print('Case 1: one record written by an interaction keyed to clock A')
HS = kron(0.35 * sz, I2)
V1 = kron(P1, sx) + 0.6 * kron(sx, I2)
g1 = pulse(np.pi / 2)
Us = propagate(lambda s: HS + g1(s) * V1) # independent of Delta here
E = kron(I2, P1) # the record reads 1
Ds, chi = packets([0.3, 1.4], 0.25, [system_state(), system_state()])
sstar = 1.0
O = lambda c: Us[at(sstar)].conj().T @ E @ Us[at(sstar)] @ c
P_phys = sum(np.vdot(O(c), O(c)).real for c in chi) * ds
def P_A(tau): # integral over t_B = tau - Delta of |E Psi(tau, t_B)|^2
return sum(np.linalg.norm(E @ Us[at(tau)] @ c)**2 for c in chi) * ds
def P_B(taup): # integral over t_A = taup + Delta of |E Psi(t_A, taup)|^2
return sum(np.linalg.norm(E @ Us[at(taup + D)] @ c)**2 for D, c in zip(Ds, chi)) * ds
print(' offsets Delta in [%.2f, %.2f]; interaction in A-time [0, 1]' % (Ds.min(), Ds.max()))
print(' P_phys = %.12f P_A(2.0) = %.12f P_A(4.0) = %.12f' % (P_phys, P_A(2.0), P_A(4.0)))
print(' P_B(%.2f) = %.12f P_B(3.0) = %.12f' % (1 - Ds.min(), P_B(1 - Ds.min()), P_B(3.0)))
print(' A-time: ' + ', '.join('P_A(%.1f) = %.6f' % (t, P_A(t)) for t in (-0.5, 0.0, 0.5, 1.0)))
print(' B-time: ' + ', '.join('P_B(%.1f) = %.6f' % (t, P_B(t)) for t in (-1.0, -0.4, 0.2, 0.6)))
print(' O is a projector on the sector: max |O^2 - O| = %.1e'
% max(np.abs(O(O(c)) - O(c)).max() for c in chi))
print('Case 2: interaction 1 (keyed to A) rotates S; interaction 2 (keyed to B) records S')
V1, V2 = kron(sy, I2), kron(P1, sx)
g1, g2 = pulse(np.pi / 8), pulse(np.pi / 2)
psi_neg, psi_pos = system_state(), system_state()
Ds, chi = packets([-1.6, 1.6], 0.25, [psi_neg, psi_pos])
UD = [propagate(lambda s, D=D: g1(s) * V1 + g2(s - D) * V2) for D in Ds]
sstar = 1.0 + Ds.max()
P_phys = sum(np.linalg.norm(E @ U[at(sstar)] @ c)**2 for U, c in zip(UD, chi)) * ds
PA = sum(np.linalg.norm(E @ U[at(4.0)] @ c)**2 for U, c in zip(UD, chi)) * ds
taup = sstar - Ds.min()
PB = sum(np.linalg.norm(E @ U[at(taup + D)] @ c)**2 for U, D, c in zip(UD, Ds, chi)) * ds
G1, G2 = expm(-1j * np.pi / 8 * V1), expm(-1j * np.pi / 2 * V2)
mu = [np.sum(np.abs(chi[Ds < 0])**2) * ds, np.sum(np.abs(chi[Ds > 0])**2) * ds]
p_rec_first = np.linalg.norm(E @ G1 @ G2 @ psi_neg)**2
p_rot_first = np.linalg.norm(E @ G2 @ G1 @ psi_pos)**2
print(' P_phys = %.12f P_A(4.0) = %.12f P_B(%.2f) = %.12f' % (P_phys, PA, taup, PB))
print(' definite orders: record first %.6f (weight %.4f), rotation first %.6f (weight %.4f)'
% (p_rec_first, mu[0], p_rot_first, mu[1]))
print(' weighted mixture of the two definite orders = %.12f'
% (mu[0] * p_rec_first + mu[1] * p_rot_first))
print('Case 3: the same interactions, an offset-coherent observable (H_S = 0)')
psi0 = system_state()
Ds, chi = packets([-1.6, 1.6], 0.25, [psi0, psi0])
UD = [propagate(lambda s, D=D: g1(s) * V1 + g2(s - D) * V2) for D in Ds]
fplus = bump((Ds + 1.6) / 0.25) + bump((Ds - 1.6) / 0.25)
fplus = fplus / np.sqrt(np.sum(fplus**2) * ds) # coherent superposition of offsets
amp_phys = sum(fp * (U[at(sstar)] @ c) for fp, U, c in zip(fplus, UD, chi)) * ds
amp_A = sum(fp * (U[at(4.0)] @ c) for fp, U, c in zip(fplus, UD, chi)) * ds
amp_B = sum(fp * (U[at(taup + D)] @ c) for fp, U, D, c in zip(fplus, UD, Ds, chi)) * ds
W1, W2 = G1 @ G2, G2 @ G1 # offset -1.6: record first; +1.6: rotation first
pred = 0.5 * (1 + np.vdot(W1 @ psi0, W2 @ psi0).real)
mix = 0.0
for sgn in (-1, 1): # incoherent mixture of the two offsets
sel = np.sign(Ds) == sgn
ch = chi * sel[:, None] / np.sqrt(np.sum(np.abs(chi[sel])**2) * ds)
a_ = sum(fp * (U[at(sstar)] @ c) for fp, U, c in zip(fplus, UD, ch)) * ds
mix += 0.5 * np.vdot(a_, a_).real
print(' P(+): clock-neutral %.12f, relative to A %.12f, relative to B %.12f'
% (np.vdot(amp_phys, amp_phys).real, np.vdot(amp_A, amp_A).real, np.vdot(amp_B, amp_B).real))
print(' (1 + Re<W1 psi, W2 psi>)/2 = %.12f; incoherent mixture of offsets gives %.12f'
% (pred, mix))
Output:
Case 1: one record written by an interaction keyed to clock A
offsets Delta in [0.06, 1.64]; interaction in A-time [0, 1]
P_phys = 0.139627690767 P_A(2.0) = 0.139627690767 P_A(4.0) = 0.139627690767
P_B(0.94) = 0.139627690767 P_B(3.0) = 0.139627690767
A-time: P_A(-0.5) = 0.000000, P_A(0.0) = 0.000000, P_A(0.5) = 0.153583, P_A(1.0) = 0.139628
B-time: P_B(-1.0) = 0.021295, P_B(-0.4) = 0.032189, P_B(0.2) = 0.145553, P_B(0.6) = 0.144791
O is a projector on the sector: max |O^2 - O| = 4.0e-15
Case 2: interaction 1 (keyed to A) rotates S; interaction 2 (keyed to B) records S
P_phys = 0.706791827009 P_A(4.0) = 0.706791827009 P_B(4.68) = 0.706791827009
definite orders: record first 0.868488 (weight 0.5000), rotation first 0.545095 (weight 0.5000)
weighted mixture of the two definite orders = 0.706791827009
Case 3: the same interactions, an offset-coherent observable (H_S = 0)
P(+): clock-neutral 0.926776695297, relative to A 0.926776695297, relative to B 0.926776695297
(1 + Re<W1 psi, W2 psi>)/2 = 0.926776695297; incoherent mixture of offsets gives 0.500000000000