"""Independent small-matrix checks for the hereditary quantum refusal theorem. Intermediate verification source; NumPy is the only nonstandard dependency. """ from __future__ import annotations import itertools,json,math from pathlib import Path import numpy as np def kron_all(xs): out=np.array([[1.0+0j]]) for x in xs: out=np.kron(out,x) return out def downset(facets,n): return {s for s in range(1<>i)&1 else 1-q[i] for i in range(n)) for s in range(1<>i)&1] for i in range(n)]) for x in range(1<1e-9 E_global=np.eye(3**n)-V[:,keep]@V[:,keep].conj().T E_local=sum((kron_all([Q[i] if (s>>i)&1 else P[i] for i in range(n)]) for s in range(1<>i)&1]).real) for i in range(n)] actual=float(np.trace(E_global@rho[x]).real) pred=probability(q,K) max_prob=max(max_prob,abs(actual-pred)); comparisons+=1 if x in K: max_allowed=max(max_allowed,abs(actual)) # Random sound effect supported in the complement must lie below E_global. null=V[:,~keep] if null.shape[1]: A=rng.normal(size=(null.shape[1],null.shape[1]))+1j*rng.normal(size=(null.shape[1],null.shape[1])) H=A@A.conj().T H/=max(1.,float(np.linalg.eigvalsh(H)[-1])) E=null@H@null.conj().T max_dom=max(max_dom,float(max(0.,-np.linalg.eigvalsh(E_global-E)[0]))) return {"trials":trials,"product_input_comparisons":comparisons, "maximum_projector_operator_norm_difference":max_proj, "maximum_probability_absolute_difference":max_prob, "maximum_false_refusal_on_allowed_word":max_allowed, "maximum_lowner_domination_violation":max_dom} def polynomial_check(): # Minimal nonfaces {1,2,3}, {3,4,5}, {1,4}; coordinates numbered 0 below. circuits=[0b00111,0b11100,0b01001] K={s for s in range(32) if not any((s&c)==c for c in circuits)} rng=np.random.default_rng(46) err=0. for _ in range(100): q=rng.uniform(size=5) term=lambda s:math.prod(q[i] for i in range(5) if (s>>i)&1) compact=term(0b01001)+term(0b00111)+term(0b11100)-term(0b01111)-term(0b11101) err=max(err,abs(probability(q,K)-compact)) return {"random_parameter_comparisons":100,"maximum_polynomial_difference":err} def pure_state_checks(seed=864126): rng=np.random.default_rng(seed) max_projector=max_probability=max_sound=0.0 cases=[]; comparisons=0 def forests_k4(): edges=list(itertools.combinations(range(4),2)); K=set() for S in range(1<>j)&1): continue u,v=find(a),find(b) if u==v: good=False; break parent[u]=v if good: K.add(S) return K circuits=[0b00111,0b11100,0b01001] specs=[('sparse_n6_t2',6,{s for s in range(64) if s.bit_count()<=2}), ('graphic_K4_forests',6,forests_k4()), ('nonmatroid_three_forbidden_supports',5, {s for s in range(32) if not any((s&c)==c for c in circuits)}), ('arbitrary_downset_n6',6,downset([0b001111,0b110011,0b101100],6)), ('full_simplex_n4',4,set(range(16))), ('zero_word_only_n4',4,{0})] for name,n,K in specs: zeros=[];ones=[];perps=[] for i in range(n): A=rng.normal(size=(2,2))+1j*rng.normal(size=(2,2)) U,_=np.linalg.qr(A) b=rng.uniform(.2,.9); phase=np.exp(1j*rng.uniform(-np.pi,np.pi)) zeros.append(U[:,0]);perps.append(U[:,1]) ones.append(b*phase*U[:,0]+math.sqrt(1-b*b)*U[:,1]) def state(x,excitation=False): out=np.array([1.+0j]) for i in range(n): v=(perps[i] if excitation else ones[i]) if (x>>i)&1 else zeros[i] out=np.kron(out,v) return out Phi=np.column_stack([state(x) for x in sorted(K)]) U,sv,_=np.linalg.svd(Phi,full_matrices=False) keep=sv>1e-10 E_global=np.eye(1<>i)&1 else 0. for i in range(n)] pred=probability(q,K) local_err=max(local_err,abs(actual-pred)); comparisons+=1 if x in K:max_sound=max(max_sound,abs(actual)) max_probability=max(max_probability,local_err) cases.append({'name':name,'coordinates':n,'allowed_words':len(K), 'projector_difference':error,'maximum_probability_difference':local_err}) return {'cases':cases,'word_comparisons':comparisons, 'maximum_projector_operator_norm_difference':max_projector, 'maximum_probability_absolute_difference':max_probability, 'maximum_allowed_false_refusal':max_sound},specs def link_bound_checks(specs,seed=29596): rng=np.random.default_rng(seed) worst_violation=0.;saturated_error=0.;checks=0 for name,n,K in specs: full=(1<>i)&1 else eps[i] for i in range(n)] actual=probability(qs,K) outside=full^S # Probability of forbidden S union N, N has outside independent clicks. upper=0. for N in range(1<>i)&1 else 1-eps[i] for i in range(n) if (outside>>i)&1) worst_violation=max(worst_violation,actual-upper) saturated=[1. if (S>>i)&1 else eps[i] for i in range(n)] saturated_error=max(saturated_error,abs(probability(saturated,K)-upper)) checks+=1 return {'allowed_support_checks':checks, 'maximum_false_positive_bound_violation':worst_violation, 'maximum_saturated_equality_error':saturated_error} def full_rank_background_check(): # Any full-rank all-zero allowed state rules out nonzero sound effect. rho0=np.diag([.9,.1]); allowed=np.kron(rho0,rho0) w,V=np.linalg.eigh(allowed) E=np.eye(4)-V[:,w>1e-12]@V[:,w>1e-12].conj().T return {'minimum_allowed_state_eigenvalue':float(w.min()), 'maximal_refusal_effect_operator_norm':float(np.linalg.norm(E,2))} if __name__=='__main__': pure,specs=pure_state_checks() result={'pure_state_filter':pure,'mixed_state_filter':matrix_checks(), 'five_vertex_overlap_polynomial':polynomial_check(), 'false_positive_link_bound':link_bound_checks(specs), 'full_rank_background_obstruction':full_rank_background_check()} assert pure['maximum_projector_operator_norm_difference'] < 1e-9 assert pure['maximum_probability_absolute_difference'] < 1e-9 assert result['mixed_state_filter']['maximum_projector_operator_norm_difference'] < 1e-9 assert result['mixed_state_filter']['maximum_probability_absolute_difference'] < 1e-9 assert result['false_positive_link_bound']['maximum_false_positive_bound_violation'] < 1e-12 assert result['false_positive_link_bound']['maximum_saturated_equality_error'] < 1e-12 assert result['five_vertex_overlap_polynomial']['maximum_polynomial_difference'] < 1e-12 assert result['full_rank_background_obstruction']['maximal_refusal_effect_operator_norm'] < 1e-12 result['all_numeric_checks_passed']=True out=Path(__file__).with_name('downset_checks.json') out.write_text(json.dumps(result,indent=2)+'\n') print(json.dumps(result,indent=2))