# Quantum Reference Frames and Single-Outcome Dynamics — A Compatibility Framework for Generative Presentism Part 39 of 40 (1,588 words) def coherent_states(alphas, nc): C = np.zeros((len(alphas), nc), complex) C[:, 0] = np.exp(-np.abs(alphas)**2 / 2) for n in range(1, nc): C[:, n] = C[:, n - 1] * alphas / np.sqrt(n) return C dp, npg = 0.12, 81 pg = (np.arange(npg) - npg // 2) * dp def pointer_states(cu, cw, M, sg, chirp): """field vector for each pair of pointer momenta after the coupling U, pointers prepared in Gaussians of widths sg, pre-chirped by exp(i M_kk p_k^2/2) if chirp""" P = np.stack([a.ravel() for a in np.meshgrid(pg, pg, indexing='ij')], 1) amp = np.prod((2 * sg**2 / np.pi)**0.25 * np.exp(-sg**2 * P**2), axis=1) ph = -0.5 * np.einsum('gk,kl,gl->g', P, M, P) + (0.5 * P**2 @ np.diag(M) if chirp else 0) alph = (P @ cw - 1j * (P @ cu)) / np.sqrt(2) # exp(-i p.phi)|0> is coherent return (amp * np.exp(1j * ph))[:, None] * coherent_states(alph, ncut) def chi_pointer(F, n1, n2): """E exp(i s.y) = sum_p for s = (n1, n2) grid steps""" G = F.reshape(npg, npg, -1) A = G[max(n1, 0):npg + min(n1, 0), max(n2, 0):npg + min(n2, 0)] B = G[max(-n1, 0):npg + min(-n1, 0), max(-n2, 0):npg + min(-n2, 0)] return np.sum(A.conj() * B) * dp * dp def chi_sequential(order, eigs, sg, s): def ms(m, w_): dd, mm = w_[:, None] - w_[None, :], (w_[:, None] + w_[None, :]) / 2 return np.exp(-dd**2 / (8 * sg[m]**2) + 1j * s[m] * mm - s[m]**2 * sg[m]**2 / 2) return run_order(order, eigs, ms) shifts = [(1, 0), (0, 1), (2, -1), (3, 2), (-2, 4), (5, 3), (-4, -3)] sg = np.array([0.8, 0.9]) res = {} for name, pts in (('overlap', np.array([[0.3, 0.1], [0.55, 0.25]])), ('one-way', np.array([[0.0, 0.0], [1.2, 0.3]]))): R, E = geometry(pts, dl, scale) M = Mret(pts, dl, scale) r = {'E': E, 'M': M, 'R': R, 'split': np.abs(M - M.T - E).max()} T0 = {nn: ordered_sums(pts, dl, scale, nn, 0.0) for nn in (32, 64)} T1 = ordered_sums(pts, dl, scale, 64, 0.6) r['fol'] = np.abs(T1 - T0[64]).max() r['disc'] = [np.abs(T0[nn] - M).max() for nn in (32, 64)] cu, cw = realize(E) Sig = 0.5 * np.outer(cu, cu) + 0.5 * np.outer(cw, cw) D = np.diag(1 / (4 * sg**2)) Mo = M - np.diag(np.diag(M)) r['C'] = Sig + np.diag(sg**2) + Mo @ D @ Mo.T r['Cfull'] = Sig + np.diag(sg**2) + M @ D @ M.T r['Cord'] = cov_closed([0, 1], E, Sig, sg) eigs = [eigh(cu[i] * qo + cw[i] * po) for i in range(2)] dev = {} for chirp in (True, False): F = pointer_states(cu, cw, M, sg, chirp) C = r['C'] if chirp else r['Cfull'] dev[chirp] = max(abs(chi_pointer(F, *n) - np.exp(-0.5 * dp**2 * np.array(n) @ C @ np.array(n))) for n in shifts) if chirp: r['seq'] = max(abs(chi_pointer(F, *n) - chi_sequential([0, 1], eigs, sg, dp * np.array(n))) for n in shifts) rho_p = (F.T @ F.conj()) * dp * dp rho0 = np.zeros((ncut, ncut), complex) rho0[0, 0] = 1 rho_s = dephase(dephase(rho0, cu[0] * qo + cw[0] * po, sg[0]), cu[1] * qo + cw[1] * po, sg[1]) r['unread'] = np.abs(rho_p - rho_s)[:60, :60].max() r['trace'] = np.trace(rho_p).real r['dev'] = dev res[name] = r ov, ow = res['overlap'], res['one-way'] print(' (a) E = Gamma - Gamma^T: max dev %.1e (overlapping pair), %.1e (one-way pair)' % (ov['split'], ow['split'])) print(' overlapping pair: Gamma_12 = %.6f, Gamma_21 = %.6f, Gamma_11 = Gamma_22 = %.6f' % (ov['M'][0, 1], ov['M'][1, 0], ov['M'][0, 0])) print(' phase of the time-ordered product, two