{ "schema_version": "1.0.0", "record_id": "qft.data.quantum-information.relative-entropy-recovery-benchmarks", "title": "Recovery-map and energy-constrained channel benchmarks", "generated_by": { "path": "scripts/generate-relative-entropy-recovery-benchmarks.mjs", "sha256": "ce0e7e439d458a23d54e8ffe874e7d83691c4655e460e716dc8aac097cc86e38", "runtime": "Node.js standard library only" }, "scope": { "claim": "Finite-regulator recovery maps and a one-mode cutoff channel are tested under explicit support, modular, locality, Hamiltonian, and energy assumptions.", "nonclaim": "The finite free-fermion calculation is not a continuum-local implementation, and the cutoff certificate is not an unconstrained diamond-norm or capacity result." }, "sources": [ { "role": "conditional-mutual-information recovery theorem", "citation": "Fawzi and Renner, Communications in Mathematical Physics 340 (2015), Theorem 5.1", "url": "https://doi.org/10.1007/s00220-015-2466-x" }, { "role": "explicit universal beta0-twirled recovery map and fidelity remainder", "citation": "Junge, Renner, Sutter, Wilde, and Winter, Annales Henri Poincare 19 (2018), Theorem 2.1 and Remark 2.2, Eqs. (17)-(21)", "url": "https://doi.org/10.1007/s00023-018-0716-0" }, { "role": "Petz equality and sufficiency map", "citation": "Petz, Communications in Mathematical Physics 105 (1986), pp. 123-131", "url": "https://doi.org/10.1007/BF01212345" }, { "role": "energy-constrained diamond norm and strong channel convergence", "citation": "Shirokov, Problems of Information Transmission 54 (2018), pp. 20-33", "url": "https://doi.org/10.1134/S0032946018010027" }, { "role": "energy-constrained continuity and bosonic channel applications", "citation": "Winter, Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities, arXiv:1712.10267, Sections II-IV", "url": "https://doi.org/10.48550/arXiv.1712.10267" } ], "recovery": { "model": { "description": "Five-site open spinless free-fermion Gibbs chain; a finite lattice regulator of a quadratic field model.", "hamiltonian": "H=sum_ij h_ij c_i^dagger c_j, h_ii=mu+(-1)^i m, h_i,i+1=h_i+1,i=-kappa", "site_count": 5, "hopping_kappa": 1, "staggered_mass_m": 0.7, "chemical_potential_mu": 0.15, "inverse_temperature_beta": 1.4, "logarithm": "natural", "fidelity": "root fidelity F(rho,sigma)=||sqrt(rho)sqrt(sigma)||_1", "tensor_order": [ "A: site 0", "B: sites 1-3", "C: site 4" ], "spectral_support_policy": "The base finite-temperature states are full rank: invert every strictly positive eigenvalue and apply no truncation or eigenvalue floor. The beta=8 stress test declares its separate clamp explicitly.", "edge_only_recovery_map": "id on sites 0-2 tensor Petz map from site 3 to sites 3-4" }, "theorem": { "conditional_mutual_information_nats": 0.0003645078306273586, "universal_root_fidelity_floor": 0.9998177626919222, "universal_remainder_statement": "I(A:C|B) >= -2 ln F(rho_ABC, (id_A tensor Rbar_B_to_BC)(rho_AB))", "beta0_density": "beta_0(t)=pi/[2(cosh(pi t)+1)]", "quadrature_change_of_variable": "u=[1+tanh(pi t/2)]/2, so beta_0(t) dt=du and t=(2/pi) artanh(2u-1)" }, "map_comparison": { "ordinary_petz": { "raw_trace": 1.0000000000000029, "root_fidelity": 0.9999963357929647, "fidelity_remainder_nats": 0.000007328427497141284, "trace_norm_error": 0.004942160134121786, "minimum_eigenvalue": 0.00002542125111509482 }, "symmetric_rotated_pair_theorem_variable_t_plus_minus_1": { "modular_parameters_t_over_2": [ -0.5, 0.5 ], "raw_trace": 1.0000000000000024, "root_fidelity": 0.9999831052205927, "fidelity_remainder_nats": 0.000033789844251287116, "trace_norm_error": 0.010629093444417495, "minimum_eigenvalue": 0.00002560609450064461 }, "universal_beta0_twirl": { "quadrature_nodes": 64, "raw_trace": 1.000000000000003, "root_fidelity": 0.9999988099429362, "fidelity_remainder_nats": 0.0000023801155438017193, "trace_norm_error": 0.0027828383891175487, "minimum_eigenvalue": 0.000025497910419547155, "diagnostics": { "quadrature_nodes": 64, "imaginary_residual_max_abs": 7.115076756936123e-20, "hermiticity_residual_max_abs": 1.3877787807814457e-17, "maximum_component_hermiticity_residual": 8.326672684688674e-17 } }, "edge_only_petz": { "raw_trace": 1.0000000000000007, "root_fidelity": 0.9974519101518537, "fidelity_remainder_nats": 0.005102683508714844, "trace_norm_error": 0.1322355160192941, "minimum_eigenvalue": 0.00003672077005443149 }, "interpretation": { "ordinary_petz": "Diagnostic performance only; the Fawzi-Renner lower bound is not asserted for bare Petz in general.", "rotated": "Equal mixture of the t=+1 and t=-1 theorem rotations; this symmetric two-point channel is not the beta0 average.", "universal": "The beta0-twirled map depends only on rho_BC and the partial-trace channel and is the theorem-backed map tested here.", "edge_only": "A valid but deliberately spatially restricted recovery channel; failure to reach the full-B guarantee does not contradict the theorem." } }, "twirl_convergence": { "rows": [ { "quadrature_nodes": 16, "raw_trace": 1.000000000000003, "root_fidelity": 0.9999987498147331, "fidelity_remainder_nats": 0.000002500372096695704, "trace_norm_error": 0.002861201240507016, "minimum_eigenvalue": 0.000025481028603140746, "diagnostics": { "quadrature_nodes": 16, "imaginary_residual_max_abs": 1.3552527156068805e-19, "hermiticity_residual_max_abs": 1.3877787807814457e-17, "maximum_component_hermiticity_residual": 5.551115123125783e-17 } }, { "quadrature_nodes": 32, "raw_trace": 1.000000000000003, "root_fidelity": 0.9999988438394456, "fidelity_remainder_nats": 0.00000231232244551878, "trace_norm_error": 0.002749536984390129, "minimum_eigenvalue": 0.00002549009668658953, "diagnostics": { "quadrature_nodes": 32, "imaginary_residual_max_abs": 6.776263578034403e-20, "hermiticity_residual_max_abs": 6.938893903907228e-18, "maximum_component_hermiticity_residual": 1.3877787807814457e-16 } }, { "quadrature_nodes": 64, "raw_trace": 1.000000000000003, "root_fidelity": 0.9999988099429362, "fidelity_remainder_nats": 0.0000023801155438017193, "trace_norm_error": 0.0027828383891175487, "minimum_eigenvalue": 0.000025497910419547155, "diagnostics": { "quadrature_nodes": 64, "imaginary_residual_max_abs": 7.115076756936123e-20, "hermiticity_residual_max_abs": 1.3877787807814457e-17, "maximum_component_hermiticity_residual": 8.326672684688674e-17 } } ], "root_fidelity_envelope": 9.402471246033883e-8, "uncertainty_kind": "deterministic midpoint-quadrature convergence envelope, not a statistical error bar or rigorous all-orders bound" }, "conditioning": { "inverse_temperature_sweep": [ { "inverse_temperature": 0.5, "rho_b_minimum_eigenvalue": 0.04293366439094895, "rho_b_condition_number": 6.190309550280819, "inverse_square_root_norm": 4.826152283538204, "conditional_mutual_information_nats": 3.0412949314495563e-7, "petz_trace_residual": 8.881784197001252e-16 }, { "inverse_temperature": 1, "rho_b_minimum_eigenvalue": 0.01286879647106994, "rho_b_condition_number": 32.20480963983188, "inverse_square_root_norm": 8.815176937947912, "conditional_mutual_information_nats": 0.000045261950429864584, "petz_trace_residual": 1.5543122344752192e-15 }, { "inverse_temperature": 2, "rho_b_minimum_eigenvalue": 0.0017545148650498445, "rho_b_condition_number": 341.18517511149065, "inverse_square_root_norm": 23.873795748102282, "conditional_mutual_information_nats": 0.002153937235070069, "petz_trace_residual": 