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arXiv:2601.17936

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

quant-phmath.NASubmitted Jan 25, 2026• Announced Jan 27, 2026• 0 citations• Revised Mar 2, 2026 (v2)
v1 · Jan 25, 2026v2 · Mar 2, 2026
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Abstract

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.

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On Golden Gates and Discrepancy

Steven B. Damelin, Qiwei Liang, B. A. W. Mode

A 1-qubit gate is defined as an element of the unitary group $U(2)$. The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of $PSU(2)$ to approximate an arbitrary element of $U(2)$. Notably, this problem has been recently studied by Sarnak in [8]. We will work with a universal gate set $T$, a subset of $SU(2)$ such that the group generated by the elements of $T$ is dense in $SU(2)$. For an arbitrary $\varepsilon > 0$, we define $t_{\varepsilon}$ as the minimum reduced word length such that every point of $SU(2)$ lies within a ball of radius $\varepsilon$ centered at the points in the dense subgroup generated by $T$. We define a measure of efficiency on $T$, denoted $K(T)$, construct a $t_{\varepsilon}'$ such that most points in $PSU(2)$ have optimally efficient approximation and we conjecture that $\varepsilon \leq f(t_{\varepsilon})5^{-t_{\varepsilon}/4}$ for a function $f:(0,\infty)\to (1,\infty)$ satisfying: \begin{equation*} \lim_{ t_{\varepsilon} \to \infty} log(f(t_{\varepsilon}))/t_{\varepsilon} \end{equation*} exists with value $0$. Our conjecture can be formulated to mean the following: Let $\nu(5^{t_{\varepsilon}})$ denote the set of integer solutions of the quadratic form: $x_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}$. Let $M\equiv M_{S^3}(\mathcal{N})$ denote the covering radius of the points $\mathcal{N}=\nu(5^{t_{\varepsilon}})\cup\nu(5^{t_{\varepsilon}-1})$ on the sphere $S^{3}$ in $\mathbb{R}^{4}$. Then $M \sim f(\log N)N^{-1/4}$. Here $N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2$. This is a weaker conjecture than that proposed in [2], namely: $M\geq N^{{-1/4}+o(1)}$. We prove that this conjecture implies that $K(T)=\dfrac{4}{3}$. This answers a question of Sarnak in [8].

•Jun 18, 2015•0 citations

Algebraic and Mathematical Foundations of Quantum Computation: Hilbert Spaces, Group Structures, and Quantum Algorithms

Bhavesh B. Prajapati

Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.

Communications on Applied Nonlinear Analysis•Jan 24, 2025•0 citations

Deforming the Trail: Baseline Quantum Circuitry for $\text{SU(2)}_k$ Lattice Gauge Theory

Zoë Webb-Mack, Natalie Klco

Quantifying quantum resources for simulating the fundamental forces of Nature is sensitive to the mapping of gauge fields onto finite quantum computational architectures. When locally truncating lattice gauge theories in the irreducible representation basis, it has been proposed to further deform the theory via quantum groups. The purpose of this deformation is (1) to provide an infinite tower of finite-dimensional ($d = k+1$) groups systematically approximating the infinite-dimensional gauge links and (2) to restore the physical unitarity of a plaquette operator diagonalization procedure analytically derived from the field continuum by recontracting vertex pairs. For the SU(2)$_k$ Yang-Mills pure-gauge theory, we provide a constructive strategy of gauge-variant completions to extend this unitarity to the entire computational Hilbert space, leading to well-defined time evolution unitaries as targets for optimized circuit synthesis. Leveraging basic circuit decompositions and symmetries of the diagonalized plaquette operator, we report resource upper-bounds on the generalized-controlled-X two-qudit gates for arbitrary local truncation $d$, reducing estimates and scaling relative to the non-deformed theory by three polynomial powers from $O(d^8)$ to $O(d^5)$. Examining the stronger q-deformed gauge constraint, which softens the total flux at vertices, we show that the physical Hilbert space dimension of the deformed plaquette operator scales equivalently to its non-deformed counterpart with a constant factor $0.2563(5)$. Thus, despite affecting interactions at all scales as exemplified by the observed flux hierarchy inversion symmetry, q-deformation continues to pass scrutiny as a reliable truncation offering advantages in quantum circuit synthesis.

