import numpy as np

X=np.array([[0,1],[1,0]],dtype=complex); Y=np.array([[0,-1j],[1j,0]],dtype=complex); Z=np.array([[1,0],[0,-1]],dtype=complex)
I=np.eye(2,dtype=complex)
paulis={'I':I,'X':X,'Y':Y,'Z':Z}

def state(phi):
    psi = np.zeros(16, dtype=complex)
    psi[0b0011] = np.cos(phi)
    psi[0b1100] = np.sin(phi)
    return psi

def kron4(a,b,c,d):
    return np.kron(np.kron(np.kron(a,b),c),d)

def corr(rho, labels):
    O = kron4(*[paulis[l] for l in labels])
    return np.trace(rho@O).real

for phi_deg in [10, 40, -10, -40, 100, 130, 200]:
    phi = np.deg2rad(phi_deg)
    psi = state(phi)
    rho = np.outer(psi, psi.conj())
    # build the 9x9 raw M_AB block (source AB active, target CD active) and get its true nuclear norm
    axes = [X,Y,Z]
    M = np.zeros((9,9))
    for ia in range(3):
        for ib in range(3):
            for ic in range(3):
                for id_ in range(3):
                    O = kron4(axes[ia],axes[ib],axes[ic],axes[id_])
                    M[3*ia+ib, 3*ic+id_] = np.trace(rho@O).real
    nn_true = np.linalg.svd(M, compute_uv=False).sum() / 3.0   # normalize by sqrt(3*3)=3
    pred_with_abs = 1/3 + 4/3*abs(np.sin(2*phi))
    pred_no_abs = 1/3 + 4/3*np.sin(2*phi)
    print(f"phi={phi_deg:+4d} deg: true ||M||_*={nn_true:.6f}  pred_with_abs={pred_with_abs:.6f}  pred_no_abs={pred_no_abs:.6f}")