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:2606.02361 · Revista Brasileira de Ensino de Física

Spin Correlations in Two-Particle Systems: A Pedagogically Motivated Comparison of Computational Approaches

S. Martins-Filho

physics.ed-phquant-phSubmitted Jun 1, 2026• Announced Jun 2, 2026• 0 citations
v1 · Jun 1, 2026v2 · Jun 17, 2026
arXivHTMLPDF

Abstract

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.

Journal reference
Rev. Bras. Ens. Fis. 48, e20260134 (2026)
DOI
10.1590/1806-9126-RBEF-2026-0134
Comments
24 pages, 3 figures, accepted for publication in Revista Brasileira de Ensino de Física (RBEF)

Related papers

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

Sokol Andoni

Abstract A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.

•Jan 20, 2022•0 citations

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

Sokol Andoni

Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector and spin state. Under the spin-position decoupling approximation it consists of the ordered sum of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) is oriented due to ordering; (3) reproduces all standard expectation values, including the total one-particle spin modulus A; (4) constrains basis opposite spins to have same orientation; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled spin pairs have opposite orientation and precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of orientation; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). The cross-product terms in the expression for the squared total spin of two particles relates to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.

•Mar 26, 2021•0 citations

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

Sokol Andoni

Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one-particle spin modulus 𝑆tot = (ℏ/2)√3; (4) constrains basis opposite spins to have same handedness; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs have precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of handedness; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). Cross-product terms in the expression for the squared total spin of two particles relate to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as done in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.

•Sep 3, 2021•0 citations

Spin 1/2 one‐ and two‐particle systems in physical space without eigen‐algebra or tensor product

Sokol Andoni

A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum S=(√3 ℏ)⁄2; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.

arXiv:2212.09463•Mathematical Methods in the Applied Sciences•Dec 13, 2022•5 citations

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

Sokol Andoni

Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standardPauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists ofthree orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) isHermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one particlespin modulus 𝑆tot = √3ℏ/2; (4) constrains basis opposite spins to have same handedness; (5)allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs haveprecisely related gauge phases and can be of same or opposite handedness; (2) the dimensionality of the spinspace doubles due to variation of handedness; (3) the four maximally entangled states are naturally definedby the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) andinversion (singlet). The cross-product terms in the expression for the squared total spin of two particlesrelates to experiment and they yield all standard expectation values after measurement. Here spin is directlydefined and transformed in 3D orientation space, without use of eigen algebra and tensor product as in thestandard formulation. The formalism allows working with whole geometric objects instead of onlycomponents, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘onlab’ (uncontrolled gauge phase) spin transformations.

•Apr 8, 2021•0 citations

Beyond Quantum Mechanics: Local QM Space‑Time Algebra Prediction of the Singlet State — Computational Notebooks and Simulations Package

Carl F. Diether

I derive the quantum mechanical prediction $-\mathbf{a}\cdot\mathbf{b}$ for the singlet spin state using local measurement functions within the Space-Time Algebra (STA) framework. I establish a compact and computationally tractable STA representation of the two-particle singlet state, $\Psi = \tfrac{1}{2}(I\sigma_2^{(1)} - I\sigma_2^{(2)})$, which is simpler than previously published forms but is not required for the correlation calculation. The analysis shows that STA naturally generates both scalar dot-product terms ($\mathbf{a}\cdot\mathbf{s}_1$ and $\mathbf{s}_2\cdot\mathbf{b}$) and bivector wedge-product terms ($\mathbf{a}\wedge\mathbf{s}_1$ and $\mathbf{s}_2\wedge\mathbf{b}$) from local spin–detector interactions, whereas standard quantum mechanics retains only the scalar contributions through expectation-value projection. Because the standard quantum formalism represents correlations via operator expectation values, the antisymmetric (cross‑product) contributions are averaged out at the level of observables. In contrast, the STA formulation keeps these bivector terms explicit in the intermediate geometric products, raising the question of whether potentially meaningful geometric structure is being hidden by the quantum averaging procedure. I verify the analytical derivation using computational simulations in Mathematica with a Clifford algebra package, confirming that local measurement functions reproduce the standard quantum correlation $-\cos\theta$, where $\theta$ is the angle between measurement directions $\mathbf{a}$ and $\mathbf{b}$. The correlation $-{\bf a}\cdot {\bf b}$ is the scalar coordinate of 3-sphere rotors.

