gates act as canonical transformations (Hamiltonian flows for Pauli, nonlinear automorphisms for Hadamard, correlated diffeomorphisms for entangling gates). The work further interprets the construction via Kähler geometry on the Segal-Bargmann space, Segre embedding for entanglement, and U(1)^N bundle structure for topological protection.

Significance. If the explicit operators and exact subspace preservation hold, the paper supplies a concrete geometric realization of quantum gates within the Bargmann-Schwinger holomorphic framework. This links standard quantum information primitives to symplectic and Kähler structures, offering potential for analytic study of gate flows and entanglement geometry without additional parameters. The approach builds directly on established constructions (Schwinger bosons, Bargmann realization, Segre embedding), yielding derivations that are algebraic rather than numerical.

major comments (1)

[Abstract and §3] Abstract and §3: the claim that CNOT, CZ and SWAP preserve the multi-qubit homogeneity constraint exactly is asserted but the explicit differential-operator expressions and the algebraic verification that they map the space of functions homogeneous of degree one in each pair onto itself are not displayed; without these steps the preservation statement for entangling gates cannot be checked directly from the given constructions.

minor comments (2)

[§4] The transition from the full Segal-Bargmann space to the |z|=1 torus is stated without an explicit projection operator or measure; a short paragraph clarifying how the restriction is implemented while retaining the differential operators would improve readability.
[§2] Notation for the per-qubit pairs (z_{a_j}, z_{b_j}) and the associated homogeneity constraint is introduced in the abstract but not restated with an equation number when the operators are applied; adding a displayed equation for the constraint would aid cross-reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestion to strengthen the presentation of the entangling gates is well taken, and we address it directly below.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3: the claim that CNOT, CZ and SWAP preserve the multi-qubit homogeneity constraint exactly is asserted but the explicit differential-operator expressions and the algebraic verification that they map the space of functions homogeneous of degree one in each pair onto itself are not displayed; without these steps the preservation statement for entangling gates cannot be checked directly from the given constructions.

Authors: We agree that the explicit differential-operator expressions for CNOT, CZ, and SWAP, together with the algebraic verification of homogeneity preservation, were not displayed in sufficient detail. In the revised manuscript we will add the full closed-form expressions for these operators (derived from the Schwinger-boson realization) and include the direct algebraic steps showing that each maps the space of functions homogeneous of degree one in every pair (z_{a_j}, z_{b_j}) onto itself. These additions will make the subspace-preservation claim for the entangling gates verifiable by direct inspection. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper begins with the standard Schwinger boson embedding of qubits into homogeneous holomorphic functions of degree one, a well-established construction independent of the present work. Gate operators are then expressed as differential operators in the Bargmann realization; their exact preservation of the physical subspace follows algebraically from the degree-preserving action of creation and annihilation operators on the homogeneous sector, without any fitted parameters or self-definitional steps. Geometric interpretations (torus flows, Kähler structure, Segre embedding) are derived as consequences of this representation rather than smuggled assumptions. No load-bearing self-citations, uniqueness theorems from prior author work, or renamings of known results appear; the central claim remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard embedding of qubits into homogeneous holomorphic functions of Schwinger bosons and the properties of the Segal-Bargmann space; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Physical qubit subspace embeds exactly into holomorphic functions homogeneous of degree one in each Schwinger boson pair
Stated as the starting framework in the abstract.

pith-pipeline@v0.9.0 · 5495 in / 1094 out tokens · 37114 ms · 2026-05-15T22:14:38.316065+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

physical qubit states correspond to functions that are homogeneous of degree one in each bosonic pair (z_aj, z_bj) ... (z_aj ∂/∂z_aj + z_bj ∂/∂z_bj) f(z) = f(z)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Pauli operators generate Hamiltonian flows ... Hadamard gate induces a nonlinear automorphism ... entangling gates produce correlated diffeomorphisms

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

, sN) runs over all 2 N computational basis states

The multi-indexs= (s 1, . . . , sN) runs over all 2 N computational basis states. In this representation, the logical basis states map to simple monomials: |0⟩j 7→z aj ,(2.11) |1⟩j 7→z bj .(2.12) Superpositions and entangled states correspond to linear combinations of such monomials, with the homogeneity condition (2.9) ensuring that the total degree in e...

work page
[2]

