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:2111.07208 · Quantum Information and Computation

Symmetric states and dynamics of three quantum bits

Francesca Albertini, Domenico D’Alessandro

Published May 1, 2022• 3 citations
arXivHTMLPDF

Abstract

The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.

DOI
10.26421/qic22.7-8-1

Related papers

Subspace Controllability and Clebsch-Gordan Decomposition of Symmetric Quantum Networks

Domenico D’Alessandro

We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, {\it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group $S_n$. Because of the symmetry, the underlying Hilbert space, ${\cal H}=(\mathbb{C}^d)^{\otimes n}$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(d^n)$, denoted here by $u^{S_n}(d^n)$. The dynamical Lie algebra ${\cal L}$, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra $u^{S_n}(d^n)$. If ${\cal L}$ acts as $su\left( \dim(V) \right)$ on each of the invariant subspaces $V$, the system is called {\it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space ${\cal H}$ coincides with the {\it Clebsch-Gordan} splitting of $(\mathbb{C}^d)^{\otimes n}$ into {\it irreducible representations} of $su(d)$. In this view, $u^{S_n}(d^n)$, is the direct sum of certain $su(n_j)$ for some $n_j$'s we shall specify, and its {\it center} which is the Abelian (Lie) algebra generated by the {\it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general $n$ and $d$ but we recast previous results on $n$ qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of $n=3$, $d=3$, that is, {\it three qutrits}.

arXiv:2307.12908•Jul 24, 2023•0 citations

Subspace Controllability and Clebsch–Gordan Decomposition of Symmetric Quantum Networks

Domenico D’Alessandro

Abstract. This paper describes a framework for the controllability analysis of networks of [Formula: see text] quantum systems of an arbitrary dimension [Formula: see text] , qudits, with dynamics determined by Hamiltonians that are invariant under the permutation group [Formula: see text]. Because of the symmetry, the underlying Hilbert space, [Formula: see text], splits into invariant subspaces for the Lie algebra of [Formula: see text]-invariant elements in [Formula: see text], denoted here by [Formula: see text]. The dynamical Lie algebra [Formula: see text], which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra [Formula: see text]. If [Formula: see text] acts as [Formula: see text] on each of the invariant subspaces [Formula: see text], the system is called subspace controllable. The described approach is based on recognizing that such a splitting of the Hilbert space [Formula: see text] coincides with the Clebsch–Gordan splitting of [Formula: see text] into irreducible representations of [Formula: see text]. In this view, [Formula: see text] is the direct sum of certain [Formula: see text] for some [Formula: see text]’s we shall specify and its center, which is the Abelian (Lie) algebra generated by the Casimir operators. Most of the results presented are for general [Formula: see text] and [Formula: see text], but the paper provides a complete treatment and proof of subspace controllability for the new case of a system with [Formula: see text], [Formula: see text], that is, three qutrits. The results are motivated by recent great interest in symmetric quantum states and systems (see, e.g., [K. Eckert et al., Ann. Phys., 299 (2001), pp. 88–127], [A. W. Harrow, “The Church of the Symmetric Subspace,” https://arxiv.org/abs/1308.6595 , 2013], [P. Migdal, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A, 88 (2013), 021335]) both at the theoretical and experimental levels and by recent proposals in geometric quantum machine learning [M. Cerezo et al., Nature Comput. Sci., 2 (2022), pp. 567–576], [Q. T. Nguyen et al., PRX Quant., 5 (2024), 020328] to exploit symmetries in the data and in quantum circuits to improve the performance of learning protocols.

SIAM Journal on Control and Optimization•Dec 13, 2024•1 citation

Entanglement of Quantum States which are Zero on the Symmetric Sector

Domenico D’Alessandro

We consider a quantum system of n qudits and the Clebsch-Gordan decomposition of the associated Hilbert space. In this decomposition, one of the subspaces is the so-called symmetric subspace or symmetric sector, that is, the subspace of all states that are invariant under the action of the symmetric group. We prove that any separable state must have a nonzero component on the symmetric sector, or, equivalently, any state which has zero component on the symmetric sector must be entangled. For the cases of n=2,3 particles, and in arbitrary dimension d, this result can be refined by providing sharp lower bounds on the size of the component of separable states on the symmetric sector. This leads us to identify a class of entanglement witnesses for these systems. We provide an example showing that in the multipartite case, this class of witnesses detects PPT entangled states.

arXiv:2311.17260•Nov 28, 2023•0 citations

Deformations of the symmetric subspace of qubit chains

Ángel Ballesteros, Iván Gutiérrez-Sagredo, José de Ramón +1 more

Abstract The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group SU(2), whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group U q ( su ( 2 ) ) . We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.

arXiv:2503.23554•Journal of Physics A Mathematical and Theoretical•Sep 10, 2025•3 citations

Maximally entangled mixed symmetric states of two qubits

Eduardo Serrano-Ensástiga, John Martin

In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.

