# ---- 15. equal ensembles, different realized-history measures (two unravellings)
plus = np.array([1, 1])/np.sqrt(2); minus = np.array([1, -1])/np.sqrt(2)
I1 = [np.diag([1, 0]), np.diag([0, 1])]
I2 = [np.eye(2)/np.sqrt(2), np.diag([1, -1])/np.sqrt(2)]
Xm = [np.outer(plus, plus), np.outer(minus, minus)]
rho = rq.normal(size=(2, 2)) + 1j*rq.normal(size=(2, 2))
rho = rho @ rho.conj().T; rho /= np.trace(rho)
ch = lambda I, r: sum(K @ r @ K.conj().T for K in I)
same_channel = np.allclose(ch(I1, rho), ch(I2, rho))
def jd(I):
return np.array([[np.linalg.norm(M @ K @ plus)**2 for M in Xm] for K in I])
J1, J2 = jd(I1), jd(I2)
def mi(Jm):
pa, pb = Jm.sum(1), Jm.sum(0)
return sum(Jm[i, j]*np.log2(Jm[i, j]/(pa[i]*pb[j])) for i in range(2) for j in range(2)
if Jm[i, j] > 1e-15)
print(f"15. same channel: {same_channel}; joint law of (first event, later X outcome):")
print(f" {J1.round(3).tolist()} vs {J2.round(3).tolist()};")
print(f" mutual information {mi(J1):.3f} vs {mi(J2):.3f} bits")
# ---- 16. imperfect quantum clocks: identity of events vs clock-relative timing
Mc, th = 32, 0.9
Sc = np.roll(np.eye(Mc), 1, axis=0)
Uc = Sc @ expm(-1j*th*(Sc + Sc.T))
step = np.abs(Uc[:, 0])**2 # law of one recorded clock step
signed = [(m if m <= Mc//2 else m - Mc) for m in range(Mc)]
p_inv = sum(step[m] for m in range(Mc) if signed[m] < 0)
p_stall = step[0]
from scipy.special import jv
bes_inv = (1 - jv(0, 2*th)**2 - 2*jv(1, 2*th)**2)/2; bes_stall = jv(1, 2*th)**2
far = sum(step[m] for m in range(Mc) if abs(signed[m]) >= 8)
print(f"16. drifting quantum clock, exact on the 32-site ring (signed displacement in -15..16):")
print(f" P(backward) = {p_inv:.9f}, P(no advance) = {p_stall:.9f};")
print(f" infinite-lattice Bessel values {bes_inv:.9f}, {bes_stall:.9f}")
print(f" max difference = {max(abs(p_inv-bes_inv), abs(p_stall-bes_stall)):.1e}; "
f"P(|displacement| >= 8) = {far:.1e}")
per = [n % 2 for n in range(5)]
print(f" period-2 counter readings at stages 0-4: {per};")
print(" superposed counter: reading at stage n is n or n+4, each with probability 0.5")
# ---- 17. gapless detectors on a free field: three-region (Sorkin-type) signaling test
# two-mode realization of the commutation pattern E_AB = 0, E_AC = E_CB = 1:
# phi_A = q1, phi_C = p1 + q2, phi_B = p2; vacuum state; Fock cutoff nc per mode
def sorkin(nc, lamA=0.9, lamC=0.8, lamB=0.7, kick=1.5):
am = np.diag(np.sqrt(np.arange(1, nc)), 1); I_ = np.eye(nc)
qm = (am + am.T)/np.sqrt(2); pm = 1j*(am.T - am)/np.sqrt(2)
phA, phC, phB = np.kron(qm, I_), np.kron(pm, I_) + np.kron(I_, qm), np.kron(I_, pm)
v0 = np.zeros(nc*nc, complex); v0[0] = 1
def fn(Hm, f):
w, V = np.linalg.eigh(Hm); return (V * f(w)) @ V.conj().T
det = lambda ph, l: [fn(ph, lambda x: np.cos(l*x)), 1j*fn(ph, lambda x: np.sin(l*x))]
gg = lambda x: 0.5*(1 + np.tanh(2*x))
lud = lambda ph: [fn(ph, lambda x: np.sqrt(gg(x))), fn(ph, lambda x: np.sqrt(1 - gg(x)))]
def bob(KA, KC, KB):
pb = 0.0
for Ka in KA:
