{
  "slug": "2026-09-09-isolated-quantum-thermalization",
  "two_qubit_example": {
    "Hamiltonian": "|01><10|+|10><01|",
    "initial": "|10>",
    "purity_minimum": 0.5,
    "recurrence_time_pi": true,
    "purpose": "An exact example of entanglement and local mixedness without equilibration."
  },
  "bose_hubbard": {
    "sites": 6,
    "particles": 6,
    "dimension": 462,
    "J": 1.0,
    "U": 1.5606060606060606,
    "initial_state": [
      1,
      1,
      1,
      1,
      1,
      1
    ],
    "initial_energy_J": 0.0,
    "initial_energy_variance_J2": 20.000000000000004,
    "beta_times_J": 0.26503061661206995,
    "gibbs_global_purity": 0.008100126744076128,
    "gibbs_energy_variance_J2": 18.33456728248445,
    "minimum_level_spacing_J": 3.894129463333229e-11,
    "near_degenerate_energy_blocks_for_sensitivity_only": [
      {
        "energy_J": 24.195164733423013,
        "multiplicity": 2,
        "initial_weight": 1.7414379583897525e-18
      }
    ],
    "spectral_residual_max": 6.493826989992347e-14,
    "time_average_definition": "Diagonal ensemble after resolving all nonzero spacings against eigenpair residuals; sensitivity to merging the closest pair checked separately, not assumed exactly degenerate.",
    "grouped_vs_resolved_time_average_difference": 1.0031201343273213e-16,
    "trace_distance_diagonal_to_gibbs_single": 0.017208809560732775,
    "trace_distance_diagonal_to_gibbs_two": 0.06579502145247454,
    "mean_instantaneous_two_site_distance_to_gibbs_10_20": 0.18139192046928596,
    "distance_of_time_averaged_two_site_state_to_gibbs_10_20": 0.08905849342386932,
    "checks": {
      "norm_error": 2.4424906541753444e-15,
      "energy_error": 1.2760637694807095e-14,
      "variance_error": 9.947598300641403e-14,
      "matrix_exponential_difference": 5.926904719271459e-14,
      "reversal_error": 1.0313163007509863e-13,
      "partial_trace_cross_check": 2.776077874333006e-17
    },
    "limits": "Ideal finite chain with exact product initial state; no experiment-specific ramp or measurement correction. Canonical reference is reduced from the fixed-N full chain. Finite-time averages are not the exact diagonal ensemble; neither an ETH proof nor a thermodynamic-limit extrapolation."
  }
}