•Jan 1, 2021•0 citations

Local unitary classes of states invariant under permutation subgroups

David W. Lyons, Jesse R. Arnold, Ashley F. Swogger

The study of entanglement properties of multiqubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.

arXiv:2109.06921•Physical Review A•Mar 25, 2022•10 citations
The unitary group acting on the Hilbert space H:=(C2)⊗3{\cal H}:=(C^2)^{\otimes 3}H:=(C2)⊗3 of three quantum bits admits a Lie subgroup, US3(8)U^{S_3}(8)US3​(8), of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space H{\cal H}H splits into three invariant subspaces of dimensions 444, 222 and 222 respectively, each corresponding to an irreducible representation of su(2)su(2)su(2). The subspace of dimension 444 is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under US3(8)U^{S_3}(8)US3​(8). This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup US3(8)U^{S_3}(8)US3​(8). As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin 12\frac{1}{2}21​ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.
We describe a framework for the controllability analysis of networks of nnn quantum systems of an arbitrary dimension ddd, {\it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group SnS_nSn​. Because of the symmetry, the underlying Hilbert space, H=(Cd)⊗n{\cal H}=(\mathbb{C}^d)^{\otimes n}H=(Cd)⊗n, splits into invariant subspaces for the Lie algebra of SnS_nSn​-invariant elements in u(dn)u(d^n)u(dn), denoted here by uSn(dn)u^{S_n}(d^n)uSn​(dn). The dynamical Lie algebra L{\cal L}L, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra uSn(dn)u^{S_n}(d^n)uSn​(dn). If L{\cal L}L acts as su(dim⁡(V))su\left( \dim(V) \right)su(dim(V)) on each of the invariant subspaces VVV, the system is called {\it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space H{\cal H}H coincides with the {\it Clebsch-Gordan} splitting of (Cd)⊗n(\mathbb{C}^d)^{\otimes n}(Cd)⊗n into {\it irreducible representations} of su(d)su(d)su(d). In this view, uSn(dn)u^{S_n}(d^n)uSn​(dn), is the direct sum of certain su(nj)su(n_j)su(nj​) for some njn_jnj​'s we shall specify, and its {\it center} which is the Abelian (Lie) algebra generated by the {\it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general nnn and ddd but we recast previous results on nnn qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of n=3n=3n=3, d=3d=3d=3, that is, {\it three qutrits}.
Abstract. This paper describes a framework for the controllability analysis of networks of [Formula: see text] quantum systems of an arbitrary dimension [Formula: see text] , qudits, with dynamics determined by Hamiltonians that are invariant under the permutation group [Formula: see text]. Because of the symmetry, the underlying Hilbert space, [Formula: see text], splits into invariant subspaces for the Lie algebra of [Formula: see text]-invariant elements in [Formula: see text], denoted here by [Formula: see text]. The dynamical Lie algebra [Formula: see text], which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra [Formula: see text]. If [Formula: see text] acts as [Formula: see text] on each of the invariant subspaces [Formula: see text], the system is called subspace controllable. The described approach is based on recognizing that such a splitting of the Hilbert space [Formula: see text] coincides with the Clebsch–Gordan splitting of [Formula: see text] into irreducible representations of [Formula: see text]. In this view, [Formula: see text] is the direct sum of certain [Formula: see text] for some [Formula: see text]’s we shall specify and its center, which is the Abelian (Lie) algebra generated by the Casimir operators. Most of the results presented are for general [Formula: see text] and [Formula: see text], but the paper provides a complete treatment and proof of subspace controllability for the new case of a system with [Formula: see text], [Formula: see text], that is, three qutrits. The results are motivated by recent great interest in symmetric quantum states and systems (see, e.g., [K. Eckert et al., Ann. Phys., 299 (2001), pp. 88–127], [A. W. Harrow, “The Church of the Symmetric Subspace,” https://arxiv.org/abs/1308.6595 , 2013], [P. Migdal, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A, 88 (2013), 021335]) both at the theoretical and experimental levels and by recent proposals in geometric quantum machine learning [M. Cerezo et al., Nature Comput. Sci., 2 (2022), pp. 567–576], [Q. T. Nguyen et al., PRX Quant., 5 (2024), 020328] to exploit symmetries in the data and in quantum circuits to improve the performance of learning protocols.
We consider a quantum system of n qudits and the Clebsch-Gordan decomposition of the associated Hilbert space. In this decomposition, one of the subspaces is the so-called symmetric subspace or symmetric sector, that is, the subspace of all states that are invariant under the action of the symmetric group. We prove that any separable state must have a nonzero component on the symmetric sector, or, equivalently, any state which has zero component on the symmetric sector must be entangled. For the cases of n=2,3 particles, and in arbitrary dimension d, this result can be refined by providing sharp lower bounds on the size of the component of separable states on the symmetric sector. This leads us to identify a class of entanglement witnesses for these systems. We provide an example showing that in the multipartite case, this class of witnesses detects PPT entangled states.
Abstract The symmetric subspace of multi-qubit systems, that is, the space of states invariant under permutations, is commonly encountered in applications in the context of quantum information and communication theory. It is known that the symmetric subspace can be described in terms of irreducible representations of the group SU(2), whose representation spaces form a basis of symmetric states, the so-called Dicke states. In this work, we present deformations of the symmetric subspace as deformations of this group structure, which are promoted to a quantum group U q ( su ( 2 ) ) . We see that deformations of the symmetric subspace obtained in this manner correspond to local deformations of the inner product of each spin, in such a way that departure from symmetry can be encoded in a position-dependent inner product. The consequences and possible extensions of these results are also discussed.
In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.
The study of entanglement properties of multiqubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.