{ "assertions": [ { "label": "generator source uses canonical LF with no carriage returns", "observed": "0", "status": "PASS" }, { "label": "independent checker source uses canonical LF with no carriage returns", "observed": "0", "status": "PASS" }, { "label": "fermion N=128: exactly half the modes are occupied", "observed": 64.0, "status": "PASS" }, { "label": "fermion N=128: chosen particle number has even global parity", "observed": "64", "status": "PASS" }, { "label": "fermion N=256: exactly half the modes are occupied", "observed": 128.0, "status": "PASS" }, { "label": "fermion N=256: chosen particle number has even global parity", "observed": "128", "status": "PASS" }, { "label": "fermion N=512: exactly half the modes are occupied", "observed": 256.0, "status": "PASS" }, { "label": "fermion N=512: chosen particle number has even global parity", "observed": "256", "status": "PASS" }, { "label": "fermion: Fourier and Hamiltonian-eigensolver correlations agree", "observed": 2.5235848015918e-14, "status": "PASS", "tolerance": 2e-11 }, { "label": "fermion: zero-temperature full correlations are projectors", "observed": 9.1783826470427e-13, "status": "PASS", "tolerance": 2e-11 }, { "label": "fermion: full correlations are Hermitian", "observed": 0.0, "status": "PASS", "tolerance": 2e-11 }, { "label": "fermion: APBC leaves a nonzero finite-size one-body gap", "observed": 0.024543538596617, "status": "PASS" }, { "label": "fermion: one-particle dimer has log negativity ln(2)", "observed": 0.0, "status": "PASS", "tolerance": 2e-10 }, { "label": "fermion: product state has zero log negativity", "observed": 0.0, "status": "PASS", "tolerance": 2e-10 }, { "label": "fermion: infinite-temperature Gaussian state has zero log negativity", "observed": 0.0, "status": "PASS", "tolerance": 2e-10 }, { "label": "fermion: translation invariance", "observed": 0.0, "status": "PASS", "tolerance": 5e-07 }, { "label": "fermion: A1/A2 exchange invariance", "observed": 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neither is reported as an m->0 extrapolation.", "measure": "E=sum_j max(0,-ln(2*nu_tilde_j))", "partial_transpose": "Transpose A2 by reversing its momenta only: P_tilde=R P_A R with R=diag(+1 on A1,-1 on A2).", "stable_spectrum": "For X_A=L L^T from Cholesky, nu_tilde_j^2 are eigenvalues of the real symmetric matrix L^T P_tilde L." }, "common": { "cft_abscissa": "x=ln(tan(pi*ell/N))", "expected_coefficient": "c/4=1/4 because each benchmark has c=1", "lattice_spacing": 1.0, "logarithm": "natural", "nonuniversal_constant": "The free-fermion fit includes an intercept. The boson calculation subtracts the ell=N/4 value, where x=0, before a through-origin fit.", "subsystems": "A1={0,...,ell-1}, A2={ell,...,2ell-1}; B is the complement; the intervals do not wrap." }, "fermion": { "Gamma_x_spectrum": "Use every eigenvalue of the generally non-Hermitian Gamma_x, verify its imaginary part is numerical zero, then use the real value; do not Hermitianize.", "algebra_center": "Fermionic mode bipartition in the parity-even CAR algebra; no Jordan-Wigner qubit transpose, local-parity projection, edge-center insertion, or averaging over alternative centers is performed.", "correlation_spectrum": "Use every eigenvalue of Hermitian C_A.", "fock_basis_order": "|n_0...n_(N-1)>=product_(j increasing)(f_j^dagger)^(n_j)|0>; inside A, every A1 mode precedes every A2 mode.", "majorana_order": "For each site j use gamma_(2j)=f_j+f_j^dagger then gamma_(2j+1)=i(f_j-f_j^dagger); order sites increasingly, all A1 first.", "measure": "E=ln Tr sqrt(rho_A^R1 (rho_A^R1)^dagger). 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"qft.quantum-information.negativity-chain-benchmarks.v1", "schema_version": 1, "source_locators": [ { "authors": "Hassan Shapourian, Ken Shiozaki, and Shinsei Ryu", "doi": "https://doi.org/10.1103/PhysRevB.95.165101", "locators": [ "Sec. II.B.1, Eqs. (6)-(10), printed p. 3: coherent-state and occupation-basis rule", "Sec. II.B.2, Eqs. (11)-(15), printed pp. 3-4: Majorana order, even parity, i^p rule, and no order reversal", "Eq. (20), printed p. 4: singular-value trace norm for the generally non-Hermitian transform", "Sec. V.A, Eq. (63), printed p. 11: half-filled anti-periodic hopping-chain realization", "Sec. V.A, Eqs. (81), (83)-(84), and (86)-(87), printed pp. 12-13: adjacent-interval finite-circle and c/4 scaling" ], "open_pdf": "https://arxiv.org/pdf/1611.07536v3", "publication": "Physical Review B 95, 165101 (2017)", "title": "Partial time-reversal transformation and entanglement negativity in fermionic systems" }, { "authors": "Sara Murciano, Riccarda Bonsignori, and Pasquale Calabrese", "doi": "https://doi.org/10.21468/SciPostPhys.10.5.111", "locators": [ "Sec. 2.1, Eqs. (8)-(17), printed pp. 5-7: fermionic partial-time-reversal conventions", "Appendix A, Eqs. (140)-(145), printed pp. 36-37: hopping ring, C, Gamma_pm, Gamma_x, and eigenvalue formula" ], "open_pdf": "https://arxiv.org/pdf/2102.10054", "publication": "SciPost Physics 10, 111 (2021)", "title": "Symmetry decomposition of negativity of massless free fermions" }, { "authors": "Pasquale Calabrese, John Cardy, and Erik Tonni", "doi": "https://doi.org/10.1088/1742-5468/2013/02/P02008", "locators": [ "Sec. 4.2, Eqs. (47)-(51), printed p. 13: adjacent-interval twist-field scaling on the infinite line", "Sec. 4.3, Eqs. (56)-(57), printed p. 14: finite periodic adjacent-interval formula", "Secs. 7 and 7.1, Eqs. (168) and (174)-(180), printed pp. 29-31: periodic harmonic-chain Hamiltonian, modes, covariances, and zero-mode regulator", "Sec. 7.3, Eqs. (186)-(193), printed pp. 32-33: covariance reduction, momentum-sign partial transpose, symplectic spectrum, and trace norm", "Sec. 7.5, Eqs. (194)-(199), printed pp. 35-36: equal adjacent intervals, finite-size formula, N/4 subtraction, and parameter-free c/4 slope" ], "open_pdf": "https://arxiv.org/pdf/1210.5359", "publication": "Journal of Statistical Mechanics: Theory and Experiment P02008 (2013)", "title": "Entanglement negativity in extended systems: a field theoretical approach" }, { "authors": "Koenraad Audenaert, Jens Eisert, Martin B. Plenio, and Reinhard F. Werner", "doi": "https://doi.org/10.1103/PhysRevA.66.042327", "locators": [ "Sec. III, printed pp. 3-4: covariance partial transpose and trace norm" ], "open_pdf": "https://arxiv.org/pdf/quant-ph/0205025", "publication": "Physical Review A 66, 042327 (2002)", "title": "Entanglement properties of the harmonic chain" } ], "status": "PASS", "uncertainty": { "boson_cft_slope": { "central_estimator": "N=512, m=1e-7 centered through-origin slope", "central_slope": 0.24999586750737, "combination_rule": "conservative linear sum of deterministic systematic components", "components": { "finite_size_spread_at_primary_mass": 1.4019359149686e-06, "fit_form_residual_slope_scale": 9.8660240611467e-06, "mass_regulator_shift_at_N512": 3.6273926186797e-05, "symplectic_numerical_route": 7.4440231756512e-11 }, "expected_c_over_4": 0.25, "statistical_uncertainty": 0.0, "total_systematic": 4.7541960603144e-05 }, "fermion_cft_slope": { "central_estimator": "all-window slopes extrapolated linearly in 1/N", "central_infinite_size_slope": 0.24993514102463, "combination_rule": "conservative linear sum of deterministic systematic components", "components": { "correlation_construction_numerical_route": 8.2436138720832e-08, "finite_size_extrapolation_ansatz": 0.00126815476497, "fit_window": 2.0826130440021e-05 }, "expected_c_over_4": 0.25, "note": "The OLS standard errors stored with each finite-N fit diagnose curvature only; they are not sampling uncertainties.", "statistical_uncertainty": 0.0, "total_systematic": 0.0012890633315487 } } }