A state in a 2D Hilbert Space (with basis states denoted as IO) and |1)) can be represented by a point in the Bloch sphere. A general density matrix can be char- acterized by a vector p, where 16 = %(1 + p - 0'). Find the vector p for the following two cases: (a) (5 Pt) A pure ensemble described by a ket vector W') = sin9 IO) + 6\"" cos 6 ll) (b) (5 Pt) An ensemble represented by the following density operator: 9 = % [I0)(0l + |1)(1| + (a: + iy)l0)(ll + (w - 3".IJ)|1)(0|] where a: and y are real numbers. What constraints one needs to put on 9:, y? Under what condition the state represented by this density operator is a pure ensemble