{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Quantum States \u2192 Density Matrices", "refinement": "This problem involves understanding the representation of quantum states in the Bloch sphere, particularly how density matrices correspond to points in this geometric space.", "specific_scope": "The density matrix given represents a mixed state of a qubit, specifically a state that is an equal mixture of the |+\u27e9 and |\u2212\u27e9 states.", "goal": "Identify the geometric representation of the given density matrix in the Bloch sphere coordinates."}, "summary": "The density matrix \u03c1 = 1/2(|+\u27e9\u27e8+| + |\u2212\u27e9\u27e8\u2212|) represents a mixed state of a qubit that is an equal superposition of the |+\u27e9 and |\u2212\u27e9 states. In the context of the Bloch sphere, this corresponds to the point at the origin, r = (0, 0, 0), indicating a completely mixed state with no preference for either |+\u27e9 or |\u2212\u27e9. Thus, the correct answer is r = (0, 0, 0)."}