arXiv:2609.15302

Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction

Tim Heib, Pérola Milman

quant-phmath-phSubmitted Sep 14, 2026• Announced Sep 15, 2026• 0 citations

Abstract

Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles---including bosons and symmetric collections of spin-$1/2$ particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of $\mathfrak{su}_{\mathbb{R}}(2)$, contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-$n/2$ representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto $\mathfrak{su}_{\mathbb{R}}(n+1)$ and $\mathfrak{u}_{\mathbb{R}}(n+1)$, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.

Comments: 50 pages (16+34), 1 figure, comments welcome

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Dec 19, 20251 citation

Nonuniversality from conserved superoperators in unitary circuits

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Universal Entanglers for Bosonic and Fermionic Systems

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Universality of swap for qudits: a representation theory approach

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arXiv:2609.15302

Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction

Tim Heib, Pérola Milman

quant-phmath-phSubmitted Sep 14, 2026• Announced Sep 15, 2026• 0 citations

arXiv HTML PDF

Abstract

1/2

particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of

\mathfrak{su}_{\mathbb{R}}(2)

, contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-

n/2

representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto

\mathfrak{su}_{\mathbb{R}}(n+1)

and

\mathfrak{u}_{\mathbb{R}}(n+1)

, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.

Comments: 50 pages (16+34), 1 figure, comments welcome

Related papers

Non-Universality from Conserved Superoperators in Unitary Circuits

Marco Lastres, Frank Pollmann, Sanjay Moudgalya

An important result in the theory of quantum control is the "universality" of $2$ -local unitary gates, i.e. the fact that any global unitary evolution of a system of $L$ qudits can be implemented by composition of $2$ -local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using $k$ -local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.An important result in the theory of quantum control is the "universality" of $2$-local unitary gates, i.e. the fact that any global unitary evolution of a system of $L$ qudits can be implemented by composition of $2$-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using $k$-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.

Dec 19, 20251 citation

Nonuniversality from conserved superoperators in unitary circuits

Marco Lastres, Frank Pollmann, Sanjay Moudgalya

An important result in the theory of quantum control is the “universality” of 2-local unitary gates, i.e., the fact that any global unitary evolution of a system of L qudits can be implemented by composition of 2-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using k -local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of nonuniversality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.An important result in the theory of quantum control is the “universality” of 2-local unitary gates, i.e., the fact that any global unitary evolution of a system of L qudits can be implemented by composition of 2-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using k -local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of nonuniversality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.

arXiv:2409.11407Physical review. B./Physical review. BDec 9, 20252 citations

Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator

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math-phCross-listedarXiv:2610.16317Oct 13, 2026

Finite-dimensional Lie algebras in bosonic quantum dynamics: The single-mode case

Tim Heib, Andreea Silvia Goia, Sona Baghiyan +2 more

We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra $\hat{A}_1$ , defined as the real subalgebra of the Weyl algebra $A_1$ consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for $\hat{A}_1$ , which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in $\hat{A}_1$ ? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of $\hat{A}_1$ that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in $\hat{A}_1$ , and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra $\hat{A}_1$, defined as the real subalgebra of the Weyl algebra $A_1$ consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for $\hat{A}_1$, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in $\hat{A}_1$? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of $\hat{A}_1$ that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in $\hat{A}_1$, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.

quant-pharXiv:2511.06940Nov 11, 2025

Universal Entanglers for Bosonic and Fermionic Systems

Joel Klassen, Jianxin Chen, Bei Zeng

A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$ , a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$ . It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ . The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq 8$ ). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$, a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq 8$). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.

arXiv:1305.7489Leibniz international proceedings in informaticsJan 1, 20131 citation

Universality of swap for qudits: a representation theory approach

James R. van Meter

An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on $d$ -state systems for $d>2$ . This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each $d$ -state system can be identified with the fundamental representation of $SU(d)$ , and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universality in terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on $d$-state systems for $d>2$. This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each $d$-state system can be identified with the fundamental representation of $SU(d)$, and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universality in terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.

arXiv:2103.12303Mar 23, 20212 citations

1/2

\mathfrak{su}_{\mathbb{R}}(2)

n/2

\mathfrak{su}_{\mathbb{R}}(n+1)

and

\mathfrak{u}_{\mathbb{R}}(n+1)

, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.

An important result in the theory of quantum control is the "universality" of

2

-local unitary gates, i.e. the fact that any global unitary evolution of a system of

L

qudits can be implemented by composition of

2

-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using

k

-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.

We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-

\tfrac12

degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series

su(1,1)

representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional

j=0

sector, complete energy-shell multiplicities, and the effect of particle interchange.

\hat{A}_1

, defined as the real subalgebra of the Weyl algebra

A_1

consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for

\hat{A}_1

, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in

\hat{A}_1

generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in

\hat{A}_1

that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in

\hat{A}_1

? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of

\hat{A}_1

that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in

\hat{A}_1

, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.

A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles

1,2

with a Hilbert space

\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}

, a UE exists when

\min{(d_1,d_2)}\geq 3

and

(d_1,d_2)\neq (3,3)

. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of

\mathbb{C}^{d}\otimes\mathbb{C}^{d}

. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of

\mathbb{C}^{d}\otimes\mathbb{C}^{d}

if and only if

d\geq 3

(resp.

d\geq 8

). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.

An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on

d

-state systems for

d>2

. This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each

d

-state system can be identified with the fundamental representation of

SU(d)

, and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universality in terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.