{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Quantum Information \u2192 Qubit Representation", "refinement": "This problem involves understanding the representation of quantum states using density matrices and their corresponding points on the Bloch sphere.", "specific_scope": "The focus is on determining the Bloch sphere coordinates of a given density matrix for a qubit, specifically the matrix \u03c1 = 1/2(|0\u27e9\u27e80| + |1\u27e9\u27e81|).", "goal": "Identify the coordinates on the Bloch sphere that correspond to the given density matrix."}, "summary": "In this problem, we analyze the density matrix \u03c1 = 1/2(|0\u27e9\u27e80| + |1\u27e9\u27e81|), which represents a completely mixed state of a qubit. This state corresponds to the center of the Bloch sphere, which is represented by the coordinates (0,0,0). The task is to recognize that this density matrix indicates an equal probability of measuring either |0\u27e9 or |1\u27e9, leading to the conclusion that the answer is (0,0,0), indicating a maximally mixed state."}