Under this restriction, the physical state space be- comes a torusT 2N parameterized by the phase angles {ϕaj , ϕbj }N j=1. The basis states correspond to: |0⟩j ≡ |↑⟩ j 7→e iϕaj ,(4.2) |1⟩j ≡ |↓⟩ j 7→e iϕbj .(4.3) The differential operators transform according to the chain rule: ∂ ∂zaj = ∂ϕaj ∂zaj ∂ ∂ϕaj =− i zaj ∂ ∂ϕaj =−ie −iϕaj ∂ ∂ϕaj , (4.4) and simil...

work page
[3]

Hamiltonian Flow Definition To establish the geometric framework rigorously, we first define the Hamiltonian flow structure onT 2N. Definition 1(Hamiltonian Flow onT 2N).Let(T 2N , ω) be the2N-dimensional torus equipped with the canonical symplectic form ω= NX j=1 dϕaj ∧dϕ bj .(4.12) 5 For any smooth real-valued functionH:T 2N →R(the Hamiltonian), the ass...

work page
[4]

Pauli Gates as Hamiltonian Flows Each Pauli operator generates a Hamiltonian flow on the toroidal phase spaceT 2N, governed by the symplectic form in Eq. (4.12). The Pauli operators correspond to specific Hamiltonian functions: •Z j gate (Phase rotation):The Hamiltonian is HZj =ϕ aj −ϕ bj = ∆ϕ j. From Eq. (4.14), the equa- tions of motion are: ˙ϕaj = ∂HZj...

work page
[5]

This map is: •Nonlinear:The output phases depend nontrivially on ∆ϕj via trigonometric functions

Hadamard as a Nonlinear Automorphism The Hadamard transformation induces a diffeomor- phismH j :T 2 →T 2 on the 2-torus associated with qubit j: ϕ′ aj ϕ′ bj = arg 1 +e i∆ϕj arg 1−e i∆ϕj + Σϕj 2 1 1 ,(4.23) where Σϕ j =ϕ aj +ϕ bj and ∆ϕ j =ϕ aj −ϕ bj. This map is: •Nonlinear:The output phases depend nontrivially on ∆ϕj via trigonometric functions. 6 •Singu...

work page
[6]

For example: •SWAPacts as a permutation of toroidal coor- dinates: (ϕ aj , ϕbj , ϕak , ϕbk)7→(ϕ ak , ϕbk , ϕaj , ϕbj), which is a global isometry ofT 2N

Multi-qubit Gates as Entangling Diffeomorphisms Gates like CNOT and SWAP induce correlations be- tween distinct toroidal factors. For example: •SWAPacts as a permutation of toroidal coor- dinates: (ϕ aj , ϕbj , ϕak , ϕbk)7→(ϕ ak , ϕbk , ϕaj , ϕbj), which is a global isometry ofT 2N. •CNOTimplements a conditional flow: the vector field on the target qubit’...

work page
[7]

Topological Considerations This geometric perspective reveals that quantum com- putation can be viewed as the controlled navigation of a point (or wavefunction) on a high-dimensional torus, where gates correspond to carefully engineered flows and deformations. The topological properties ofT 2N, such as its nontrivial fundamental groupπ 1(T2N) =Z 2N, under...

work page
[8]

Strocchi,Complex Coordinates and Quantum Mechan- ics, Rev

F. Strocchi,Complex Coordinates and Quantum Mechan- ics, Rev. Mod. Phys.38, 36 (1966)

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W. P. Schleich,Quantum Optics in Phase Space(Wiley- VCH, Berlin, 2001)

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C. K. Zachos, D. B. Fairlie, T. L. Curtright,Quantum mechanics in phase space: an overview with selected pa- pers, World Scientific 2005

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S. L. Braunstein and P. van Loock,Quantum informa- tion with continuous variables, Rev. Mod. Phys.77, 513 (2005)

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Miquel, J

C. Miquel, J. P. Paz, M. Saraceno,Quantum computers in phase space, Phys. Rev. A65, 062309 (2002)

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I. E. Segal,Mathematical problems of relativistic physics, in Kac, M. (ed.), Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II, Lectures in Applied Mathematics, American Mathematical Society (1963)

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Bargmann,On a Hilbert space of analytic functions and an associated integral transform part I, Commun