3.774758283725532e-15 }, { "inverse_temperature": 4, "rho_b_minimum_eigenvalue": 0.0001938823665170574, "rho_b_condition_number": 3565.4601731243206, "inverse_square_root_norm": 71.8175927551049, "conditional_mutual_information_nats": 0.013883984969312646, "petz_trace_residual": 3.552713678800501e-15 }, { "inverse_temperature": 8, "rho_b_minimum_eigenvalue": 0.0000063247339972271685, "rho_b_condition_number": 111384.35860623048, "inverse_square_root_norm": 397.6297476451573, "conditional_mutual_information_nats": 0.022124173722821917, "petz_trace_residual": 1.5765166949677223e-14 } ], "inverse_floor_policy": "For rho_B^(-1/2), replace each eigenvalue lambda by max(lambda, lambda_floor) before inversion; this clamps small positive eigenvalues and does not project them away.", "inverse_floor_stress_test_at_beta_8": [ { "floor_fraction_of_rho_b_max": 0, "absolute_floor": 0, "clamped_rho_b_eigenvalue_count": 0, "raw_trace": 1.0000000000000158, "trace_preservation_defect": 1.5765166949677223e-14, "minimum_output_eigenvalue": 4.8315131100344264e-17 }, { "floor_fraction_of_rho_b_max": 1e-8, "absolute_floor": 7.044764396361685e-9, "clamped_rho_b_eigenvalue_count": 0, "raw_trace": 1.0000000000000158, "trace_preservation_defect": 1.5765166949677223e-14, "minimum_output_eigenvalue": 4.8315131100344264e-17 }, { "floor_fraction_of_rho_b_max": 0.000001, "absolute_floor": 7.044764396361684e-7, "clamped_rho_b_eigenvalue_count": 0, "raw_trace": 1.0000000000000158, "trace_preservation_defect": 1.5765166949677223e-14, "minimum_output_eigenvalue": 4.8315131100344264e-17 }, { "floor_fraction_of_rho_b_max": 0.0001, "absolute_floor": 0.00007044764396361684, "clamped_rho_b_eigenvalue_count": 3, "raw_trace": 0.9999662608978527, "trace_preservation_defect": 0.00003373910214732323, "minimum_output_eigenvalue": -1.827653792192088e-17 } ], "interpretation": "As rho_B approaches rank deficiency, ||rho_B^(-1/2)|| grows. An undocumented eigenvalue clamp changes the map and can destroy trace preservation even though no supported eigenspace is removed." }, "diagnostics": { "hamiltonian_reconstruction_residual_max_abs": 1.2656542480726785e-14, "gaussian_one_body_residual_max_abs": 7.771561172376096e-16, "petz_trace_preservation_residual_max_abs": 7.831235659949698e-14, "petz_reference_recovery_residual_max_abs": 3.6290415117434804e-15, "rotated_reference_recovery_residual_max_abs": 3.651592916931179e-15, "relative_entropy_loss_identity_residual": 1.0325074129013956e-14, "marginal_traces": { "a": 1.0000000000000004, "b": 1.0000000000000004, "c": 1.0000000000000002, "ab": 1.0000000000000004, "bc": 1.0000000000000004, "abc": 1.0000000000000004 } } }, "bosonic_cutoff_channel": { "channel_pair": { "ideal": "identity channel on one bosonic wavepacket mode", "cutoff": "C_N(rho)=P_N rho P_N+Tr(Q_N rho)|0><0|, P_N=sum_{n=0}^N |n>=sqrt(1-delta)|0>+sqrt(delta)|N+1>, giving trace-norm error 2 sqrt(delta)", "upper_bound": "min(2,2 sqrt(delta)+delta), from ||rho-P rho P||_1<=2 sqrt(delta) plus the replacement output of trace delta", "scaling": "Theta(sqrt(E/(N+1))) while E/(N+1) tends to zero" }, "energy_sweep_at_N_31": [ { "energy_budget_E": 0.25, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.0078125, "coherent_two_level_witness_lower_bound": 0.1767766952966369, "gentle_replacement_upper_bound": 0.1845891952966369 }, { "energy_budget_E": 0.5, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.015625, "coherent_two_level_witness_lower_bound": 0.25, "gentle_replacement_upper_bound": 0.265625 }, { "energy_budget_E": 1, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.03125, "coherent_two_level_witness_lower_bound": 