foliations (relative rapidity 0.6):') print(' max difference %.1e and %.1e;' % (ov['fol'], ow['fol'])) print(' deviation from Gamma, 32 and 64 nodes per axis: %.1e, %.1e (overlapping),' % tuple(ov['disc'])) print(' %.1e, %.1e (one-way)' % tuple(ow['disc'])) print(' (b) overlapping pair, explicit pointers, characteristic function vs closed form:') print(' pre-chirped pointers %.1e; without the pre-chirp, full Gamma, %.1e' % (ov['dev'][True], ov['dev'][False])) print(' (c) unread field channel vs two Gaussian collapse channels: max dev %.1e;' % ov['unread']) print(' trace %.12f' % ov['trace']) print(' (d) one-way pair: pointers vs Gaussian collapses applied in causal order, direct') print(' characteristic functions: max dev %.1e; closed-form covariances: max diff %.1e' % (ow['seq'], np.abs(ow['C'] - ow['Cord']).max())) dC = ov['C'] - ov['Cord'] print(' overlapping pair, pointer law minus the law of collapses in centre-time order:') print(' Var(y1) %+.6f, Var(y2) %+.6f, Cov(y1, y2) %+.6f;' % (dC[0, 0], dC[1, 1], dC[0, 1])) print(' direct characteristic functions differ by up to %.4f' % ov['seq']) Output: C1. Random-kick form of the unread Gaussian collapse channel Kraus integral over y vs exact multiplier: max dev 2.4e-14 Gaussian mixture of kicks vs exact multiplier: max dev 7.8e-16 C2. Energy-momentum cost (a) cutoff 140: ground 0.150060024010, random 0.150060024010; 1/(8 m sig^2) = 0.150060024010 (a) cutoff 180: ground 0.150060024010, random 0.150060024010; 1/(8 m sig^2) = 0.150060024010 (b) two modes: vacuum 0.126367187500, random 0.126367187500; predicted 0.126367187500 (c) lattice: canonical 0.041729335207, Fourier formula 0.041729335207, relative diff. 5.0e-16 (d) Delta P(h) = (0.00710022, 0.00104472), squared invariant mass 0.00004932 boost of rapidity 0.5: |Delta P(Lambda h) - Lambda Delta P(h)| = 1.3e-16 C3. Poisson-located Gaussian collapses, massless field in 1+1 dimensions (a) counts, area 16, intensity 6: mean 96.00, variance 96.35 (expected 96 and 96) (b) intensity 1.5, 40000 samples per box: side P(orderable) bound of Prop. 13.22 1.0 0.59905 0.79757 1.5 0.25833 0.60115 2.0 0.08060 0.40465 2.5 0.01510 0.24325 (c) 300 orderable configurations, 1265 collapses, 1176 linear extensions: max covariance difference between linear extensions 1.8e-15 (d) 4 three-collapse configurations, one-mode realization, Fock cutoff 140: characteristic function, direct vs closed form: max dev 1.8e-14 64-cell binned history laws: max difference between linear extensions 4.8e-16, normalization error 3.1e-14 (e) overlapping pair, E(h1, h2) = 1.099141: increase of Var(y2) caused by collapse 1: 0.4719185385; E^2/(4 sigma1^2) = 0.4719185385 binned joint laws for the two orders: total-variation distance 0.0570 unread channels in the two orders: max difference of the final states 6.8e-15 (f) 300 non-orderable configurations ordered by centre time in two frames (relative rapidity 0.6): laws differ in 234, max covariance difference 0.701 (g) centre-time order is a linear extension of the causal order in 300 of the 300 orderable configurations of item (c) (h) order j, c1, c2, y by t; c1 and c2 overlap; E(h_j, h_c1) = 0.2334, E(h_c1, h_y) = 0.2334, E(h_c1, h_c2) = 1.2221, E(h_j, h_y) = 0.0 cutoff 36: Gaussian c1: change of the target law by a kick at j 1.6e-15, by a collapse at j 1.1e-16; target law sums