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"PASS", "tolerance": 2e-09, "unit": "natural_log", "value": 3.6604053121891e-13 }, { "comparison": "greater_than_or_equal", "id": "central_bond_retained_adversary_breaks_entanglement_invariance", "note": "Retaining the central spring at half strength leaves post-quench A-B coupling and invalidates exact entanglement invariance.", "status": "PASS", "tolerance": 0.0001, "unit": "natural_log", "value": 0.15041244483344 }, { "comparison": "less_than_or_equal", "id": "NPT_classification_consistency", "note": "Positive stored log negativity agrees with at least one PT symplectic eigenvalue below 1/2.", "status": "PASS", "tolerance": 0.0, "unit": "boolean_violation", "value": 0.0 }, { "comparison": "equals", "id": "multimode_PPT_caveat_encoded", "note": "For the N/2 by N/2 multimode split, zero Gaussian logarithmic negativity means PPT and is not asserted to prove separability.", "status": "PASS", "tolerance": 1, "unit": "boolean", "value": 1 } ], "central_bond_retained_adversary": { 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"finite_open_harmonic_chain_gaussian_benchmark_with_a_central_spring_cut_supports_record_conditioned_versus_unconditional_logarithmic_negativity_comparisons_for_the_declared_homodyne_unravelings_and_frozen_regulator_perfect_efficiency_no_knowledge_monitoring_is_a_record_known_local_random_unitary_and_preserves_the_cut_entanglement_while_discarding_the_record_recovers_the_same_diffusive_unconditional_channel_positive_logarithmic_negativity_certifies_NPT_entanglement_but_zero_logarithmic_negativity_does_not_prove_separability_for_this_multimode_split_no_postselection_no_feedback_based_decoherence_cancellation_no_measurement_induced_transition_continuum_limit_universal_scaling_or_interacting_QFT_claim_central_spring_removal_is_required_for_the_local_unitary_entanglement_invariance", "data_files": { "complete_csv": { "bytes": 3598105, "path": "public/figures/quantum-information/monitored-free-boson-unraveling-entanglement.csv", "records": 8784, "sha256": 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(kappa/J)/2." ], "correction_to_proposed_protocol": "none", "method": "Derived first and centered second moments directly from d rho_c=Lindblad_dimensionless(rho_c)d tau+sqrt(eta)H[e^(-i theta)l^T R]rho_c dW_tau.", "status": "PASS" }, "equations": { "conditional_covariance": "dot(Vc)=A Vc+Vc A^T+D-eta c_theta(Vc)c_theta(Vc)^T", "conditional_mean": "d(mu_c)=A mu_c d tau+sqrt(eta)c_theta(Vc)dW_tau", "diffusion": "D=Omega l l^T Omega^T, l=sqrt(kappa/J)e_Q(N/2)", "drift": "dR/dtau=A R with A=Omega G_dimensionless and G_dimensionless=diag(K_post,I)", "eta1_purity": "V_c Omega V_c=Omega/4", "ground_covariance": "V0=1/2 diag(K0^(-1/2),K0^(1/2))", "innovation": "c_theta(V)=2 cos(theta) V l+sin(theta) Omega l", "mean_covariance": "dot(M)=A M+M A^T+eta c_theta(Vc)c_theta(Vc)^T, M=Cov(mu_c)", "no_knowledge": "theta=pi/2 gives c=Omega l and dot(Vc)=A Vc+Vc A^T+(1-eta)D", "record_average": "Vu=Vc+M for the zero-mean initial state", "unconditional": "dot(Vu)=A Vu+Vu A^T+D", "unconditional_heating": "d(/J)/d tau=(kappa/J)/2 for L=sqrt(kappa)Q_N/2" }, "headline_metrics": { "central_bond_half_strength_adversary_maximum_log_negativity_change": 0.15041244483344, "maximum_N16_vs_N24_log_negativity_difference": 2.7416482895865e-05, "maximum_RK4_dt_vs_half_step_log_negativity_difference": 1.8462897877214e-11, "maximum_eta1_purity_matrix_residual_frobenius": 1.622252914799e-11, "maximum_inefficient_residual_diffusion_error_frobenius": 0.0, "maximum_no_knowledge_unitary_covariance_error_frobenius": 7.3707944588075e-11, "maximum_record_average_covariance_error_frobenius": 2.1029071789174e-14, "maximum_stable_raw_symplectic_spectrum_difference": 5.9952043329758e-15, "maximum_unconditional_heating_law_residual": 2.4868995751604e-14, "primary_initial_log_negativity": 0.27690881696996, "primary_kappa_0p10_tJ6": { "informative_eta1": 0.21195937281037, "no_knowledge_eta0p8": 0.23973491557267, "no_knowledge_eta1": 0.27690881696959, "unconditional": 0.16534582017413 }, "primary_kappa_0p40_tJ6": { "informative_eta1": 0.13316503551631, "no_knowledge_eta0p8": 0.17832903322723, "no_knowledge_eta1": 0.27690881696959, "unconditional": 0.084323093967922 } }, "initial_covariance_checks": { "16": { "maximum_vacuum_symplectic_error": 0.0, "pp_block_residual_frobenius": 0.0, "qq_block_residual_frobenius": 0.0, "stationarity_residual_frobenius": 0.0, "symmetry_residual": 0.0 }, "24": { "maximum_vacuum_symplectic_error": 0.0, "pp_block_residual_frobenius": 0.0, "qq_block_residual_frobenius": 0.0, "stationarity_residual_frobenius": 0.0, "symmetry_residual": 0.0 } }, "plot_encoding": { "record_code": { "strength_scan": 1, "time_series": 0 }, "series_code": { "informative_eta1": 1, "no_knowledge_eta0p8": 2, "no_knowledge_eta1": 3, "unconditional": 0 }, "size_control_markers": "N=24 strength-scan records use marker=open_ and line_style=none." }, "protocol": { "canonical_order": "R=(Q_1,...,Q_N,P_1,...,P_N)^T", "canonical_rescaling": "Q_j=sqrt(J) q_j; P_j=p_j/sqrt(J)", "chain": { "boundary": "open", "initial_hamiltonian": "H0/J=1/2 sum_j[P_j^2+(m/J)^2 Q_j^2] + 1/2 sum_{j=1}^{N-1}(Q_{j+1}-Q_j)^2", "initial_state": "exact Gaussian ground covariance of the uncut H0", "mass_over_J": 0.7, "partition": "A=sites 1..N/2; B=sites N/2+1..N", "quench": "At t=0 remove only the spring between one-based sites N/2 and N/2+1." }, "commutator": "[R_a,R_b]=i Omega_ab", "entanglement": { "formula": "E_N=sum_j max(0,-ln(2 tilde_nu_j))", "logarithm": "natural", "partial_transpose": "flip P_j for every j in B", "scope": "E_N>0 certifies NPT entanglement; E_N=0 does not certify separability for this multimode bipartition.", "symplectic_spectrum_method": "Cholesky V=C C^T followed by Hermitian eigvalsh(i C^T Omega C); raw eigvals(i Omega V) is retained only as a comparison and no clipping is used." }, "monitoring": { "dimensionless_SME_rate": "gamma=kappa/J", "kappa_over_J": [ 0.0, 0.02, 0.05, 0.1, 0.2, 0.4 ], "measured_site": "one-based N/2, the last site of A", "physical_lindblad_operator": "L=sqrt(kappa) Q_{N/2}", "unravelings": [ { "eta": 1.0, "id": "informative_eta1", "theta": 0.0 }, { "eta": 1.0, "id": "no_knowledge_eta1", "theta": 1.5707963267948966 }, { "eta": 0.8, "id": "no_knowledge_eta0p8", "theta": 1.5707963267948966 } ] }, "numerics": { "floating_point": "IEEE-754 binary64 through NumPy", "half_step_control": { "N": 16, "dt_J": 0.00125 }, "integrator": "fixed-step classical RK4", "primary": { "N": 16, "dt_J": 0.0025 }, "sample_times_tJ": { "count": 61, "increment": 0.1, "maximum": 6.0, "minimum": 0.0 }, "size_control": { "N": 24, "dt_J": 0.0025 } }, "units": { "J_energy": 1.0, "dimensionless_rate": "gamma=kappa/J", "dimensionless_time": "tau=tJ", "hbar": 1.0, "lattice_spacing": 1.0 }, "vacuum_symplectic_eigenvalue": 0.5 }, "reproduction": { "check": "python figures-src/quantum-information/generate-monitored-free-boson-unraveling-entanglement.py --check", "determinism": "No random sampling is used; --check regenerates every byte and compares all three frozen outputs.", "write": "python figures-src/quantum-information/generate-monitored-free-boson-unraveling-entanglement.py" }, "resolution_metrics": { "half_step_control": { "maximum_eta1_purity_matrix_residual": 1.0146512629418e-12, "maximum_eta_residual_diffusion_error": 0.0, "maximum_informative_purity_error": 1.1013412404282e-13, "maximum_no_knowledge_unitary_covariance_error": 4.6065688957901e-12, "maximum_npt_classification_violation": 0.0, "maximum_record_average_covariance_error": 2.1029071789174e-14, "maximum_stable_raw_symplectic_spectrum_difference": 5.9952043329758e-15, "maximum_symmetry_residual": 0.0, "maximum_unconditional_heating_law_residual": 1.9539925233403e-14, "minimum_partial_transpose_symplectic_eigenvalue": 0.38097249663124, "minimum_physical_symplectic_eigenvalue": 0.49999999999959 }, "primary": { "maximum_eta1_purity_matrix_residual": 1.622252914799e-11, "maximum_eta_residual_diffusion_error": 0.0, "maximum_informative_purity_error": 1.6908696665041e-12, 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"minimum_partial_transpose_symplectic_eigenvalue": 0.38097430248327, "minimum_physical_symplectic_eigenvalue": 0.4999999999934 } }, "rights_basis": "original_QFT_org_reproducible_numerical_calculation_repository_artifact_license_not_yet_specified", "schema_id": "qft.quantum-information.monitored-free-boson-unraveling-entanglement.v1", "schema_version": 1, "title": "Monitored free-boson unraveling and entanglement benchmark" }