{
  "paper_id": "qqc-001-foundations",
  "notation": [
    {
      "symbol": "|ψ⟩",
      "name": "normalized state vector",
      "latex": "\\ket{\\psi}",
      "meaning": "A normalized representative of a single-qubit pure state in C^2.",
      "domain": "quantum mechanics",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": ["α", "β", "CP^1"]
    },
    {
      "symbol": "α, β",
      "name": "spinor components",
      "latex": "\\alpha, \\beta",
      "meaning": "Complex amplitudes of a single-qubit state vector |ψ⟩ = (α, β)^T.",
      "domain": "quantum mechanics",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": ["|ψ⟩", "S^3"]
    },
    {
      "symbol": "C^2",
      "name": "two-dimensional complex vector space",
      "latex": "\\C^2",
      "meaning": "Ambient complex vector space for single-qubit state vectors before projectivization.",
      "domain": "linear algebra",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": ["|ψ⟩", "S^3", "R^4"]
    },
    {
      "symbol": "S^3",
      "name": "three-sphere",
      "latex": "S^3",
      "meaning": "The unit sphere in R^4; in this paper it appears both as the normalized-spinor sphere and as the manifold underlying the unit quaternions.",
      "domain": "geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.1",
      "related_symbols": ["C^2", "CP^1", "Sp(1)"]
    },
    {
      "symbol": "CP^1",
      "name": "complex projective line",
      "latex": "\\CP^1",
      "meaning": "Projective pure-state space of a single qubit, obtained by quotienting normalized spinors by global phase.",
      "domain": "geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.2",
      "related_symbols": ["S^2", "|ψ⟩", "global phase"]
    },
    {
      "symbol": "S^2",
      "name": "Bloch sphere",
      "latex": "S^2",
      "meaning": "The unit 2-sphere used to represent pure single-qubit states after quotienting global phase.",
      "domain": "quantum geometry",
      "type": "space",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.2",
      "related_symbols": ["CP^1", "r(ψ)", "SO(3)"]
    },
    {
      "symbol": "r(ψ)",
      "name": "Bloch vector",
      "latex": "\\mathbf{r}(\\psi)",
      "meaning": "The unit vector in S^2 associated with a normalized qubit state through the Bloch-sphere map.",
      "domain": "quantum geometry",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.2",
      "related_symbols": ["|ψ⟩", "S^2", "σ_x, σ_y, σ_z"]
    },
    {
      "symbol": "U(2)",
      "name": "single-qubit unitary group",
      "latex": "\\mathrm{U}(2)",
      "meaning": "Group of 2 x 2 unitary matrices; this is the full group in which single-qubit gates live before quotienting global phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.3",
      "related_symbols": ["SU(2)", "U", "V"]
    },
    {
      "symbol": "SU(2)",
      "name": "special unitary group",
      "latex": "\\mathrm{SU}(2)",
      "meaning": "Group of 2 x 2 unitary matrices with determinant 1; used as a representative model for single-qubit gates modulo global phase.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.3",
      "related_symbols": ["U(2)", "SO(3)", "Sp(1)"]
    },
    {
      "symbol": "U, V",
      "name": "single-qubit gate and SU(2) representative",
      "latex": "U, V",
      "meaning": "U denotes a gate in U(2), while V denotes a determinant-1 representative in SU(2) after removing global phase.",
      "domain": "quantum gates",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.3",
      "related_symbols": ["U(2)", "SU(2)", "q"]
    },
    {
      "symbol": "H",
      "name": "quaternion algebra",
      "latex": "\\mathbb{H}",
      "meaning": "Hamilton's quaternion algebra over R with basis 1, i, j, k.",
      "domain": "algebra",
      "type": "algebra",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": ["q", "i, j, k", "Sp(1)"]
    },
    {
      "symbol": "q",
      "name": "quaternion or unit-quaternion gate representative",
      "latex": "q",
      "meaning": "A quaternion a + b i + c j + d k; when |q| = 1 it models an element of SU(2).",
      "domain": "algebra / quantum gates",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": ["H", "|q|", "Φ"]
    },
    {
      "symbol": "i, j, k",
      "name": "quaternion basis units",
      "latex": "\\mathbf{i}, \\mathbf{j}, \\mathbf{k}",
      "meaning": "Imaginary quaternion units satisfying i^2 = j^2 = k^2 = ijk = -1.",
      "domain": "algebra",
      "type": "constant",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": ["H", "q", "σ_x, σ_y, σ_z"]
    },
    {
      "symbol": "Sp(1)",
      "name": "unit-quaternion group",
      "latex": "\\mathrm{Sp}(1)",
      "meaning": "Group of unit quaternions; a Lie-group model of SU(2).",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 2.4",
      "related_symbols": ["H", "SU(2)", "S^3"]
    },
    {
      "symbol": "Φ",
      "name": "chosen quaternion-matrix identification",
      "latex": "\\Phi",
      "meaning": "The explicit map from H to 2 x 2 complex matrices used in the paper, chosen so that i, j, k align with the x, y, z Pauli axes.",
      "domain": "representation theory",
      "type": "map",
      "standard_or_project_specific": "variant",
      "first_use": "Section 3.1",
      "related_symbols": ["q", "SU(2)", "σ_x, σ_y, σ_z"]
    },
    {
      "symbol": "σ_x, σ_y, σ_z",
      "name": "Pauli matrices",
      "latex": "\\sigma_x, \\sigma_y, \\sigma_z",
      "meaning": "Standard Pauli matrices used to express axis-angle rotations in SU(2).",
      "domain": "quantum mechanics",
      "type": "matrix",
      "standard_or_project_specific": "standard",
      "first_use": "Section 3.1",
      "related_symbols": ["SU(2)", "Φ", "r(ψ)"]
    },
    {
      "symbol": "n̂",
      "name": "unit rotation axis",
      "latex": "\\hat{\\mathbf{n}}",
      "meaning": "Unit vector in S^2 specifying a rotation axis in axis-angle form.",
      "domain": "geometry",
      "type": "vector",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.1",
      "related_symbols": ["θ", "q", "S^2"]
    },
    {
      "symbol": "θ",
      "name": "rotation angle",
      "latex": "\\theta",
      "meaning": "Rotation angle appearing in both the SU(2) and quaternionic axis-angle formulas.",
      "domain": "geometry",
      "type": "scalar",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.1",
      "related_symbols": ["n̂", "q", "SU(2)"]
    },
    {
      "symbol": "ρ",
      "name": "double-cover map to SO(3)",
      "latex": "\\rho",
      "meaning": "Homomorphism from Sp(1) to SO(3) defined by conjugation on pure quaternions.",
      "domain": "Lie groups",
      "type": "map",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.3",
      "related_symbols": ["Sp(1)", "SO(3)", "q"]
    },
    {
      "symbol": "SO(3)",
      "name": "rotation group in three dimensions",
      "latex": "\\mathrm{SO}(3)",
      "meaning": "Group of proper rotations of R^3; the physical Bloch-sphere action factors through this group.",
      "domain": "Lie groups",
      "type": "group",
      "standard_or_project_specific": "standard",
      "first_use": "Section 4.3",
      "related_symbols": ["SU(2)", "Sp(1)", "S^2"]
    }
  ]
}