Why is this Hamiltonian matrix diagonal?

Question

I've only recently started using density matrices in my work but I am confused with the following code that I have whether I am getting the right matrix:

def Hamiltonian(alpha,h):

    Sx = np.array([[0,1],[1,0]])
    Sy = np.array([[0,-1j],[1j,0]])
    Sz = np.array([[1,0],[0,-1]])
    I  = np.array([[1,0],[0,1]])

    H =  (alpha*np.kron(np.kron(Sx,Sx),I)) 
    H =+ (alpha*np.kron(np.kron(Sy,Sy),I)) 
    H =+ (alpha*np.kron(np.kron(I,Sx),Sx)) 
    H =+ (alpha*np.kron(np.kron(I,Sy),Sy)) 
    H =+ (h*np.kron(np.kron(I,Sz),I))

    return H

So the above gives me my Hamiltonian Function, where alpha is a real number and h is a magnetization parameter acting on one of my qubits.

I have tried the following:

H = Hamiltonian(1,0.5)
print(H)

$$\begin{bmatrix} 0.5&0&0&0&0&0&0&0 \\ 0&0.5&0&0&0&0&0&0 \\ 0&0&-0.5&0&0&0&0&0 \\ 0&0&0&-0.5&0&0&0&0 \\ 0&0&0&0&0.5&0&0&0 \\ 0&0&0&0&0&0.5&0&0 \\ 0&0&0&0&0&0&-0.5&0 \\ 0&0&0&0&0&0&0&-0.5 \end{bmatrix}$$

Why is it diagonal?

Answer 1

Your matrix is diagonal because the + signs are on the wrong sides of the equals signs. The code below will give you the correct matrix:

alpha=1;h=0.5;

x=[0  1;  1 0 ];
y=[0 -1i; 1i 0];
z=[1  0;  0 -1];
I=eye(2);

H = alpha*kron(kron(x,x),I)+...
    alpha*kron(kron(y,y),I)+...
    alpha*kron(kron(I,x),x)+...
    alpha*kron(kron(I,y),y)+...
    h*kron(kron(I,z),I)

Here is the result:

The eigenvectors and eigenvlaues are much more complicated than what you have.

If you don't have Octave, these commands will install it, then open it:

sudo apt-get install octave
octave