{"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 for mixed states represented by density matrices.", "specific_scope": "The task is to analyze the given density matrix to determine its corresponding point in the Bloch sphere representation of qubit states.", "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 mixed state that is equally weighted between the |0\u27e9 and |1\u27e9 states. In the Bloch sphere representation, this corresponds to the center of the sphere, which is the point (0,0,0). Therefore, the correct geometrical position of the density matrix in qubit space is r = (0,0,0)."}