Problem 3: Two-level system and density matriceSuppose a 2 x 2 matrix X (not necessarily Hermitian or unitary) is written asX = a000 + a.σ,where ao and ak, k = 1, 2, 3, are numbers, 0o = 1 is the identity matrix and o are thePauli matrices.(a)How are ao and a related to tr(X) and tr(OX)?Obtain ao and ak in terms of the matrix elements Xij.Assume that ao, ak ER such that X is Hermitian and could beinterpreted as a Hamiltonian, what are the eigenvalues of X?

(a)Let us first write down all the sigma_k, where k=1,2,3 ........(1) You can see from the matrices…

Problem 3: Two-level system and density matrice Suppose a 2 x 2 matrix X (not necessarily Hermitian or unitary) is written as X = aooo + a.o, where ao and ak, k = 1, 2, 3, are numbers, 0 = 1 is the identity matrix and o are the Pauli matrices. (a) (b) How are ao and a related to tr(X) and tr(okX)? Obtain ao and a in terms of the matrix elements Xij.

Problem 3: Two-level system and density matrice Suppose a 2 x 2 matrix X (not necessarily Hermitian or unitary) is written as X = aooo + a.o, where ao and ak, k = 1, 2, 3, are numbers, 0 = 1 is the identity matrix and o are the Pauli matrices. (a) (b) How are ao and a related to tr(X) and tr(okX)? Obtain ao and a in terms of the matrix elements Xij.