{
  "schema_version": "1.0.0",
  "generated_on": "2026-09-01",
  "generated_by": "scripts/generate-relative-entropy-operational-benchmarks.mjs",
  "deterministic": true,
  "logarithm_base": "e",
  "information_units": "nats",
  "scope": {
    "model": "one normalized bosonic wavepacket mode of a finite-volume, UV-regulated free field",
    "algebra": "the type-I algebra of that selected mode; continuum claims require a separately controlled limit",
    "copy_resource": "independent preparations are used only in the explicitly n-copy rows",
    "energy": "mean occupation for H=a^dagger a with vacuum energy subtracted"
  },
  "hypothesis_testing": {
    "physical_model": "one detector-selected bosonic wavepacket mode, independently prepared on every run",
    "states": "sigma=tau_N and rho=D(alpha)tau_ND(alpha)^dagger with N=1/2 and real alpha=1",
    "measurement": "homodyne measurement of q=(a+a^dagger)/sqrt(2), a smeared field quadrature of the selected mode",
    "parameters": {
      "thermal_mean_occupation_N": 0.5,
      "displacement_amplitude_alpha": 1,
      "quadrature_variance": 1,
      "rho_quadrature_mean": 1.4142135623730951,
      "sigma_quadrature_mean": 0,
      "type_I_tolerance_epsilon": 0.1,
      "standard_normal_quantile": -1.2815515655446006,
      "rho_energy_in_units_of_mode_frequency_including_vacuum": 2,
      "sigma_energy_in_units_of_mode_frequency_including_vacuum": 1
    },
    "measured_distributions": "p_rho=N(sqrt(2),1) and p_sigma=N(0,1)",
    "sufficient_statistic": "the sample mean of n independent homodyne outcomes",
    "exact_expressions": {
      "measured_relative_entropy": "D_q=(Delta q)^2/(2v)=1 nat",
      "measured_relative_entropy_variance": "V_q=(Delta q)^2/v=2 nat^2",
      "full_quantum_relative_entropy": "D(rho||sigma)=|alpha|^2 ln[(N+1)/N]=ln 3 nats",
      "full_quantum_relative_entropy_variance": "V(rho||sigma)=|alpha|^2(2N+1){ln[(N+1)/N]}^2=2(ln 3)^2 nats^2",
      "homodyne_type_II_error": "beta_n=Phi[-sqrt(2n)-Phi^(-1)(epsilon)]",
      "measured_second_order": "D_(H,q)^epsilon(n)=nD_q+sqrt(nV_q) Phi^(-1)(epsilon)+O(ln n)",
      "detector_blind_control": "if both outcome laws are identical, D_H^epsilon=-ln(1-epsilon) for every n and the per-copy exponent tends to zero"
    },
    "values_nats": {
      "measured_relative_entropy": 1.0000000000000002,
      "measured_relative_entropy_variance": 2.0000000000000004,
      "full_quantum_relative_entropy": 1.0986122886681096,
      "full_quantum_relative_entropy_variance": 2.4138979216251633,
      "detector_blind_hypothesis_testing_divergence": 0.10536051565782628
    },
    "finite_copy_rows": [
      {
        "copies": 1,
        "sample_mean_threshold": 0.13266199682849455,
        "achieved_type_I_error_alpha": 0.10000000000000003,
        "optimal_homodyne_type_II_error_beta": 0.447230349640647,
        "homodyne_hypothesis_testing_divergence_nats": 0.8046814935213004,
        "measured_stein_first_order_nD_nats": 1.0000000000000002,
        "measured_second_order_approximation_nats": -0.8123876048736465,
        "exact_per_copy_nats": 0.8046814935213004,
        "measured_first_order_per_copy_nats": 1.0000000000000002,
        "full_quantum_first_order_per_copy_nats": 1.0986122886681096,
        "exact_minus_measured_second_order_nats": 1.6170690983949467
      },
      {
        "copies": 4,
        "sample_mean_threshold": 0.7734377796007949,
        "achieved_type_I_error_alpha": 0.10000000000000003,
        "optimal_homodyne_type_II_error_beta": 0.0609466278214072,
        "homodyne_hypothesis_testing_divergence_nats": 2.7977567515516975,
        "measured_stein_first_order_nD_nats": 4.000000000000001,
        "measured_second_order_approximation_nats": 0.3752247902527075,
        "exact_per_copy_nats": 0.6994391878879244,
        "measured_first_order_per_copy_nats": 1.0000000000000002,
