85 lines
2.8 KiB
Python
85 lines
2.8 KiB
Python
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"""optimize_cluster_witness.py --
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(1) optimize the individual cluster map ||M_AB||_* over different biseparable cut types,
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showing that its universal ceiling (5.0) is reached not just by the "home" cut AB|CD
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but also by a state biseparable across the UNRELATED cut AC|BD (Bell_AC x Bell_BD);
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(2) optimize min(||M_AB||_*, ||M_AC||_*, ||M_AD||_*) over 1|3-biseparable pure states,
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finding a naive ceiling of 7/3 -- later shown (via mixture_search.py) to be beatable
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by proper MIXTURES, since min() is not convex.
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"""
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import numpy as np
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from scipy.optimize import minimize
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from core import state_1_3, state_2_2
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from core2 import full_tensor
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from cluster import cluster_map, nuc
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def triple(psi):
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C = full_tensor(psi)
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return nuc(cluster_map(C, 0, 1)), nuc(cluster_map(C, 0, 2)), nuc(cluster_map(C, 0, 3))
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# --- witness ||M_AB||_* over various biseparable cut types ---
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def obj_A_BCD(params):
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psi = state_1_3(params, 0)
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C = full_tensor(psi)
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return -nuc(cluster_map(C, 0, 1))
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def obj_C_ABD(params):
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psi = state_1_3(params, 2)
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C = full_tensor(psi)
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return -nuc(cluster_map(C, 0, 1))
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def obj_AC_BD(params):
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psi = state_2_2(params, (0, 2), (1, 3))
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C = full_tensor(psi)
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return -nuc(cluster_map(C, 0, 1))
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def obj_AB_CD(params):
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psi = state_2_2(params, (0, 1), (2, 3))
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C = full_tensor(psi)
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return -nuc(cluster_map(C, 0, 1))
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# --- min(M_AB,M_AC,M_AD) over 1|3-biseparable pure states ---
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def neg_min_A_BCD(params):
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psi = state_1_3(params, 0)
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return -min(triple(psi))
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def neg_min_B_ACD(params):
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psi = state_1_3(params, 1)
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return -min(triple(psi))
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def run(obj, nparams, args, n_restarts, seed0, label, maxiter=2500):
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best = -np.inf
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bx = None
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for i in range(n_restarts):
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rng = np.random.default_rng(seed0 + i)
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x0 = rng.normal(size=nparams)
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res = minimize(obj, x0, args=args, method='Powell',
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options={'maxiter': maxiter, 'xtol': 1e-9, 'ftol': 1e-11})
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v = -res.fun
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if v > best:
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best = v
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bx = res.x
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print(f'{label}: best value = {best:.8f} ({n_restarts} restarts)')
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return best, bx
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if __name__ == "__main__":
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print("=== ||M_AB||_* over various biseparable cut types (universal ceiling = 5) ===")
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run(obj_A_BCD, 18, (), 8, 10, "biseparable A|BCD")
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run(obj_C_ABD, 18, (), 8, 20, "biseparable C|ABD")
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run(obj_AC_BD, 16, (), 8, 30, "biseparable AC|BD <-- reaches 5 (Bell_AC x Bell_BD)")
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run(obj_AB_CD, 16, (), 6, 40, "biseparable AB|CD (home cut, sanity <= 1)")
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print()
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print("=== min(M_AB,M_AC,M_AD) over 1|3-biseparable states (naive ceiling = 7/3) ===")
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run(neg_min_A_BCD, 18, (), 10, 300, "max over biseparable A|BCD")
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run(neg_min_B_ACD, 18, (), 10, 400, "max over biseparable B|ACD")
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print("For comparison: GHZ4 / connected graph states also give min = 7/3 =", 7 / 3)
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