Preliminaries Single-qubit Pauli operators are defined as x=[0110],y=[0ii0],z=[1001], and i=1 Problem 1 Express the following vectors or matrices using Dirac notation (a.k.a bra-ket notation) in the computational basis. (a) [2121] (b) [3231] (c) 6131021 (d) [21002i] (e) [acbd] Problem 2 A Hermitian matrix satisfies A=A. Prove that Hermitian matrices can only have real eigenvalues. Problem 3 Problem 4 Write the eigenvectors of the single-qubit Pauli operators in the computational basis using Dirac notation. Problem 5 Let =00+11, where C,C, and 2+2=1. Express the following expected value of an observable in terms of and . (a) zI (b) Iy (c) xx