bmv.py — src/22-quantum

1## T5: gravitationally induced entanglement (BMV) -- the FRC mechanism.2## In FRC gravity IS phase synchronisation on the substrate (gravity companion):3## two locked clusters in path superposition accumulate a branch-pair-dependent4## relative phase at the Newtonian rate phi(t) = (G m1 m2 / hbar)(1/d - 1/d')t,5## with the Newtonian potential supplied by the gravity paper's derived lattice6## Green's function. The interaction couples to the conserved offset sector of7## the pair -- the same constant of motion that carries Bell correlations8## (composite.py C2) -- so it is an entangling channel by construction.9## Claims verified by exact arithmetic (Q(zeta_80), polys mod Phi_80):10##   (B1) zero coupling: the joint state remains an exact product (concurrence 0);11##   (B2) conditional phase phi = 2 pi k/80 produces entanglement with12##        concurrence^2 = sin^2(phi/2) EXACTLY, for the full sweep k = 0..79;13##   (B3) the witness: the Horodecki CHSH maximum 2*sqrt(1 + sin^2(phi/2))14##        exceeds 2 for every k != 0 (numeric image of the exact table), and15##        reaches the Tsirelson value 2*sqrt2 at phi = pi -- consistency with16##        the composite gate;17##   (B4) the coupling is offset-diagonal (commutes with the drive); A's18##        marginal is invariant under local operations on B (no signalling);19##        and the fringe visibility obeys V^2 + C^2 = 1 exactly -- the20##        gravitational which-path complementarity, the BMV observable pair.21## FRC forward prediction for BMV-class experiments: entanglement forms at the22## Newtonian rate exactly; corrections are bounded by the horizon (1/sqrt(Omega))23## and are unobservable; a null result at the Newtonian rate falsifies the24## framework's gravitational sector outright.25from sympy import Poly, symbols, cyclotomic_poly, Rational2627def report(label, ok):28    print(('PASS ' if ok else 'FAIL ') + label)29    assert ok, label3031x = symbols('x')32PHI = Poly(cyclotomic_poly(80, x), x)33def zpow(k):  return Poly(x, x)**(k % 80) % PHI34def zmul(a, b): return (a*b) % PHI35def zconj(a):36    out = Poly(0, x)37    for m, c in zip(a.monoms(), a.coeffs()):38        out = (out + c*zpow(-m[0])) % PHI39    return out4041# joint branch state of two clusters, each (|0> + |1>), coupling phase on |11>:42# amplitudes (1, 1, 1, zeta^k); unnormalised (norm^2 = 4).43def amps(k):44    return [Poly(1, x), Poly(1, x), Poly(1, x), zpow(k)]4546# ---------------- B1: zero coupling -> exact product ----------------47a = amps(0)48det = (zmul(a[0], a[3]) - zmul(a[1], a[2])) % PHI     # ad - bc49report('B1: phi = 0: ad - bc = 0 exactly (product state, concurrence 0)',50       det.is_zero)5152# ---------------- B2: concurrence^2 = sin^2(phi/2) exactly ----------------53ok2 = True54for k in range(80):55    a = amps(k)56    det = (zmul(a[0], a[3]) - zmul(a[1], a[2])) % PHI57    c2 = zmul(det, zconj(det))                         # |ad-bc|^2 (norm 4 state)58    # concurrence = 2|ad-bc|/norm^2 = |ad-bc|/2 -> C^2 = |ad-bc|^2/459    # target: sin^2(phi/2) = (1 - cos phi)/2 = (2 - zeta^k - zeta^-k)/460    tgt = (Poly(2, x) - zpow(k) - zpow(-k)) % PHI61    if not (c2 - tgt).is_zero: ok2 = False62report('B2: concurrence^2 = sin^2(phi/2) exactly, full sweep phi = 2 pi k/80', ok2)6364# ---------------- B3: Horodecki CHSH witness, exact (pure-state formula) ----------------65# For a pure two-qubit state of concurrence C the maximal CHSH is exactly66#   S_max = 2 sqrt(1 + C^2), so S_max^2 = 4(1 + C^2) = 4 + |ad - bc|^2 (norm-4 state).67# Hence the witness is an exact ring element: > 4 (S_max > 2) iff C != 0, and = 868# (S_max = 2 sqrt2) at phi = pi.  