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Entropy

Electron Spin Correlations: Probabilistic Description and Geometric Representation

Ana Marı́a Cetto

Published Oct 9, 2022• 9 citations
Publisher

Abstract

The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which helps clarify the concepts of contextuality and measurement dependence. The dependence of the spin correlation on conditional probabilities allows for a clear separation between system state and measurement context; the latter determines how the probability space should be partitioned in calculating the correlation. A probability distribution function ρ(ϕ) is then proposed, which reproduces the quantum correlation for a pair of single-particle spin projections and is amenable to a simple geometric representation that gives meaning to the variable ϕ. The same procedure is shown to be applicable to the bipartite system in the singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical picture of the electron spin, as discussed at the end of the paper.

DOI
10.3390/e24101439

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The electron spin correlation is shown to be expressible in terms of a bona fide probability distribution function with an associated geometric representation. With this aim, an analysis is presented of the probabilistic features of the spin correlation within the quantum formalism, which helps clarify the concepts of contextuality and measurement dependence. The dependence of the spin correlation on conditional probabilities allows for a clear separation between system state and measurement context; the latter determines how the probability space should be partitioned in calculating the correlation. A probability distribution function ρ(ϕ) is then proposed, which reproduces the quantum correlation for a pair of single-particle spin projections and is amenable to a simple geometric representation that gives meaning to the variable ϕ. The same procedure is shown to be applicable to the bipartite system in the singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical picture of the electron spin, as discussed at the end of the paper.
An analysis is made within the quantum formalism of the probabilistic features of the electron spin correlation, with the purpose of clarifying the concepts of contextuality and measurement dependence. The quantum formulas for the spin correlation are then derived on the basis of a probability distribution function and its associated geometric representation, both for a pair of projections of a single-particle spin and for the bipartite system in singlet spin state. This endows the spin correlation with a clear probabilistic meaning and leaves the door open for a possible physical image of the electron spin, as discussed at the end of the paper.
The quantum formula for the spin correlation of the bipartite singlet spin state, CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b), is derived on the basis of a probability distribution ρ(ϕ)ρ(ϕ)ρ(ϕ) that is generic, i. e., independent of (a,b)(\boldsymbol{a},\boldsymbol{b})(a,b). In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product A=αβA=αβA=αβ of the individual spin projections ααα and βββ onto a\boldsymbol{a}a and b\boldsymbol{b}b. A specific partitioning and a corresponding set of realizations {$ ϕ$} are associated with every measurement setting (a,b)(\boldsymbol{a},\boldsymbol{b})(a,b); this precludes the transfer of ααα or βββ from CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b) to CQ(a,b′)C_{Q}(\boldsymbol{a},\boldsymbol{b'})CQ​(a,b′), for b′≠b.\boldsymbol{b'}\neq\boldsymbol{b}.b′=b. A geometric model that reproduces the spin correlation serves to validate our approach, giving a concrete meaning to the quantum result in terms of a (local random variable) probability distribution.
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.
The quantum formula for the spin correlation of the bipartite singlet spin state, CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b), is derived on the basis of a probability distribution ρ(ϕ)\rho(\phi)ρ(ϕ) that is generic, i. e., independent of (a,b)(\boldsymbol{a},\boldsymbol{b})(a,b). In line with a previous result obtained within the framework of the quantum formalism, the probability space is partitioned according to the sign of the product A=αβA=\alpha\betaA=αβ of the individual spin projections α\alphaα and β\betaβ onto a\boldsymbol{a}a and b\boldsymbol{b}b; this precludes the transfer of α\alphaα or β\betaβ from CQ(a,b)C_{Q}(\boldsymbol{a},\boldsymbol{b})CQ​(a,b) to CQ(a,b′)C_{Q}(\boldsymbol{a},\boldsymbol{b'})CQ​(a,b′), for b′≠b.\boldsymbol{b'}\neq\boldsymbol{b}.b′=b. A specific physical model that reproduces the quantum spin correlation serves to validate our approach.
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.