{
  "schema_version": "1.0.0",
  "generated_on": "2026-09-01",
  "generated_by": "scripts/generate-relative-entropy-channel-benchmarks.mjs",
  "deterministic": true,
  "entropy_units": "nats",
  "scope": {
    "model": "one bosonic normal mode of a finite-volume, UV-regulated free scalar field",
    "regulator": "finite set of spatial normal modes; no photon-number truncation enters the analytic results",
    "comparison_rule": "the identical channel acts on both members of each state pair"
  },
  "formulas": {
    "umegaki_relative_entropy": "D(rho||sigma) is the extended Umegaki functional when supp(rho)<=supp(sigma) and may still be +Infinity in infinite dimensions; support failure gives +Infinity",
    "displaced_thermal_pair": "D(D(alpha)tau_nD(alpha)^dagger||tau_n)=|alpha|^2 ln((n+1)/n), n>0",
    "classical_relative_entropy": "d(r||s)=sum_i r_i ln(r_i/s_i)"
  },
  "additive_noise_benchmark": {
    "parameters": {
      "thermal_mean_occupation_n": 1,
      "additive_noise_mean_occupation_nu": 1,
      "displacement_magnitude_squared": 2
    },
    "state_pair": {
      "reference": "tau_n",
      "excitation": "D(alpha) tau_n D(alpha)^dagger",
      "thermal_spectrum": "(n+1)^(-1) [n/(n+1)]^k for k=0,1,..."
    },
    "channel": {
      "definition": "N_nu(X)=integral d^2z/(pi nu) exp(-|z|^2/nu) D(z) X D(z)^dagger",
      "picture": "Schrodinger",
      "normal": true,
      "completely_positive": true,
      "trace_preserving": true,
      "displacement_covariant": true,
      "covariance_convention": "[q,p]=i; vacuum covariance=(1/2) I_2",
      "covariance_map": "V -> V + nu I_2",
      "mean_map": "d -> d"
    },
    "exact_expressions": {
      "input_relative_entropy": "2 ln 2",
      "output_relative_entropy": "2 ln(3/2)",
      "contraction_gap": "2 ln(4/3)"
    },
    "values_nats": {
      "input_relative_entropy": 1.3862943611198906,
      "output_relative_entropy": 0.8109302162163288,
      "contraction_gap": 0.5753641449035618
    },
    "equality_scope": "For finite n, equality occurs only when nu=0 or |alpha|^2=0.",
    "residuals": {
      "input_vs_2_ln_2": 0,
      "output_vs_2_ln_3_over_2": 0,
      "gap_vs_2_ln_4_over_3": 1.1102230246251565e-16,
      "covariance_update": 0
    }
  },
  "postselection_benchmark": {
    "parameters": {
      "epsilon": 0.1,
      "p": 0.75,
      "q": 0.25
    },
    "input_states_diagonal": {
      "rho": [
        0.075,
        0.025,
        0.9
      ],
      "sigma": [
        0.025,
        0.075,
        0.9
      ]
    },
    "projective_instrument": {
      "success_projector_diagonal": [
        1,
        1,
        0
      ],
      "failure_projector_diagonal": [
        0,
        0,
        1
      ],
      "complete": true,
      "success_probability_rho": 0.1,
      "success_probability_sigma": 0.1
    },
    "exact_expressions": {
      "input_relative_entropy": "ln(3)/20",
      "normalized_success_relative_entropy": "ln(3)/2",
      "apparent_amplification_factor": "10",
      "full_flagged_relative_entropy": "ln(3)/20"
    },
    "values_nats": {
      "input_relative_entropy": 0.054930614433405495,
      "normalized_success_relative_entropy": 0.549306144334055,
      "apparent_amplification_factor": 10
    },
    "full_flagged_instrument": {
      "flattened_outcome_labels": [
        "success:0",
        "success:1",
        "failure:2"
      ],
      "rho_distribution": [
        0.075,
        0.025,
        0.9
      ],
      "sigma_distribution": [
        0.025,
        0.075,
        0.9
      ],
      "outcome_relative_entropy": 0,
      "success_conditional_relative_entropy": 0.549306144334055,
      "failure_conditional_relative_entropy": 0,
      "probability_weighted_success_term": 0.0549306144334055,
      "probability_weighted_failure_term": 0,
      "total_relative_entropy": 0.0549306144334055,
      "data_processing_holds": true
    },
    "normalized_branch": {
      "rho_distribution": [
        0.75,
        0.25
      ],
      "sigma_distribution": [
        0.25,
        0.75
      ],
      "map": "X -> P X P / Tr(P X)",
      "broken_hypothesis": "state-dependent normalization makes the branch map nonlinear and not trace preserving",
      "nonlinearity_witness": {
        "first_state": [
          0.8,
          0.1,
          0.1
        ],
        "second_state": [
          0.1,
          0.1,
          0.8
        ],
        "equal_mixture_after_normalization": [
          0.8181818181818181,
          0.18181818181818182
        ],
        "equal_mixture_of_normalized_outputs": [
          0.6944444444444444,
          0.3055555555555556
        ],
        "l1_difference": 0.24747474747474746
      }
    },
    "residuals": {
      "rho_normalization": 0,
      "sigma_normalization": 0,
      "equal_success_probabilities": 0,
      "input_vs_ln_3_over_20": 1.3877787807814457e-17,
      "success_vs_ln_3_over_2": 2.220446049250313e-16,
      "flagged_chain_rule": 6.938893903907228e-18,
      "flattened_flagged_distribution": 0
    }
  },
  "positivity_failures": {
    "nonpositive_trace_preserving_map": {
      "definition": "Lambda(X)=2 X - Tr(X) I_2/2",
      "linear": true,
      "trace_preserving": true,
      "positive": false,
      "input_density_matrix": [
        [
          1,
          0
        ],
        [
          0,
          0
        ]
      ],
      "output_matrix": [
        [
          1.5,
          0
        ],
        [
          0,
          -0.5
        ]
      ],
      "output_eigenvalues": [
        -0.5,
        1.5
      ],
      "consequence": "The output is not a state, so its quantum relative entropy is undefined."