to 1.000000000000 unsharp Lueders instrument at j (not Weyl-diagonal): change 2.8e-16 unsharp Lueders c1 (negative control): change by the kick, largest bin 0.0008100 cutoff 44: Gaussian c1: change of the target law by a kick at j 8.9e-16, by a collapse at j 1.1e-16; target law sums to 1.000000000000 unsharp Lueders instrument at j (not Weyl-diagonal): change 1.1e-16 unsharp Lueders c1 (negative control): change by the kick, largest bin 0.0008100 C4. Gapless pointer realization of the collapses (Proposition 13.31) (a) E = Gamma - Gamma^T: max dev 2.2e-16 (overlapping pair), 0.0e+00 (one-way pair) overlapping pair: Gamma_12 = -0.060881, Gamma_21 = -1.160022, Gamma_11 = Gamma_22 = -0.500000 phase of the time-ordered product, two foliations (relative rapidity 0.6): max difference 0.0e+00 and 0.0e+00; deviation from Gamma, 32 and 64 nodes per axis: 2.8e-02, 9.0e-03 (overlapping), 1.9e-03, 4.8e-04 (one-way) (b) overlapping pair, explicit pointers, characteristic function vs closed form: pre-chirped pointers 7.3e-14; without the pre-chirp, full Gamma, 2.0e-13 (c) unread field channel vs two Gaussian collapse channels: max dev 1.7e-14; trace 1.000000000000 (d) one-way pair: pointers vs Gaussian collapses applied in causal order, direct characteristic functions: max dev 1.8e-13; closed-form covariances: max diff 0.0e+00 overlapping pair, pointer law minus the law of collapses in centre-time order: Var(y1) +0.001144, Var(y2) +0.053727, Cov(y1, y2) +0.000000; direct characteristic functions differ by up to 0.0049 C.2 Two Ideal Clocks # Verification of Proposition 14.5 (Appendix C.2): two ideal clocks, interactions in the # constraint. Physical states are Psi(t_A, t_B) = U_Delta(t_A) chi(Delta), Delta = t_A - t_B, # where U_Delta solves i dU/ds = H_Delta(s) U from s0 = -2.5, with # H_Delta(s) = H_S + sum_{k keyed to A} g_k(s) V_k + sum_{k keyed to B} g_k(s - Delta) V_k. import numpy as np from scipy.linalg import expm sx = np.array([[0, 1], [1, 0]], complex) sy = np.array([[0, -1j], [1j, 0]]) sz = np.diag([1.0, -1.0]).astype(complex) P1 = np.diag([0.0, 1.0]).astype(complex) I2 = np.eye(2) kron = np.kron bump = lambda z: np.where(np.abs(z) < 1, np.exp(-1 / np.maximum(1 - z**2, 1e-300)), 0.0) ds, s0, s1 = 0.01, -2.5, 7.0 sgrid = s0 + ds * np.arange(int(round((s1 - s0) / ds)) + 1) mids = sgrid[:-1] + ds / 2 def pulse(area): """switching function supported in [0, 1]; its midpoint sum on the grid equals area""" g = lambda s: bump((s - 0.5) / 0.5) norm = np.sum(g(np.arange(0, 1, ds) + ds / 2)) * ds return lambda s: area * g(s) / norm def propagate(Hfun): """U(s) from s0 at every grid point""" U = np.eye(4, dtype=complex) out = [U] for m in mids: U = expm(-1j * ds * Hfun(m)) @ U out.append(U) return out at = lambda s: int(round((s - s0) / ds)) rng = np.random.default_rng(5) def system_state(): v = rng.normal(size=2) + 1j * rng.normal(size=2) return np.kron(v / np.linalg.norm(v), [1, 0]) # record starts in |0> def packets(centres, width, states): """chi(Delta) on a grid of offsets: wave packets in Delta carrying the given states""" Ds = np.round(np.arange(-3.0, 3.0 + ds / 2, ds), 10) chi = np.zeros((len(Ds), 4), complex) for c, st in zip(centres, states): chi += bump((Ds - c) / width)[:, None] * st[None, :] chi /= np.sqrt(np.sum(np.abs(chi)**2) * ds) keep = np.abs(chi).sum(1) > 0 return Ds[keep], chi[keep]