        "full_quantum_first_order_per_copy_nats": 1.0986122886681096,
        "exact_minus_measured_second_order_nats": 2.42253196129899
      },
      {
        "copies": 16,
        "sample_mean_threshold": 1.093825670986945,
        "achieved_type_I_error_alpha": 0.10000000000000003,
        "optimal_homodyne_type_II_error_beta": 0.000006063204571948977,
        "homodyne_hypothesis_testing_divergence_nats": 12.013272090400028,
        "measured_stein_first_order_nD_nats": 16.000000000000004,
        "measured_second_order_approximation_nats": 8.750449580505418,
        "exact_per_copy_nats": 0.7508295056500017,
        "measured_first_order_per_copy_nats": 1.0000000000000002,
        "full_quantum_first_order_per_copy_nats": 1.0986122886681096,
        "exact_minus_measured_second_order_nats": 3.26282250989461
      },
      {
        "copies": 64,
        "sample_mean_threshold": 1.2540196166800202,
        "achieved_type_I_error_alpha": 0.10000000000000003,
        "optimal_homodyne_type_II_error_beta": 5.504253785589157e-24,
        "homodyne_hypothesis_testing_divergence_nats": 53.5565210229887,
        "measured_stein_first_order_nD_nats": 64.00000000000001,
        "measured_second_order_approximation_nats": 49.50089916101084,
        "exact_per_copy_nats": 0.8368206409841984,
        "measured_first_order_per_copy_nats": 1.0000000000000002,
        "full_quantum_first_order_per_copy_nats": 1.0986122886681096,
        "exact_minus_measured_second_order_nats": 4.055621861977855
      }
    ],
    "scope_warning": "The exact rows optimize threshold tests only within the declared homodyne outcome algebra. The unrestricted quantum Stein rate is ln 3, while the homodyne-restricted rate is 1. The second-order expression is an asymptotic approximation, not a finite-n bound."
  },
  "fidelity_and_chernoff": {
    "physical_model": "one normalized bosonic wavepacket mode with states sigma=tau_N and rho=D(Delta alpha)tau_ND(Delta alpha)^dagger",
    "conventions": {
      "commutator": "[q,p]=i",
      "vacuum_covariance": "(1/2) I_2",
      "displacement_gap": "|Delta alpha|^2=1",
      "fidelity": "F(rho,sigma)=||sqrt(rho)sqrt(sigma)||_1 (unsquared)"
    },
    "parameters": {
      "displacement_gap_magnitude_squared": 1,
      "warm_thermal_mean_occupation": 0.5
    },
    "exact_expressions": {
      "equal_covariance_fidelity": "F(rho,sigma)=exp[-|Delta alpha|^2/{2(2N+1)}]",
      "coherent_squared_fidelity": "F^2=exp(-|Delta alpha|^2)",
      "coherent_chernoff_exponent": "xi_QCB=|Delta alpha|^2",
      "coherent_n_copy_Helstrom_error": "p_err(n)={1-sqrt[1-exp(-n|Delta alpha|^2)]}/2",
      "equal_covariance_chernoff_exponent": "xi_QCB=|Delta alpha|^2(1-sqrt(r))/(1+sqrt(r)), r=N/(N+1), with minimizing s=1/2"
    },
    "zero_temperature_displaced_thermal_limit": {
      "name": "vacuum and a unit-displaced coherent state",
      "state_pair": "vacuum |0> and coherent state |Delta alpha> with |Delta alpha|^2=1",
      "unsquared_fidelity": 0.6065306597126334,
      "squared_fidelity": 0.36787944117144233,
      "chernoff_overlap": 0.36787944117144233,
      "chernoff_exponent_nats": 1,
      "finite_copy_rows": [
        {
          "copies": 1,
          "unsquared_fidelity": 0.6065306597126334,
          "squared_fidelity": 0.36787944117144233,
          "trace_distance_half_norm": 0.7950600976206501,
          "exact_equal_prior_Helstrom_error": 0.10246995118967495,
          "chernoff_exponential_scale": 0.36787944117144233
        },
        {
          "copies": 2,
          "unsquared_fidelity": 0.36787944117144233,
          "squared_fidelity": 0.1353352832366127,
          "trace_distance_half_norm": 0.9298734950321937,
          "exact_equal_prior_Helstrom_error": 0.03506325248390313,
          "chernoff_exponential_scale": 0.1353352832366127