    },
    "positive_not_completely_positive_ancilla_check": {
      "map": "identity on the first qubit, tensored with matrix transpose on the second",
      "single_system_map_positive": true,
      "single_system_map_trace_preserving": true,
      "completely_positive": false,
      "bell_density_matrix": [
        [
          0.5,
          0,
          0,
          0.5
        ],
        [
          0,
          0,
          0,
          0
        ],
        [
          0,
          0,
          0,
          0
        ],
        [
          0.5,
          0,
          0,
          0.5
        ]
      ],
      "partial_transpose_matrix": [
        [
          0.5,
          0,
          0,
          0
        ],
        [
          0,
          0,
          0.5,
          0
        ],
        [
          0,
          0.5,
          0,
          0
        ],
        [
          0,
          0,
          0,
          0.5
        ]
      ],
      "partial_transpose_eigenvalues": [
        -0.5,
        0.5,
        0.5,
        0.5
      ],
      "consequence": "The negative eigenvalue exposes failure of positivity after adjoining an ancilla."
    }
  },
  "support_limit": {
    "fixed_parameters": {
      "additive_noise_mean_occupation_nu": 1,
      "displacement_magnitude_squared": 2
    },
    "interpretation": "As n tends to zero from above, the input reference loses faithfulness and the input relative entropy diverges; additive noise with nu=1 makes the output reference faithful.",
    "rows": [
      {
        "thermal_mean_occupation_n": 1,
        "reference_faithful": true,
        "input_support_condition": "satisfied",
        "input_relative_entropy_nats": 1.3862943611198906,
        "input_extended_value": "finite",
        "output_thermal_mean_occupation": 2,
        "output_reference_faithful": true,
        "output_relative_entropy_nats": 0.8109302162163288,
        "contraction_gap_nats": 0.5753641449035618
      },
      {
        "thermal_mean_occupation_n": 0.1,
        "reference_faithful": true,
        "input_support_condition": "satisfied",
        "input_relative_entropy_nats": 4.795790545596741,
        "input_extended_value": "finite",
        "output_thermal_mean_occupation": 1.1,
        "output_reference_faithful": true,
        "output_relative_entropy_nats": 1.2932543298501047,
        "contraction_gap_nats": 3.5025362157466366
      },
      {
        "thermal_mean_occupation_n": 0.01,
        "reference_faithful": true,
        "input_support_condition": "satisfied",
        "input_relative_entropy_nats": 9.23024103368252,
        "input_extended_value": "finite",
        "output_thermal_mean_occupation": 1.01,
        "output_reference_faithful": true,
        "output_relative_entropy_nats": 1.3763687824356323,
        "contraction_gap_nats": 7.853872251246887
      },
      {
        "thermal_mean_occupation_n": 0.001,
        "reference_faithful": true,
        "input_support_condition": "satisfied",
        "input_relative_entropy_nats": 13.81750955863044,
        "input_extended_value": "finite",
        "output_thermal_mean_occupation": 1.001,
        "output_reference_faithful": true,
        "output_relative_entropy_nats": 1.3852951105370257,
        "contraction_gap_nats": 12.432214448093415
      },
      {
        "thermal_mean_occupation_n": 0.0001,
        "reference_faithful": true,
        "input_support_condition": "satisfied",
        "input_relative_entropy_nats": 18.420880733953034,
        "input_extended_value": "finite",
        "output_thermal_mean_occupation": 1.0001,
        "output_reference_faithful": true,
        "output_relative_entropy_nats": 1.386194368619307,
        "contraction_gap_nats": 17.034686365333727
      },
      {
        "thermal_mean_occupation_n": 0,
        "reference_faithful": false,
        "input_support_condition": "fails for nonzero displacement",
        "input_relative_entropy_nats": null,
        "input_extended_value": "+Infinity",
        "output_thermal_mean_occupation": 1,
        "output_reference_faithful": true,
        "output_relative_entropy_nats": 1.3862943611198906,
        "output_exact_expression": "2 ln 2"
      }
    ]
  },
  "sources": [
    {
      "citation": "Caruso, Eisert, Giovannetti, and Holevo, New Journal of Physics 10 (2008) 083030",
      "doi": "10.1088/1367-2630/10/8/083030",
      "locator": "Eqs. (16)-(17), pp. 5-6; additive classical noise, Section 2.4, p. 18"
    },
    {
      "citation": "K. R. Parthasarathy, A Pedagogical Note on the Computation of Relative Entropy of Two n-Mode Gaussian States, arXiv:2102.06708",
      "url": "https://arxiv.org/abs/2102.06708",
      "locator": "Weyl and covariance conventions, Eqs. (2.1) and (2.9)-(2.18), pp. 2-3; thermal state Eq. (4.1), p. 5; Lemma 3 and Eq. (5.8), p. 6"
    }
  ],
  "self_check": {
    "tolerance": 2e-14,
    "count": 86,
    "all_passed": true,
    "checks": [
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "additive noise is nonnegative",
        "passed": true
      },
