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

Proceedings of the Steklov Institute of Mathematics

On a Complete Basis in the Space of Rotationally Invariant Operators of $$N$$ Quantum Spins $$1/2$$

Filipp Gennadevich Uskov

Published Jul 1, 2021• 1 citation
Publisher

Abstract

Systems of quantum spins $$1/2$$ with isotropic Heisenberg interaction play an important role in physics. In studying such systems, it may be useful to have a complete, yet non-overcomplete, basis of operators each of which has the symmetry of the Hamiltonian, i.e., is invariant with respect to rotations (global $$\mathrm{SU}(2)$$ transformations of the Pauli matrices). This paper presents an algorithm for constructing such a basis. The algorithm is implemented in Wolfram Mathematica.

DOI
10.1134/s0081543821020243

Related papers

Combined use of translational and spin-rotational invariance for spin systems

Tjark Heitmann, Jürgen Schnack

Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry ${C}_{N}$ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry $SU(2)$ can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both ${C}_{N}$ and $SU(2)$, mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.

arXiv:1902.02093•Physical review. B./Physical review. B•Apr 2, 2019•24 citations

Generalized exchange operators for a system of spin-1 particles

Charlie Jeudy, Michel Rouleux

Abstract The irreps ( S U ( 2 ) , H , U ) of SU(2) of dimension (2S + 1)N, i.e. operators acting on the space H = H N = C ( 2 S + 1 ) N of N identical particles with spin S, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case S = 1/2, Dirac [1] discovered that there is another rep given by ( S ( N ) , H , V ) where S ( N ) is the permutation group, Thus, the standard “linear” Hamiltonian, or Heisenberg interaction Hamiltonian H 0 = ∑ 1 ≤ i ≤ N S → i ⋅ S → j , where σ → i = 2 S → i is the vector of Pauli matrices, can be interpreted as the sum of the “Exchange Operators” Pij between particles i and j. Schrödinger [3] generalized to higher spin numbers S the Exchange Operator

arXiv:2411.03952•Journal of Physics Conference Series•Dec 1, 2024•0 citations

Generalized Spin Bases for Quantum Chemistry and Quantum Information

M. Kibler

Symmetry-adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a prime number. A connection of this work to the unitary group U(d) and the Pauli group is briefly underlined.

arXiv:0807.1850•Collection of Czechoslovak Chemical Communications•Jan 1, 2008•3 citations

Transformation of spin in quantum reference frames

Marion Mikusch, Luis C. Barbado, Časlav Brukner

In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum $j$ and introduce the transformations between them by upgrading the Euler angles occurring in classical $\text{SO}(3)$ spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large $j$, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.

arXiv:2103.05022•Physical Review Research•Nov 29, 2021•23 citations

Pauli Spin Matrices

Carl W David

The Pauli Spin Matrix representation of Spin Operators in Quantum Mechanics are explicitly demonstrated and illustrated in detail for one and two spin systems.

•Jan 1, 2006•0 citations

Spin correlations in two-particle systems: a pedagogically motivated comparison of computational approaches

S. Martins-Filho

In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-$1/2$ particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form $\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle$, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of $2\times2$ complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.

physics.ed-phCross-listedarXiv:2606.02361•Revista Brasileira de Ensino de Física•Jun 2, 2026•0 citations
On a Complete Basis in the Space of Rotationally Invariant Operators of NNN Quantum Spins 1/21/21/2
Systems of quantum spins 1/21/21/2 with isotropic Heisenberg interaction play an important role in physics. In studying such systems, it may be useful to have a complete, yet non-overcomplete, basis of operators each of which has the symmetry of the Hamiltonian, i.e., is invariant with respect to rotations (global SU(2)\mathrm{SU}(2)SU(2) transformations of the Pauli matrices). This paper presents an algorithm for constructing such a basis. The algorithm is implemented in Wolfram Mathematica.
Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry CN{C}_{N}CN​ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2)SU(2)SU(2) can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both CN{C}_{N}CN​ and SU(2)SU(2)SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.
Abstract The irreps ( S U ( 2 ) , H , U ) of SU(2) of dimension (2S + 1)N, i.e. operators acting on the space H = H N = C ( 2 S + 1 ) N of N identical particles with spin S, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case S = 1/2, Dirac [1] discovered that there is another rep given by ( S ( N ) , H , V ) where S ( N ) is the permutation group, Thus, the standard “linear” Hamiltonian, or Heisenberg interaction Hamiltonian H 0 = ∑ 1 ≤ i ≤ N S → i ⋅ S → j , where σ → i = 2 S → i is the vector of Pauli matrices, can be interpreted as the sum of the “Exchange Operators” Pij between particles i and j. Schrödinger [3] generalized to higher spin numbers S the Exchange Operator
Symmetry-adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a prime number. A connection of this work to the unitary group U(d) and the Pauli group is briefly underlined.
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum jjj and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3)\text{SO}(3)SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large jjj, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.
In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-1/21/21/2 particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form ⟨ψ∣Su^(1)Sv^(2)∣ψ⟩\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle⟨ψ∣Su^(1)​Sv^(2)​∣ψ⟩, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of 2×22\times22×2 complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.