{"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, specifically for a mixed state represented by a density matrix.", "specific_scope": "The density matrix given is a specific case of a mixed state for a qubit, which can be analyzed to determine its position in the Bloch sphere representation.", "goal": "Identify the correct geometrical position of the given density matrix in the qubit space."}, "summary": "The density matrix \u03c1 = 1/2 (|0\u27e9\u27e80| + |1\u27e9\u27e81|) represents a completely mixed state for a qubit, which corresponds to a uniform distribution over the states |0\u27e9 and |1\u27e9. In the Bloch sphere representation, this state is located at the center of the sphere, which corresponds to the point r = (0, 0, 0). Therefore, the correct answer is option 1: r = (0, 0, 0)."}