#not OK to post homework
#Nick Belov, Feb 9nd 2025, Assignment 5

read "C5.txt":
#PM(): the three famous Pauli matrices sigma_x,sigma_y, sigma_z
#RHM(n,K): a random n by n Hermitian matrix with entriries from -K to K
#EV(L): inputs a Hermitian matrix (say n by n) and outputs the LIST of eigenvalues
#EVC1(L,g): inputs a matrix L and an eigenvalue g finds the NORMALIZED eigenvector
#EVC(L): inputs a Hermitian matrix and outputs the pair ListOfEigenvalues, ListOfEigenvectors
#ProbDist(L,A): Given an obervalble represented by the Hermitian matrix L and a state vector A outputs first all the eigenvectors (pure states) and the prob. of winding in the respective pure states followed by the expected value of A (i.e. if you make the observation L and the state is A the average in the long run

#UM(alpha,beta,gamma,delta)
#that implements box 1.1. on p. 20 of the The Nielsen-Chuang book [Note the superflous comma]
#For random alpha, beta, gamma, delta apply it to IsUM(A). Also, a litle bit of challenge, prove it for geneal (symbolic) alpha,beta,gamma, delta
UM := proc(alpha, beta, gamma, delta)
    local ealph, ebetaminus, ebetaplus, edeltaminus, edeltaplus;
    local c, s;
    local f1, f2, f3;
    local M1, M2, Mfinal;

    ealph := exp(I*alpha);
    ebetaminus := exp(-I*beta/2);
    ebetaplus := exp( I*beta/2);
    edeltaminus := exp(-I*delta/2);
    edeltaplus := exp( I*delta/2);

    # Trig pieces
    c := cos(gamma/2);
    s := sin(gamma/2);

    f1 := Matrix([[ebetaminus,0],
                  [0,ebetaplus]]);
    f2 := Matrix([[c,-s],
                  [s,c]]);
    f3 := Matrix([[edeltaminus, 0],
                  [0,edeltaplus]]);

    M1 := f1 . f2;
    M2 := M1 . f3;

    Mfinal := ealph * M2;

    return [
      [ Mfinal[1,1], Mfinal[1,2] ],
      [ Mfinal[2,1], Mfinal[2,2] ]
    ];
end: