{ "schema_version": "qft.relative-entropy-correlation-benchmarks.v1", "generated_on": "2026-09-01", "generated_by": "scripts/generate-relative-entropy-correlation-benchmarks.mjs", "deterministic_generation": { "random_seed": null, "stochastic_steps": false, "arithmetic": "IEEE-754 binary64 JavaScript Number arithmetic", "eigensolver": "deterministic cyclic Jacobi rotations for real symmetric matrices", "output_order": "fixed parameter order encoded in the generator", "generator_source_sha256": "fdd99b083dccbbda42d3b24df0c212c7df993cfcd409eb02690112d916c1237e" }, "conventions": { "logarithms": "natural logarithms; all entropies and information quantities are in nats", "lattice_endpoints": "sites are cell centers and every physical interval endpoint is a cell edge", "uncertainty_policy": "roundoff diagnostics are reported separately from cutoff and finite-volume variation" }, "mutual_information_free_massive_scalar": { "question": "Does one regulated free-scalar state give finite two-interval mutual information and obey the bounded-Weyl correlation lower bound?", "geometry": "two equal intervals of physical length R separated by distance L on a periodic circle of circumference L_box", "regulator": { "model": "periodic massive real-scalar harmonic chain with mR=1", "hamiltonian": "H=(1/2) sum_n [p_n^2+m^2 q_n^2+(q_(n+1)-q_n)^2/a^2]", "entropy_method": "restrict X and P, diagonalize X_R^(1/2) P_R X_R^(1/2), and sum the bosonic entropy function of its symplectic eigenvalues", "smeared_coordinates": "Q_A=N_A^(-1/2) sum_(j in A) q_j and Q_B=N_A^(-1/2) sum_(j in B) q_j", "bounded_observables": "X_t=exp(i t Q_A) and Y_t=exp(-i t Q_B) are commuting norm-one Weyl unitaries on the two lattice regions", "gaussian_connected_correlation": "C_W(t)=exp[-t^2(V-C)]-exp[-t^2 V], with V== and C=", "optimized_parameter": "t_*^2=C^(-1) log[V/(V-C)] for 0= |C_W(t)|^2/2 for natural logarithms" }, "fixed_ratio_refinement": { "fixed_parameters": "mR=1, L/R=1, L_box/R=16", "rows": [ { 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different cutoffs are mixed?", "model": { "hamiltonian": "H=(1/2) sum_n [p_n^2+m^2 q_n^2+(q_(n+1)-q_n)^2/a^2], periodic q_N=q_0", "commutator": "[q_n,p_m]=i delta_nm", "frequency": "omega_k=sqrt[m^2+4 sin^2(pi k/N)/a^2]", "covariance": "X_ij=(1/(2N)) sum_k cos[2 pi k(i-j)/N]/omega_k and P_ij=(1/(2N)) sum_k omega_k cos[2 pi k(i-j)/N]", "entropy_method": "nu_j=sqrt(lambda_j(X_R^(1/2) P_R X_R^(1/2))); S(R)=sum_j[(nu_j+1/2)log(nu_j+1/2)-(nu_j-1/2)log(nu_j-1/2)]" }, "common_regulator_refinement": { "fixed_parameters": { "circumference": 32, "mass": 1, "lengths": [ 2, 1, 2 ] }, "rows": [ { "site_count": 256, "circumference": 32, "lattice_spacing": 0.125, "mass": 1, "physical_lengths_A_B_C": [ 2, 1, 2 ], "site_lengths_A_B_C": [ 16, 8, 16 ], "entropy_AB_nats": 0.6968832387916177, "entropy_BC_nats": 0.6968832387916177, "entropy_B_nats": 0.6820980581897209, "entropy_ABC_nats": 0.6970379906774752, "conditional_mutual_information_nats": 0.014630428716039368, "diagnostics": { 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The negative value diagnoses an invalid cross-regulator subtraction, not a violation of SSA." } }, "conditional_mutual_information_cft": { "question": "How does contiguous-interval CMI compare between the vacuum and a thermal state, and how do common-cutoff cancellation and buffer width control the answer?", "scope": "unitary nonchiral 1+1-dimensional CFT on the infinite line at equal time, velocity one", "interval_lengths": "A, B, C are spatially ordered contiguous intervals of positive lengths a, b, c", "vacuum_formula": "I_0=(c_CFT/3) log[((a+b)(b+c))/(b(a+b+c))]", "thermal_formula": "I_beta=(c_CFT/3) log[sinh(pi(a+b)/beta)sinh(pi(b+c)/beta)/(sinh(pi b/beta)sinh(pi(a+b+c)/beta)]", "positivity_identity": "sinh(x+y)sinh(y+z)-sinh(y)sinh(x+y+z)=sinh(x)sinh(z)", "cutoff_cancellation": "Using one epsilon and one endpoint prescription cancels every (c_CFT/3)log(1/epsilon) term and the additive interval constant.", "mismatched_cutoff_shift": "delta I=(c_CFT/3) log[(epsilon_B epsilon_ABC)/(epsilon_AB epsilon_BC)]", "rows": [ { "central_charge": 1, "lengths_A_B_C": [ 1, 0.5, 1 ], "inverse_temperature_beta": null, "state": "vacuum", "conditional_mutual_information_nats": 0.19592888830070634, "vacuum_value_nats": 0.19592888830070634 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 0.5, 1 ], "inverse_temperature_beta": 8, "state": "thermal", "conditional_mutual_information_nats": 0.17998266881834618, "vacuum_value_nats": 0.19592888830070634 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 0.5, 1 ], "inverse_temperature_beta": 4, "state": "thermal", "conditional_mutual_information_nats": 0.14314560891256325, "vacuum_value_nats": 0.19592888830070634 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 0.5, 1 ], "inverse_temperature_beta": 2, "state": "thermal", "conditional_mutual_information_nats": 0.07179411970016576, "vacuum_value_nats": 0.19592888830070634 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 1, 1 ], "inverse_temperature_beta": null, "state": "vacuum", "conditional_mutual_information_nats": 0.09589402415059362, "vacuum_value_nats": 0.09589402415059362 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 1, 1 ], "inverse_temperature_beta": 8, "state": "thermal", "conditional_mutual_information_nats": 0.08072870827583241, "vacuum_value_nats": 0.09589402415059362 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 1, 1 ], "inverse_temperature_beta": 4, "state": "thermal", "conditional_mutual_information_nats": 0.051238282529665335, "vacuum_value_nats": 0.09589402415059362 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 1, 1 ], "inverse_temperature_beta": 2, "state": "thermal", "conditional_mutual_information_nats": 0.013505922730806455, "vacuum_value_nats": 0.09589402415059362 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 2, 1 ], "inverse_temperature_beta": null, "state": "vacuum", "conditional_mutual_information_nats": 0.03926101188546115, "vacuum_value_nats": 0.03926101188546115 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 2, 1 ], "inverse_temperature_beta": 8, "state": "thermal", "conditional_mutual_information_nats": 0.02602075088789335, "vacuum_value_nats": 0.03926101188546115 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 2, 1 ], "inverse_temperature_beta": 4, "state": "thermal", "conditional_mutual_information_nats": 0.00933228767029733, "vacuum_value_nats": 0.03926101188546115 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 2, 1 ], "inverse_temperature_beta": 2, "state": "thermal", "conditional_mutual_information_nats": 0.0005704234590894567, "vacuum_value_nats": 0.03926101188546115 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 4, 1 ], "inverse_temperature_beta": null, "state": "vacuum", "conditional_mutual_information_nats": 0.013607331506751733, "vacuum_value_nats": 0.013607331506751733 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 4, 1 ], "inverse_temperature_beta": 8, "state": "thermal", "conditional_mutual_information_nats": 0.004466733919883688, "vacuum_value_nats": 0.013607331506751733 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 4, 1 ], "inverse_temperature_beta": 4, "state": "thermal", "conditional_mutual_information_nats": 0.0003911114006974377, "vacuum_value_nats": 0.013607331506751733 }, { "central_charge": 1, "lengths_A_B_C": [ 1, 4, 1 ], "inverse_temperature_beta": 2, "state": "thermal", "conditional_mutual_information_nats": 0.0000010641524568200111, "vacuum_value_nats": 0.013607331506751733 } ], "checks": { "beta_10000000_at_lengths_1_1_1_nats": 0.09589402415058264, "vacuum_at_lengths_1_1_1_nats": 0.09589402415059362, "absolute_large_beta_difference_nats": 1.0977330155981235e-14 } }, "source_roles": [ { "role": "free-scalar covariance, entropy, mutual-information, and harmonic-chain