V. Bargmann,On a Hilbert space of analytic functions and an associated integral transform part I, Commun. Pure. Appl. Math.14(3): 187 (1961)

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Perelomov,Generalized Coherent States and Their Applications, Springer Berlin, Heidelberg (1986)

A. Perelomov,Generalized Coherent States and Their Applications, Springer Berlin, Heidelberg (1986)

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Vourdas, R

A. Vourdas, R. F. Bishop,Thermal coherent states in the Bargmann representation, Phys. Rev. A50, 3331 (1994)

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First Summer School in Analysis and Mathematical Physics

B. C. Hall,Holomorphic methods in analysis and mathe- matical physics, in “First Summer School in Analysis and Mathematical Physics” (S. P´ erez-Esteva and C. Villegas- Blas, Eds.), 1-59, Contemp. Math.260, Amer. Math. Soc., 2000

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M. W. AlMasri and M. R. B. Wahiddin,Bargmann representation of quantum absorption refrigerators, Rep. Math. Phys.89(2), Pages 185-198 (2022)

work page 2022
[20]

M. W. AlMasri and M. R. B. Wahiddin.Quantum De- composition Algorithm For Master Equations of Stochas- tic Processes: The Damped Spin Case, Mod. Phys. Lett. A,37(32), 2250216 (2022)

work page 2022
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Ashtekar, T

A. Ashtekar, T. Schilling,Geometrical formulation of quantum mechanics, in On Einstein’s Path (Springer, 1999), pp. 23-65

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[22]

Brody, L

D. Brody, L. Hughston,Geometric quantum mechanics, J. Geom. Phys.38, 19-53 (2001)

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[23]

Chabaud, S

U. Chabaud, S. Mehraban,Holomorphic representation of quantum computations, Quantum6, 831 (2022)

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[24]

Abraham and J

R. Abraham and J. E. Marsden,Foundations of Mechan- ics, 2nd edition, Benjamin/Cummings (1978)

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[25]

V. I. Arnold,Mathematical Methods of Classical Mechan- ics, 2nd edition, Springer (1989)

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[26]

Hatcher,Algebraic Topology, Cambridge University Press (2002)

A. Hatcher,Algebraic Topology, Cambridge University Press (2002)

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[27]

J. P. Provost and G. Vallee,Riemannian structure on manifolds of quantum states,Commun. Math. Phys.76, 289 (1980)

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[28]

Bengtsson and K

I. Bengtsson and K. ˙Zyczkowski,Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition, Cambridge University Press (2017)

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[29]

J. K. Pachos,Introduction to Topological Quantum Computation(Cambridge University Press, Cambridge, 2012)

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[30]

Heydari,Segre variety, conifold, Hopf fibration, and separable multi-qubit states, Quantum Information and Computation6(2006) 400-409

H. Heydari,Segre variety, conifold, Hopf fibration, and separable multi-qubit states, Quantum Information and Computation6(2006) 400-409

work page 2006
[31]

Holweck, J.-G

F. Holweck, J.-G. Luque, and J.-Y. Thibon,Geometric descriptions of entangled states by auxiliary varieties, J. Math. Phys.53(10), 102203 (2012)

work page 2012
[32]

J. R. Klauder,The action option and a Feynman quan- tization of spinor fields in terms of ordinary c-numbers, Ann. Phys. (N.Y.)11, 123 (1960)

work page 1960
[33]

W. D. Kirwin,Coherent States in Geometric Quantiza- tion, J. Geom. Phys.57(2007), no. 2, pages 531 – 548

work page 2007
[34]

S. T. Ali, J.-P. Antoine, and J.-P. Gazeau,Coherent States, Wavelets, and Their Generalizations, 2nd ed., Theoretical and Mathematical Physics (Springer, New York, 2014)

work page 2014

[1] [1]

, sN) runs over all 2 N computational basis states

The multi-indexs= (s 1, . . . , sN) runs over all 2 N computational basis states. In this representation, the logical basis states map to simple monomials: |0⟩j 7→z aj ,(2.11) |1⟩j 7→z bj .(2.12) Superpositions and entangled states correspond to linear combinations of such monomials, with the homogeneity condition (2.9) ensuring that the total degree in e...

work page

[2] [2]

