{
  "paper_id": "qqc-001-foundations",
  "glossary": [
    {
      "term": "axis-angle representation",
      "definition": "A parametrization of a rotation by an axis n-hat in S^2 and an angle theta. In this paper it appears in both SU(2) form and unit-quaternion form.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["unit quaternion", "SU(2)", "Bloch sphere"],
      "see_also": ["Section 4.1"]
    },
    {
      "term": "Bloch sphere",
      "definition": "The unit 2-sphere S^2 representing pure states of a single qubit after quotienting normalized state vectors by global phase.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["pure state", "global phase", "CP^1"],
      "see_also": ["prop-2"]
    },
    {
      "term": "double cover",
      "definition": "A two-to-one surjective map between spaces or groups. In this paper SU(2) and Sp(1) double-cover SO(3).",
      "aliases": ["two-to-one cover"],
      "paper_specific": false,
      "related_terms": ["SU(2)", "SO(3)", "spinor"],
      "see_also": ["prop-7"]
    },
    {
      "term": "global phase",
      "definition": "A common complex phase factor e^{i chi} multiplying a state vector. For pure states this phase does not change the physical state.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["pure state", "CP^1", "U(2)"],
      "see_also": ["Section 2.2", "Section 2.3"]
    },
    {
      "term": "normalized spinor",
      "definition": "A unit-norm vector in C^2 used as a representative of a single-qubit pure state before quotienting global phase.",
      "aliases": ["normalized state vector"],
      "paper_specific": false,
      "related_terms": ["spinor", "S^3", "pure state"],
      "see_also": ["prop-1"]
    },
    {
      "term": "pure quaternion",
      "definition": "A quaternion with zero real part, identified naturally with a vector in R^3.",
      "aliases": ["imaginary quaternion"],
      "paper_specific": false,
      "related_terms": ["quaternion", "SO(3)", "double cover"],
      "see_also": ["Section 4.3"]
    },
    {
      "term": "pure state",
      "definition": "A one-dimensional ray in Hilbert space. For a single qubit, pure states form CP^1 and are represented geometrically by the Bloch sphere S^2.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["global phase", "Bloch sphere", "qubit"],
      "see_also": ["prop-2"]
    },
    {
      "term": "quaternion",
      "definition": "An element of Hamilton's quaternion algebra H of the form a + b i + c j + d k with real coefficients.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["unit quaternion", "pure quaternion", "SU(2)"],
      "see_also": ["Section 2.4"]
    },
    {
      "term": "qubit",
      "definition": "A two-level quantum system whose state vectors live in C^2 and whose pure states are rays in that space.",
      "aliases": ["quantum bit"],
      "paper_specific": false,
      "related_terms": ["pure state", "Bloch sphere", "spinor"],
      "see_also": ["Section 2.1"]
    },
    {
      "term": "RQM framing",
      "definition": "The project-specific choice in this series to use the quaternion model of SU(2) as a primary expository language for single-qubit gate geometry while preserving standard mathematical distinctions.",
      "aliases": ["RQM Technologies framing"],
      "paper_specific": true,
      "related_terms": ["SU(2)", "unit quaternion", "single-qubit gate"],
      "see_also": ["int-9"]
    },
    {
      "term": "single-qubit gate",
      "definition": "A unitary operator acting on C^2. Such gates live in U(2), and after removal of overall phase may be represented by elements of SU(2).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["U(2)", "SU(2)", "unit quaternion"],
      "see_also": ["prop-3"]
    },
    {
      "term": "spinor",
      "definition": "A vector transforming under the spinorial representation of rotations, naturally associated with SU(2) rather than directly with SO(3).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["SU(2)", "double cover", "normalized spinor"],
      "see_also": ["Section 2.1", "prop-7"]
    },
    {
      "term": "SU(2)",
      "definition": "The group of 2 x 2 complex unitary matrices with determinant 1. It provides representatives for single-qubit gates modulo global phase and is isomorphic to the unit quaternions.",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["U(2)", "SO(3)", "unit quaternion"],
      "see_also": ["prop-3", "prop-5"]
    },
    {
      "term": "unit quaternion",
      "definition": "A quaternion of norm 1. Unit quaternions form the group Sp(1), which is isomorphic to SU(2).",
      "aliases": [],
      "paper_specific": false,
      "related_terms": ["quaternion", "Sp(1)", "axis-angle representation"],
      "see_also": ["prop-5"]
    }
  ]
}