0.3535533905932738, "gentle_replacement_upper_bound": 0.3848033905932738 }, { "energy_budget_E": 2, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.0625, "coherent_two_level_witness_lower_bound": 0.5, "gentle_replacement_upper_bound": 0.5625 }, { "energy_budget_E": 4, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.125, "coherent_two_level_witness_lower_bound": 0.7071067811865476, "gentle_replacement_upper_bound": 0.8321067811865476 }, { "energy_budget_E": 8, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.25, "coherent_two_level_witness_lower_bound": 1, "gentle_replacement_upper_bound": 1.25 }, { "energy_budget_E": 16, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.5, "coherent_two_level_witness_lower_bound": 1.4142135623730951, "gentle_replacement_upper_bound": 1.9142135623730951 }, { "energy_budget_E": 32, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 1, "coherent_two_level_witness_lower_bound": 2, "gentle_replacement_upper_bound": 2 } ], "cutoff_sweep_at_E_2": [ { "energy_budget_E": 2, "retained_maximum_occupation_N": 3, "tail_probability_bound_delta": 0.5, "coherent_two_level_witness_lower_bound": 1.4142135623730951, "gentle_replacement_upper_bound": 1.9142135623730951 }, { "energy_budget_E": 2, "retained_maximum_occupation_N": 7, "tail_probability_bound_delta": 0.25, "coherent_two_level_witness_lower_bound": 1, "gentle_replacement_upper_bound": 1.25 }, { "energy_budget_E": 2, "retained_maximum_occupation_N": 15, "tail_probability_bound_delta": 0.125, "coherent_two_level_witness_lower_bound": 0.7071067811865476, "gentle_replacement_upper_bound": 0.8321067811865476 }, { "energy_budget_E": 2, "retained_maximum_occupation_N": 31, "tail_probability_bound_delta": 0.0625, "coherent_two_level_witness_lower_bound": 0.5, "gentle_replacement_upper_bound": 0.5625 }, { "energy_budget_E": 2, "retained_maximum_occupation_N": 63, "tail_probability_bound_delta": 0.03125, "coherent_two_level_witness_lower_bound": 0.3535533905932738, "gentle_replacement_upper_bound": 0.3848033905932738 }, { "energy_budget_E": 2, "retained_maximum_occupation_N": 127, "tail_probability_bound_delta": 0.015625, "coherent_two_level_witness_lower_bound": 0.25, "gentle_replacement_upper_bound": 0.265625 } ], "adversaries": { "cutoff_dependent_hamiltonian": { "hamiltonian": "H_N=n/(N+1)", "cutoff_N": 63, "energy_budget_E": 1, "witness_state": "|N+1>", "witness_energy": 1, "actual_trace_norm_error": 2, "invalid_fixed_hamiltonian_upper_bound_if_reused": 0.265625 }, "uncounted_energetic_input": { "statement": "An arbitrary reference may be unbounded because the channel does not act on it; an additional channel input mode is not a reference and must be included in the constraint Hamiltonian.", "failure_case": "If the channel pair differs on an uncounted input mode, a high-energy orthogonal output witness can restore trace-norm error 2 at every reported E.", "strongest_surviving_claim": "The certificate applies only to the declared acted-on input algebra with the fixed Hamiltonian H=n; it is not an unconstrained diamond-norm or capacity bound." } } }, "verification": { "passed_checks": 301, "all_checks_passed": true, "methods": [ "cyclic Jacobi eigendecomposition with reconstruction and orthogonality diagnostics", "independent many-body versus Fermi-Dirac one-body Gaussian correlation identity", "exact partial traces plus an independently evaluated relative-entropy-loss identity for conditional mutual information", "basis-level trace-preservation test for the finite-dimensional Petz map", "beta0 quantile midpoint quadrature at 16, 32, and 64 nodes", "analytic Fock-tail lower and upper certificates with monotonicity and adversarial Hamiltonian checks" ] } }