method", "citation": "H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007", "doi": "https://doi.org/10.1088/1751-8113/42/50/504007", "locators": "Section 2.2.1, Eqs. (47), (63), and (65)-(66), and Section 3.1.8, Eqs. (197)-(199)" }, { "role": "bounded-observable mutual-information lower bound", "citation": "M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100 (2008) 070502", "doi": "https://doi.org/10.1103/PhysRevLett.100.070502", "locators": "Eq. (5) for the natural-log Pinsker correlation lower bound" }, { "role": "single-interval vacuum and thermal CFT entropy", "citation": "P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002", "doi": "https://doi.org/10.1088/1742-5468/2004/06/P06002", "locators": "Eq. (16) and its n-to-1 derivative for the vacuum interval; Eq. (18) for the thermal interval" }, { "role": "strong subadditivity theorem", "citation": "E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14 (1973) 1938-1941", "doi": "https://doi.org/10.1063/1.1666274", "locators": "Theorem 2 and Eqs. (1.5)-(1.6) in the expanded 1973 account" } ], "verification_summary": { "checks_passed": 392, "check_labels": [ "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys 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matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar gap has an integer site count", "free-scalar intervals do not wrap around the periodic box", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "free-scalar smeared position covariance lies strictly between zero and the variance", "free-scalar mutual information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "free-scalar box has an integer site count", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive 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information is nonnegative", "free-scalar mutual information obeys the optimized Weyl Pinsker bound", "linear continuum fit has nonzero cutoff variance", "linear continuum fit has nonzero cutoff variance", "linear continuum fit has nonzero cutoff variance", "free-scalar finest mutual-information benchmark agrees with the independent record", "free-scalar optimized Weyl bound agrees with the independent calculation", "harmonic-chain physical endpoints lie on cell edges", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver 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definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "translated equal-length 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symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "translated equal-length harmonic-chain blocks have equal entropy", "common-regulator harmonic-chain conditional mutual information is nonnegative", "harmonic-chain physical endpoints lie on cell edges", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "translated equal-length harmonic-chain blocks have equal entropy", "common-regulator harmonic-chain conditional mutual information is nonnegative", "harmonic-chain physical endpoints lie on cell edges", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "translated equal-length harmonic-chain blocks have equal entropy", "common-regulator harmonic-chain conditional mutual information is nonnegative", "harmonic-chain physical endpoints lie on cell edges", "harmonic-chain global covariance obeys XP=I/4", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted position covariance is positive definite", "Jacobi input is a nonempty square matrix", "Jacobi eigensolver converged", "restricted symplectic eigenvalues obey nu >= 1/2 up to roundoff", "translated equal-length harmonic-chain blocks have equal entropy", "common-regulator harmonic-chain conditional mutual information is nonnegative", "mixed-cutoff forward combination is spuriously negative", "mixed-cutoff reverse combination is spuriously positive", "thermal CFT formula approaches the vacuum formula", "CFT conditional mutual information rows are nonnegative" ] } }