va = Ka @ v0
for Kc in KC: pb += np.linalg.norm(KB[1] @ (Kc @ va))**2
return pb
KB = det(phB, lamB); Id = [np.eye(nc*nc)]; kA = [fn(phA, lambda x: np.exp(1j*kick*x))]
out = {}
for nm, KC in (('gapless', det(phC, lamC)), ('luders', lud(phC))):
base = bob(Id, KC, KB)
out[nm] = (base, bob(det(phA, lamA), KC, KB) - base, bob(kA, KC, KB) - base)
return out
s40, s50 = sorkin(40), sorkin(50)
ana = 0.5 - 0.5*np.cos(2*0.7*0.8)*np.exp(-0.7**2)
print(f"17. three-region test, Bob's P(b=1) and its shift under Alice's detector / kick:")
print(f" gapless Charlie: P = {s50['gapless'][0]:.10f} (analytic {ana:.10f}); shifts "
f"{s50['gapless'][1]:.1e}, {s50['gapless'][2]:.1e}")
print(f" unsharp Lueders Charlie: P = {s50['luders'][0]:.6f}; shifts "
f"{s50['luders'][1]:.6f}, {s50['luders'][2]:.6f}")
print(f" cutoff 40 vs 50: max change = "
f"{max(abs(s40[k][i] - s50[k][i]) for k in s40 for i in range(3)):.1e}")
# ---- 18. Gaussian collapse of a smeared field at C (Weyl-diagonal): same three-region test
def gauss_test(nc=40, lamA=0.9, lamB=0.7, kick=1.5, sig=0.6, p_hit=1.0, dy=0.02):
am = np.diag(np.sqrt(np.arange(1, nc)), 1); I_ = np.eye(nc)
qm = (am + am.T)/np.sqrt(2); pm = 1j*(am.T - am)/np.sqrt(2)
phA, phC, phB = np.kron(qm, I_), np.kron(pm, I_) + np.kron(I_, qm), np.kron(I_, pm)
v0 = np.zeros(nc*nc, complex); v0[0] = 1
def fn(Hm, f):
w, V = np.linalg.eigh(Hm); return (V * f(w)) @ V.conj().T
wC, VC = np.linalg.eigh(phC); ys = np.arange(-10, 10 + dy/2, dy)
KB1 = 1j*fn(phB, lambda x: np.sin(lamB*x))
def bob(KA):
pb = 0.0
for Ka in KA:
w = VC.conj().T @ (Ka @ v0)
if p_hit < 1: pb += (1 - p_hit)*np.linalg.norm(KB1 @ (VC @ w))**2
for y in ys:
k = (2*np.pi*sig**2)**-0.25*np.exp(-(wC - y)**2/(4*sig**2))
pb += p_hit*dy*np.linalg.norm(KB1 @ (VC @ (k*w)))**2
return pb
detA = [fn(phA, lambda x: np.cos(lamA*x)), 1j*fn(phA, lambda x: np.sin(lamA*x))]
base = bob([np.eye(nc*nc)])
ana = 0.5 - 0.5*np.exp(-lamB**2)*((1 - p_hit) + p_hit*np.exp(-lamB**2/(2*sig**2)))
shifts = (bob(detA) - base, bob([fn(phA, lambda x: np.exp(1j*kick*x))]) - base)
return base, ana, shifts
for ph in (1.0, 0.4):
b_, a_, sh = gauss_test(p_hit=ph)
print(f"18. Gaussian collapse at C (hit probability {ph}): P = {b_:.10f} "
f"(analytic {a_:.10f});")
print(f" shifts under Alice's detector / kick: {sh[0]:.1e}, {sh[1]:.1e}")
import sys, scipy
print(f"Environment: Python {sys.version.split()[0]}, NumPy {np.__version__}, "
f"SciPy {scipy.__version__}")
print("Seeds: 7 (initial state; present state of item 11); 1000-1199 (item 9); "
"2026 (items 14-15)")
Output:
1. reductions unitary; V_AB = explicit permutation: True
groupoid law V_BC V_AB = V_AC: True
2. instrument complete: True | all generators G-invariant: True
3. intertwining V K^A V^dag = K^B for W and every Kraus operator: True
4. 4096 histories of 3 stages; max |p_A-p_B| = 2.78e-17, max |p_A-p_neutral| = 1.39e-17
max(1-|<V phi_A|phi_B>|) = 6.66e-16, total prob = 1.000000000000