Under this restriction, the physical state space be- comes a torusT 2N parameterized by the phase angles {ϕaj , ϕbj }N j=1. The basis states correspond to: |0⟩j ≡ |↑⟩ j 7→e iϕaj ,(4.2) |1⟩j ≡ |↓⟩ j 7→e iϕbj .(4.3) The differential operators transform according to the chain rule: ∂ ∂zaj = ∂ϕaj ∂zaj ∂ ∂ϕaj =− i zaj ∂ ∂ϕaj =−ie −iϕaj ∂ ∂ϕaj , (4.4) and simil...

work page

[3] [3]

Hamiltonian Flow Definition To establish the geometric framework rigorously, we first define the Hamiltonian flow structure onT 2N. Definition 1(Hamiltonian Flow onT 2N).Let(T 2N , ω) be the2N-dimensional torus equipped with the canonical symplectic form ω= NX j=1 dϕaj ∧dϕ bj .(4.12) 5 For any smooth real-valued functionH:T 2N →R(the Hamiltonian), the ass...

work page

[4] [4]

Pauli Gates as Hamiltonian Flows Each Pauli operator generates a Hamiltonian flow on the toroidal phase spaceT 2N, governed by the symplectic form in Eq. (4.12). The Pauli operators correspond to specific Hamiltonian functions: •Z j gate (Phase rotation):The Hamiltonian is HZj =ϕ aj −ϕ bj = ∆ϕ j. From Eq. (4.14), the equa- tions of motion are: ˙ϕaj = ∂HZj...

work page

[5] [5]

This map is: •Nonlinear:The output phases depend nontrivially on ∆ϕj via trigonometric functions

Hadamard as a Nonlinear Automorphism The Hadamard transformation induces a diffeomor- phismH j :T 2 →T 2 on the 2-torus associated with qubit j: ϕ′ aj ϕ′ bj = arg 1 +e i∆ϕj arg 1−e i∆ϕj + Σϕj 2 1 1 ,(4.23) where Σϕ j =ϕ aj +ϕ bj and ∆ϕ j =ϕ aj −ϕ bj. This map is: •Nonlinear:The output phases depend nontrivially on ∆ϕj via trigonometric functions. 6 •Singu...

work page

[6] [6]

For example: •SWAPacts as a permutation of toroidal coor- dinates: (ϕ aj , ϕbj , ϕak , ϕbk)7→(ϕ ak , ϕbk , ϕaj , ϕbj), which is a global isometry ofT 2N

Multi-qubit Gates as Entangling Diffeomorphisms Gates like CNOT and SWAP induce correlations be- tween distinct toroidal factors. For example: •SWAPacts as a permutation of toroidal coor- dinates: (ϕ aj , ϕbj , ϕak , ϕbk)7→(ϕ ak , ϕbk , ϕaj , ϕbj), which is a global isometry ofT 2N. •CNOTimplements a conditional flow: the vector field on the target qubit’...

work page

[7] [7]

Topological Considerations This geometric perspective reveals that quantum com- putation can be viewed as the controlled navigation of a point (or wavefunction) on a high-dimensional torus, where gates correspond to carefully engineered flows and deformations. The topological properties ofT 2N, such as its nontrivial fundamental groupπ 1(T2N) =Z 2N, under...

work page

[8] [8]

Strocchi,Complex Coordinates and Quantum Mechan- ics, Rev

F. Strocchi,Complex Coordinates and Quantum Mechan- ics, Rev. Mod. Phys.38, 36 (1966)

work page 1966

[9] [9]

W. P. Schleich,Quantum Optics in Phase Space(Wiley- VCH, Berlin, 2001)

work page 2001

[10] [10]

C. K. Zachos, D. B. Fairlie, T. L. Curtright,Quantum mechanics in phase space: an overview with selected pa- pers, World Scientific 2005

work page 2005

[11] [11]

S. L. Braunstein and P. van Loock,Quantum informa- tion with continuous variables, Rev. Mod. Phys.77, 513 (2005)

work page 2005

[12] [12]

Miquel, J

C. Miquel, J. P. Paz, M. Saraceno,Quantum computers in phase space, Phys. Rev. A65, 062309 (2002)

work page 2002

[13] [13]

I. E. Segal,Mathematical problems of relativistic physics, in Kac, M. (ed.), Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II, Lectures in Applied Mathematics, American Mathematical Society (1963)

work page 1960

[14] [14]