•Feb 25, 2026•1 citation
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.
Abstract A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.
Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector and spin state. Under the spin-position decoupling approximation it consists of the ordered sum of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) is oriented due to ordering; (3) reproduces all standard expectation values, including the total one-particle spin modulus A; (4) constrains basis opposite spins to have same orientation; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled spin pairs have opposite orientation and precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of orientation; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). The cross-product terms in the expression for the squared total spin of two particles relates to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.
Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one-particle spin modulus 𝑆tot = (ℏ/2)√3; (4) constrains basis opposite spins to have same handedness; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs have precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of handedness; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). Cross-product terms in the expression for the squared total spin of two particles relate to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as done in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.
A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum S=(√3 ℏ)⁄2; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.
Abstract A novel representation of spin 1/2 combines in a single geometric object the roles of the standardPauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists ofthree orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) isHermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one particlespin modulus 𝑆tot = √3ℏ/2; (4) constrains basis opposite spins to have same handedness; (5)allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs haveprecisely related gauge phases and can be of same or opposite handedness; (2) the dimensionality of the spinspace doubles due to variation of handedness; (3) the four maximally entangled states are naturally definedby the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) andinversion (singlet). The cross-product terms in the expression for the squared total spin of two particlesrelates to experiment and they yield all standard expectation values after measurement. Here spin is directlydefined and transformed in 3D orientation space, without use of eigen algebra and tensor product as in thestandard formulation. The formalism allows working with whole geometric objects instead of onlycomponents, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘onlab’ (uncontrolled gauge phase) spin transformations.
I derive the quantum mechanical prediction −a⋅b-\mathbf{a}\cdot\mathbf{b}−a⋅b for the singlet spin state using local measurement functions within the Space-Time Algebra (STA) framework. I establish a compact and computationally tractable STA representation of the two-particle singlet state, Ψ=12(Iσ2(1)−Iσ2(2))\Psi = \tfrac{1}{2}(I\sigma_2^{(1)} - I\sigma_2^{(2)})Ψ=21​(Iσ2(1)​−Iσ2(2)​), which is simpler than previously published forms but is not required for the correlation calculation. The analysis shows that STA naturally generates both scalar dot-product terms (a⋅s1\mathbf{a}\cdot\mathbf{s}_1a⋅s1​ and s2⋅b\mathbf{s}_2\cdot\mathbf{b}s2​⋅b) and bivector wedge-product terms (a∧s1\mathbf{a}\wedge\mathbf{s}_1a∧s1​ and s2∧b\mathbf{s}_2\wedge\mathbf{b}s2​∧b) from local spin–detector interactions, whereas standard quantum mechanics retains only the scalar contributions through expectation-value projection. Because the standard quantum formalism represents correlations via operator expectation values, the antisymmetric (cross‑product) contributions are averaged out at the level of observables. In contrast, the STA formulation keeps these bivector terms explicit in the intermediate geometric products, raising the question of whether potentially meaningful geometric structure is being hidden by the quantum averaging procedure. I verify the analytical derivation using computational simulations in Mathematica with a Clifford algebra package, confirming that local measurement functions reproduce the standard quantum correlation −cos⁡θ-\cos\theta−cosθ, where θ\thetaθ is the angle between measurement directions a\mathbf{a}a and b\mathbf{b}b. The correlation −a⋅b-{\bf a}\cdot {\bf b}−a⋅b is the scalar coordinate of 3-sphere rotors.