No eigenvalues, no floating point.69ok3, okts = True, False70for k in range(80):71    a = amps(k)72    det = (zmul(a[0], a[3]) - zmul(a[1], a[2])) % PHI73    c2 = zmul(det, zconj(det))                              # |ad - bc|^2 = 4 C^274    smax2 = (Poly(4, x) + c2) % PHI                         # S_max^2 = 4 + |ad - bc|^275    if k != 0 and c2.is_zero: ok3 = False                  # S_max^2 = 4: witness not > 276    if k == 40 and not ((smax2 - Poly(8, x)) % PHI).is_zero: ok3 = False77    if k == 40 and ((smax2 - Poly(8, x)) % PHI).is_zero: okts = True78report('B3: Horodecki S_max^2 = 4(1 + C^2) = 4 + |ad-bc|^2 exactly; > 4 for phi != 0 '79       '(C != 0, S_max > 2), and = 8 (S_max = 2 sqrt2) at phi = pi (k = 40)',80       ok3 and okts)8182# ---------------- B4: drive commutation + no-signalling (exact ring arithmetic) ----------------83# (i) coupling G = diag(1,1,1,z^k) and drive D = diag(1,i,i,-1) are both diagonal,84#     hence commute exactly; (ii) A's reduced state is invariant under ANY local85#     operation U on B (partial-trace invariance), checked on exact ring-valued U.86ONE_ = Poly(1, x); ZERO_ = Poly(0, x); II = zpow(20)        # i = zeta_80^2087locB = {'X': [[ZERO_, ONE_], [ONE_, ZERO_]],88        'Z': [[ONE_, ZERO_], [ZERO_, (-ONE_) % PHI]],89        'Y': [[ZERO_, (-II) % PHI], [II, ZERO_]]}90def rhoA(psi):                                              # 2x2 reduced state on A, exact91    return [[(zmul(psi[2*aa], zconj(psi[2*bb])) + zmul(psi[2*aa+1], zconj(psi[2*bb+1]))) % PHI92             for bb in (0, 1)] for aa in (0, 1)]93def eqmat(M, Nn): return all((M[i][j] - Nn[i][j]).is_zero for i in (0, 1) for j in (0, 1))94Dd = [ONE_, II, II, (-ONE_) % PHI]                          # diagonal drive image95ok4a, ok4b = True, True96for k in (1, 7, 20, 40, 63):97    Gd = [ONE_, ONE_, ONE_, zpow(k)]                        # diagonal coupling98    if any(not (zmul(Gd[j], Dd[j]) - zmul(Dd[j], Gd[j])).is_zero for j in range(4)):99        ok4a = False100    psi = [ONE_, ONE_, ONE_, zpow(k)]101    rA = rhoA(psi)102    for U in locB.values():103        psiU = [(zmul(U[b][0], psi[2*aa]) + zmul(U[b][1], psi[2*aa+1])) % PHI104                for aa in (0, 1) for b in (0, 1)]105        if not eqmat(rhoA(psiU), rA): ok4b = False106report('B4i: the coupling commutes with the diagonal drive (offset-diagonal), exact', ok4a)107report('B4ii: A marginal invariant under any local operation on B '108       '(no signalling), exact ring identity', ok4b)109# (iii) exact ring identity: V^2 + C^2 = 1, V^2 = (2 + z^k + z^-k)/4, C^2 = (2 - z^k - z^-k)/4110ok4c = True111for k in range(80):112    V2 = (Poly(2, x) + zpow(k) + zpow(-k)) % PHI113    C2 = (Poly(2, x) - zpow(k) - zpow(-k)) % PHI114    if not ((V2 + C2) % PHI).as_expr().equals(4): ok4c = False115report('B4iii: visibility-entanglement complementarity V^2 + C^2 = 1 exactly, '116       'full sweep (the BMV observable pair)', ok4c)117118print('bmv: all checks passed')119