quant-pharXiv:2605.15076•May 15, 2026•0 citations

A Note on a Quantitative Form of the Solovay-Kitaev Theorem

Steven B. Damelin, B. A. W. Mode

The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of $SU(2)$ to approximate an arbitrary element of $SU(2)$. The Solovay-Kitaev Theorem is a well-known theorem that guarantees the existence of a finite sequence of 1-qubit quantum gates approximating an arbitrary unitary matrix in $SU(2)$ within specified accuracy $\varepsilon > 0$. In this note we study a quantitative description of this theorem in the following sense. We will work with a universal gate set $T$, a subset of $SU(2)$ such that the group generated by the elements of $T$ is dense in $SU(2)$. For $\varepsilon > 0$ small enough, we define $t_{\varepsilon}$ as the minimum reduced word length such that every point of $SU(2)$ lies within a ball of radius $\varepsilon$ centered at the points in the dense subgroup generated by $T$. For a measure of efficiency on T, which we denote $K(T)$, we prove the following theorem: Fix a $δ$ in $[0, \frac{2}{3}]$. Choose $f: (0, \infty) \rightarrow (1, \infty)$ satisfying $\lim_{\varepsilon\to 0+}\dfrac{\log(f(t_{\varepsilon}))}{t_{\varepsilon}}$ exists with value $0$. Assume that the inequality $\varepsilon \leqslant f(t_{\varepsilon})\cdot 5^{\frac{-t_{\varepsilon}}{6-3δ}}$ holds. Then $K(T) \leqslant 2-δ$. Our conjecture implies the following: Let $ν(5^{t_{\varepsilon}})$ denote the set of integer solutions of the quadratic form: $x_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}$. Let $M\equiv M_{S^3}(\mathcal{N})$ denote the covering radius of the points $\mathcal{N}=ν(5^{t_{\varepsilon}})\cupν(5^{t_{\varepsilon}-1})$ on the sphere $S^{3}$ in $\mathbb{R}^{4}$. Then $M \sim f(\log N)N^{\frac{-1}{6-3δ}}$. Here $N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2$.

arXiv:1709.03007•Sep 9, 2017•1 citation

On a Problem of Construction of Efficient Universal Sets of Quantum Gates: A preliminary report

Qingzhong Liang, Steven B. Damelin

A 1-qubit gate is defined as an element of the unitary group $U(2)$, the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group $G=PSU(2)$. Here, $PSU(2)$ is the group formed by taking all elements of $U(2)$ with determinant 1 and taking the quotient of this subgroup by the group $\{I,-I\}$ where $I$ is the identity matrix. In this http URL [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers. In this report, we study the efficiency of the approximation of $G$ by using a dense group generated by a universal subset of $G$. The measure of the efficiency $K(g)$ of an arbitrary universal subset $g$ is defined in [1] and satisfies $K(g)\geqslant 1$. $g$ is more efficient when $K(g)$ is closer to $1$. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].