      {
        "label": "additive-noise input benchmark",
        "passed": true
      },
      {
        "label": "additive-noise output benchmark",
        "passed": true
      },
      {
        "label": "additive-noise contraction gap",
        "passed": true
      },
      {
        "label": "additive-noise covariance update",
        "passed": true
      },
      {
        "label": "additive-noise data processing holds",
        "passed": true
      },
      {
        "label": "nontrivial additive noise gives strict contraction",
        "passed": true
      },
      {
        "label": "relative-entropy distributions have matching lengths",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy distributions have matching lengths",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy distributions have matching lengths",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy distributions have matching lengths",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "postselection rho normalization",
        "passed": true
      },
      {
        "label": "postselection sigma normalization",
        "passed": true
      },
      {
        "label": "postselection equal success probabilities",
        "passed": true
      },
      {
        "label": "postselection input relative entropy",
        "passed": true
      },
      {
        "label": "postselection normalized-success relative entropy",
        "passed": true
      },
      {
        "label": "full flagged instrument chain rule",
        "passed": true
      },
      {
        "label": "relative-entropy distributions have matching lengths",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "relative-entropy inputs are nonnegative",
        "passed": true
      },
      {
        "label": "flattened flagged output relative entropy",
        "passed": true
      },
      {
        "label": "postselection amplification factor",
        "passed": true
      },
      {
        "label": "normalized postselection has a nonlinearity witness",
        "passed": true
      },
      {
        "label": "Jacobi eigensolver input is square",
        "passed": true
      },
      {
        "label": "Jacobi eigensolver input is symmetric",
        "passed": true
      },
      {
        "label": "Jacobi eigensolver converged",
        "passed": true
      },
      {
        "label": "nonpositive map preserves trace",
        "passed": true
      },
      {
        "label": "nonpositive map output spectrum: matching lengths",
        "passed": true
      },
      {
        "label": "nonpositive map output spectrum: entry 0",
        "passed": true
      },
      {
        "label": "nonpositive map output spectrum: entry 1",
        "passed": true
      },
      {
        "label": "nonpositive map has a negative output eigenvalue",
        "passed": true
      },
      {
        "label": "partial-transpose matrix has bipartite size",
        "passed": true
      },
      {
        "label": "Jacobi eigensolver input is square",
        "passed": true
      },
      {
        "label": "Jacobi eigensolver input is symmetric",
        "passed": true
      },
      {
        "label": "Jacobi eigensolver converged",
        "passed": true
      },
      {
        "label": "partial transpose preserves trace",
        "passed": true
      },
      {
        "label": "Bell-state partial-transpose spectrum: matching lengths",
        "passed": true
      },
      {
        "label": "Bell-state partial-transpose spectrum: entry 0",
        "passed": true
      },
      {
        "label": "Bell-state partial-transpose spectrum: entry 1",
        "passed": true
      },
      {
        "label": "Bell-state partial-transpose spectrum: entry 2",
        "passed": true
      },
      {
        "label": "Bell-state partial-transpose spectrum: entry 3",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "support-limit DPI at n=1",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "support-limit DPI at n=0.1",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "support-limit DPI at n=0.01",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "support-limit DPI at n=0.001",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "support-limit DPI at n=0.0001",
        "passed": true
      },
      {
        "label": "thermal reference is faithful for finite formula",
        "passed": true
      },
      {
        "label": "squared displacement is nonnegative",
        "passed": true
      },
      {
        "label": "pure-reference noisy output limit",
        "passed": true
      },
      {
        "label": "support-limit input divergence is monotone at row 1",
        "passed": true
      },
      {
        "label": "support-limit input divergence is monotone at row 2",
        "passed": true
      },
      {
        "label": "support-limit input divergence is monotone at row 3",
        "passed": true
      },
      {
        "label": "support-limit input divergence is monotone at row 4",
        "passed": true
      }
    ]
  }
}