Bargmann,On a Hilbert space of analytic functions and an associated integral transform part I, Commun

V. Bargmann,On a Hilbert space of analytic functions and an associated integral transform part I, Commun. Pure. Appl. Math.14(3): 187 (1961)

work page 1961

[15] [15]

Perelomov,Generalized Coherent States and Their Applications, Springer Berlin, Heidelberg (1986)

A. Perelomov,Generalized Coherent States and Their Applications, Springer Berlin, Heidelberg (1986)

work page 1986

[16] [16]

G. B. Folland,Harmonic Analysis in Phase Space, Princeton University Press (1989)

work page 1989

[17] [17]

Vourdas, R

A. Vourdas, R. F. Bishop,Thermal coherent states in the Bargmann representation, Phys. Rev. A50, 3331 (1994)

work page 1994

[18] [18]

First Summer School in Analysis and Mathematical Physics

B. C. Hall,Holomorphic methods in analysis and mathe- matical physics, in “First Summer School in Analysis and Mathematical Physics” (S. P´ erez-Esteva and C. Villegas- Blas, Eds.), 1-59, Contemp. Math.260, Amer. Math. Soc., 2000

work page 2000

[19] [19]

M. W. AlMasri and M. R. B. Wahiddin,Bargmann representation of quantum absorption refrigerators, Rep. Math. Phys.89(2), Pages 185-198 (2022)

work page 2022

[20] [20]

M. W. AlMasri and M. R. B. Wahiddin.Quantum De- composition Algorithm For Master Equations of Stochas- tic Processes: The Damped Spin Case, Mod. Phys. Lett. A,37(32), 2250216 (2022)

work page 2022

[21] [21]

Ashtekar, T

A. Ashtekar, T. Schilling,Geometrical formulation of quantum mechanics, in On Einstein’s Path (Springer, 1999), pp. 23-65

work page 1999

[22] [22]

Brody, L

D. Brody, L. Hughston,Geometric quantum mechanics, J. Geom. Phys.38, 19-53 (2001)

work page 2001

[23] [23]

Chabaud, S

U. Chabaud, S. Mehraban,Holomorphic representation of quantum computations, Quantum6, 831 (2022)

work page 2022

[24] [24]

Abraham and J

R. Abraham and J. E. Marsden,Foundations of Mechan- ics, 2nd edition, Benjamin/Cummings (1978)

work page 1978

[25] [25]

V. I. Arnold,Mathematical Methods of Classical Mechan- ics, 2nd edition, Springer (1989)

work page 1989

[26] [26]

Hatcher,Algebraic Topology, Cambridge University Press (2002)

A. Hatcher,Algebraic Topology, Cambridge University Press (2002)

work page 2002

[27] [27]

J. P. Provost and G. Vallee,Riemannian structure on manifolds of quantum states,Commun. Math. Phys.76, 289 (1980)

work page 1980

[28] [28]

Bengtsson and K

I. Bengtsson and K. ˙Zyczkowski,Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition, Cambridge University Press (2017)

work page 2017

[29] [29]

J. K. Pachos,Introduction to Topological Quantum Computation(Cambridge University Press, Cambridge, 2012)

work page 2012

[30] [30]

Heydari,Segre variety, conifold, Hopf fibration, and separable multi-qubit states, Quantum Information and Computation6(2006) 400-409

H. Heydari,Segre variety, conifold, Hopf fibration, and separable multi-qubit states, Quantum Information and Computation6(2006) 400-409

work page 2006

[31] [31]

Holweck, J.-G

F. Holweck, J.-G. Luque, and J.-Y. Thibon,Geometric descriptions of entangled states by auxiliary varieties, J. Math. Phys.53(10), 102203 (2012)

work page 2012

[32] [32]

J. R. Klauder,The action option and a Feynman quan- tization of spinor fields in terms of ordinary c-numbers, Ann. Phys. (N.Y.)11, 123 (1960)

work page 1960

[33] [33]

W. D. Kirwin,Coherent States in Geometric Quantiza- tion, J. Geom. Phys.57(2007), no. 2, pages 531 – 548

work page 2007

[34] [34]

S. T. Ali, J.-P. Antoine, and J.-P. Gazeau,Coherent States, Wavelets, and Their Generalizations, 2nd ed., Theoretical and Mathematical Physics (Springer, New York, 2014)

work page 2014