•Jun 18, 2015•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
Elementary Quantum Gates from Lie Group Embeddings in U(2n)U(2^n)U(2n): Geometry, Universality, and Discretization
In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group U(2n)U(2^n)U(2n). Writing N=2nN=2^nN=2n, we introduce an \emph{intrinsic descriptor layer} in U(N)U(N)U(N) by declaring as primitive the motions inside faithful embedded copies of SU(2)SU(2)SU(2) (phase-free), together with a phase-inclusive U(2)U(2)U(2) variant. We describe the embedding landscape \Emb(SU(2),U(N))\Emb(SU(2),U(N))\Emb(SU(2),U(N)) as a finite union of U(N)U(N)U(N)-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by \Gr2(\CN)\Gr_2(\C^N)\Gr2​(\CN) up to a PSU(2)PSU(2)PSU(2) gauge. Equipping U(N)U(N)U(N) with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in SU(N)SU(N)SU(N) from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in U(N)U(N)U(N) by explicit abelian phase bookkeeping (equivalently, via the U(2)U(2)U(2) two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in SU(2)SU(2)SU(2) (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield U(N)U(N)U(N)-level synthesis with global operator-norm error control.
A 1-qubit gate is defined as an element of the unitary group U(2)U(2)U(2). The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of PSU(2)PSU(2)PSU(2) to approximate an arbitrary element of U(2)U(2)U(2). Notably, this problem has been recently studied by Sarnak in [8]. We will work with a universal gate set TTT, a subset of SU(2)SU(2)SU(2) such that the group generated by the elements of TTT is dense in SU(2)SU(2)SU(2). For an arbitrary ε>0\varepsilon > 0ε>0, we define tεt_{\varepsilon}tε​ as the minimum reduced word length such that every point of SU(2)SU(2)SU(2) lies within a ball of radius ε\varepsilonε centered at the points in the dense subgroup generated by TTT. We define a measure of efficiency on TTT, denoted K(T)K(T)K(T), construct a tε′t_{\varepsilon}'tε′​ such that most points in PSU(2)PSU(2)PSU(2) have optimally efficient approximation and we conjecture that ε≤f(tε)5−tε/4\varepsilon \leq f(t_{\varepsilon})5^{-t_{\varepsilon}/4}ε≤f(tε​)5−tε​/4 for a function f:(0,∞)→(1,∞)f:(0,\infty)\to (1,\infty)f:(0,∞)→(1,∞) satisfying: \begin{equation*} \lim_{ t_{\varepsilon} \to \infty} log(f(t_{\varepsilon}))/t_{\varepsilon} \end{equation*} exists with value 000. Our conjecture can be formulated to mean the following: Let ν(5tε)\nu(5^{t_{\varepsilon}})ν(5tε​) denote the set of integer solutions of the quadratic form: x12+x22+x32+x42=5tεx_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}x12​+x22​+x32​+x42​=5tε​. Let M≡MS3(N)M\equiv M_{S^3}(\mathcal{N})M≡MS3​(N) denote the covering radius of the points N=ν(5tε)∪ν(5tε−1)\mathcal{N}=\nu(5^{t_{\varepsilon}})\cup\nu(5^{t_{\varepsilon}-1})N=ν(5tε​)∪ν(5tε​−1) on the sphere S3S^{3}S3 in R4\mathbb{R}^{4}R4. Then M∼f(log⁡N)N−1/4M \sim f(\log N)N^{-1/4}M∼f(logN)N−1/4. Here N≡N(ε)=6⋅5tε−2N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2N≡N(ε)=6⋅5tε​−2. This is a weaker conjecture than that proposed in [2], namely: M≥N−1/4+o(1)M\geq N^{{-1/4}+o(1)}M≥N−1/4+o(1). We prove that this conjecture implies that K(T)=43K(T)=\dfrac{4}{3}K(T)=34​. This answers a question of Sarnak in [8].
Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.
Deforming the Trail: Baseline Quantum Circuitry for SU(2)k\text{SU(2)}_kSU(2)k​ Lattice Gauge Theory
Quantifying quantum resources for simulating the fundamental forces of Nature is sensitive to the mapping of gauge fields onto finite quantum computational architectures. When locally truncating lattice gauge theories in the irreducible representation basis, it has been proposed to further deform the theory via quantum groups. The purpose of this deformation is (1) to provide an infinite tower of finite-dimensional (d=k+1d = k+1d=k+1) groups systematically approximating the infinite-dimensional gauge links and (2) to restore the physical unitarity of a plaquette operator diagonalization procedure analytically derived from the field continuum by recontracting vertex