5. naive same-form frame-B rule: total-variation distance from correct record measure = 0.180
6. records [(0, 1, 1), (0, 2, 3)] vs [(0, 2, 3), (0, 1, 1)] -> same present ray: True
7. entanglement entropy: frame A 0.000 bits, frame B 1.000 bits
8. absolute collapse: ||(1-Pi) P_abs Psi|| / ||P_abs Psi|| = 0.894
9. continuous time: 200 trajectories, 1158 events; identical event records in A and B: True
max(1-|<V psi_A|psi_B>|) at final time = 6.7e-16
10. generic parameters, 4096 three-stage histories: exact coincidences = 0; largest overlap = 0.9852
minimum error probability for telling that pair apart (Helstrom) = 0.414
11. two different 3-stage histories, same present ray: |<end|Phi>| = 1.000000000000, 1.000000000000
history probabilities 4.15e-05, 7.62e-05; overlap of the two initial states = 0.552
12a. finite width: both branches end at the same ray (|<end|Phi>| = 1.000000000000, 1.000000000000),
probabilities 5.04e-05, 1.62e-03; next-event total variation = 0.1000 (lambda/6 = 0.1000)
12b. sharp, fixed initial state: same present ray: True; probabilities 8.254e-04, 1.238e-03;
next-event total variation = 0.1000
13. conditioned on (1, 2, 0) at stage 3: Doob forward kernel vs conditioned law, max diff = 8.3e-17
14. causally local instruments: all 136 foliations (linear extensions), 256 outcome histories;
max |p - p'| = 6.9e-18, max state deviation = 4.4e-16
instruments changed outside the predecessor set of slot 3: max change = 7.8e-16
order ideal [0, 1, 3]: law from its own instruments vs marginal of full law, max diff = 1.1e-16
non-commuting 'spacelike' pair: total variation between orders = 0.165
15. same channel: True; joint law of (first event, later X outcome):
[[0.25, 0.25], [0.25, 0.25]] vs [[0.5, 0.0], [0.0, 0.5]];
mutual information 0.000 vs 1.000 bits
16. drifting quantum clock, exact on the 32-site ring (signed displacement in -15..16):
P(backward) = 0.104042655, P(no advance) = 0.338161965;
infinite-lattice Bessel values 0.104042655, 0.338161965
max difference = 1.1e-16; P(|displacement| >= 8) = 7.4e-09
period-2 counter readings at stages 0-4: [0, 1, 0, 1, 0];
superposed counter: reading at stage n is n or n+4, each with probability 0.5
17. three-region test, Bob's P(b=1) and its shift under Alice's detector / kick:
gapless Charlie: P = 0.3665447170 (analytic 0.3665447170); shifts -2.2e-16, -6.1e-16
unsharp Lueders Charlie: P = 0.275038; shifts -0.021492, -0.046628
cutoff 40 vs 50: max change = 8.8e-08
18. Gaussian collapse at C (hit probability 1.0): P = 0.3449027185 (analytic 0.3449027185);
shifts under Alice's detector / kick: -3.3e-16, -5.0e-14
18. Gaussian collapse at C (hit probability 0.4): P = 0.2541731691 (analytic 0.2541731691);
shifts under Alice's detector / kick: -8.3e-16, -2.0e-14
Environment: Python 3.13.16, NumPy 2.5.3, SciPy 1.18.1
Seeds: 7 (initial state; present state of item 11); 1000-1199 (item 9); 2026 (items 14-15)
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