pairs. For the SU(2)k_kk​ Yang-Mills pure-gauge theory, we provide a constructive strategy of gauge-variant completions to extend this unitarity to the entire computational Hilbert space, leading to well-defined time evolution unitaries as targets for optimized circuit synthesis. Leveraging basic circuit decompositions and symmetries of the diagonalized plaquette operator, we report resource upper-bounds on the generalized-controlled-X two-qudit gates for arbitrary local truncation ddd, reducing estimates and scaling relative to the non-deformed theory by three polynomial powers from O(d8)O(d^8)O(d8) to O(d5)O(d^5)O(d5). Examining the stronger q-deformed gauge constraint, which softens the total flux at vertices, we show that the physical Hilbert space dimension of the deformed plaquette operator scales equivalently to its non-deformed counterpart with a constant factor 0.2563(5)0.2563(5)0.2563(5). Thus, despite affecting interactions at all scales as exemplified by the observed flux hierarchy inversion symmetry, q-deformation continues to pass scrutiny as a reliable truncation offering advantages in quantum circuit synthesis.
The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of SU(2)SU(2)SU(2) to approximate an arbitrary element of SU(2)SU(2)SU(2). The Solovay-Kitaev Theorem is a well-known theorem that guarantees the existence of a finite sequence of 1-qubit quantum gates approximating an arbitrary unitary matrix in SU(2)SU(2)SU(2) within specified accuracy ε>0\varepsilon > 0ε>0. In this note we study a quantitative description of this theorem in the following sense. We will work with a universal gate set TTT, a subset of SU(2)SU(2)SU(2) such that the group generated by the elements of TTT is dense in SU(2)SU(2)SU(2). For ε>0\varepsilon > 0ε>0 small enough, we define tεt_{\varepsilon}tε​ as the minimum reduced word length such that every point of SU(2)SU(2)SU(2) lies within a ball of radius ε\varepsilonε centered at the points in the dense subgroup generated by TTT. For a measure of efficiency on T, which we denote K(T)K(T)K(T), we prove the following theorem: Fix a δδδ in [0,23][0, \frac{2}{3}][0,32​]. Choose f:(0,∞)→(1,∞)f: (0, \infty) \rightarrow (1, \infty)f:(0,∞)→(1,∞) satisfying lim⁡ε→0+log⁡(f(tε))tε\lim_{\varepsilon\to 0+}\dfrac{\log(f(t_{\varepsilon}))}{t_{\varepsilon}}limε→0+​tε​log(f(tε​))​ exists with value 000. Assume that the inequality ε⩽f(tε)⋅5−tε6−3δ\varepsilon \leqslant f(t_{\varepsilon})\cdot 5^{\frac{-t_{\varepsilon}}{6-3δ}}ε⩽f(tε​)⋅56−3δ−tε​​ holds. Then K(T)⩽2−δK(T) \leqslant 2-δK(T)⩽2−δ. Our conjecture implies the following: Let ν(5tε)ν(5^{t_{\varepsilon}})ν(5tε​) denote the set of integer solutions of the quadratic form: x12+x22+x32+x42=5tεx_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}x12​+x22​+x32​+x42​=5tε​. Let M≡MS3(N)M\equiv M_{S^3}(\mathcal{N})M≡MS3​(N) denote the covering radius of the points N=ν(5tε)∪ν(5tε−1)\mathcal{N}=ν(5^{t_{\varepsilon}})\cupν(5^{t_{\varepsilon}-1})N=ν(5tε​)∪ν(5tε​−1) on the sphere S3S^{3}S3 in R4\mathbb{R}^{4}R4. Then M∼f(log⁡N)N−16−3δM \sim f(\log N)N^{\frac{-1}{6-3δ}}M∼f(logN)N6−3δ−1​. Here N≡N(ε)=6⋅5tε−2N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2N≡N(ε)=6⋅5tε​−2.
A 1-qubit gate is defined as an element of the unitary group U(2)U(2)U(2), the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group G=PSU(2)G=PSU(2)G=PSU(2). Here, PSU(2)PSU(2)PSU(2) is the group formed by taking all elements of U(2)U(2)U(2) with determinant 1 and taking the quotient of this subgroup by the group {I,−I}\{I,-I\}{I,−I} where III is the identity matrix. In this http URL [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers. In this report, we study the efficiency of the approximation of GGG by using a dense group generated by a universal subset of GGG. The measure of the efficiency K(g)K(g)K(g) of an arbitrary universal subset ggg is defined in [1] and satisfies K(g)⩾1K(g)\geqslant 1K(g)⩾1. ggg is more efficient when K(g)K(g)K(g) is closer to 111. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].
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.