        },
        {
          "copies": 5,
          "unsquared_fidelity": 0.0820849986238988,
          "squared_fidelity": 0.006737946999085468,
          "trace_distance_half_norm": 0.9966253323094464,
          "exact_equal_prior_Helstrom_error": 0.001687333845276806,
          "chernoff_exponential_scale": 0.006737946999085467
        },
        {
          "copies": 10,
          "unsquared_fidelity": 0.006737946999085467,
          "squared_fidelity": 0.00004539992976248485,
          "trace_distance_half_norm": 0.9999772997774687,
          "exact_equal_prior_Helstrom_error": 0.000011350111265628904,
          "chernoff_exponential_scale": 0.00004539992976248485
        }
      ]
    },
    "warm_displaced_thermal_pair": {
      "thermal_mean_occupation": 0.5,
      "unsquared_fidelity": 0.7788007830714049,
      "squared_fidelity": 0.6065306597126334,
      "thermal_ratio": 0.3333333333333333,
      "minimizing_s": 0.5,
      "chernoff_overlap": 0.7649466451949238,
      "chernoff_exponent_nats": 0.26794919243112275
    },
    "scope_warning": "The exact finite-copy Helstrom formula shown here uses the pure N=0 member of the displaced-thermal family. For N>0 the fidelity and Chernoff exponent remain analytic, but the finite-copy Helstrom trace norm is not replaced by a fidelity bound."
  },
  "channel_distinguishability": {
    "channels": "pure-loss channels L_eta0 and L_eta1, with L_eta(|alpha><alpha|)=|sqrt(eta)alpha><sqrt(eta)alpha|",
    "input_hamiltonian": "H=a^dagger a with ground energy zero",
    "parameters": {
      "eta_0_power_transmissivity": 1,
      "eta_1_power_transmissivity": 0.81,
      "difference_of_amplitude_transmissivities": 0.09999999999999998,
      "squared_difference": 0.009999999999999995
    },
    "exact_expressions": {
      "coherent_output_fidelity": "F_E=exp[-E(sqrt(eta_0)-sqrt(eta_1))^2/2]",
      "coherent_output_trace_distance": "T_E=sqrt{1-exp[-E(sqrt(eta_0)-sqrt(eta_1))^2]}",
      "constrained_diamond_lower_bound": "||L_eta0-L_eta1||_(diamond,E) >= 2 T_E",
      "coherent_probe_success_probability": "p_succ=(1+T_E)/2"
    },
    "rows": [
      {
        "input_mean_occupation_E": 1,
        "output_unsquared_fidelity": 0.9950124791926823,
        "output_trace_distance_half_norm": 0.09975052005293954,
        "coherent_probe_trace_norm": 0.19950104010587907,
        "equal_prior_success_probability": 0.5498752600264698
      },
      {
        "input_mean_occupation_E": 10,
        "output_unsquared_fidelity": 0.951229424500714,
        "output_trace_distance_half_norm": 0.308484330175846,
        "coherent_probe_trace_norm": 0.616968660351692,
        "equal_prior_success_probability": 0.654242165087923
      },
      {
        "input_mean_occupation_E": 100,
        "output_unsquared_fidelity": 0.6065306597126335,
        "output_trace_distance_half_norm": 0.79506009762065,
        "coherent_probe_trace_norm": 1.5901201952413,
        "equal_prior_success_probability": 0.897530048810325
      },
      {
        "input_mean_occupation_E": 400,
        "output_unsquared_fidelity": 0.1353352832366128,
        "output_trace_distance_half_norm": 0.9907998592608226,
        "coherent_probe_trace_norm": 1.9815997185216452,
        "equal_prior_success_probability": 0.9953999296304112
      }
    ],
    "limiting_statement": "For eta_0 not equal to eta_1, the coherent-probe lower bound tends to 2 as E tends to infinity. Since every channel diamond distance is at most 2, the unconstrained diamond distance is 2.",
    "scope_warning": "The tabulated value is a certified coherent-probe lower bound on the energy-constrained diamond norm, not a claim that coherent states optimize that constrained norm."
  },
  "entropy_continuity_counterexample": {
    "states": "rho=|0><0| and sigma_d=(1-delta)|0><0|+(delta/d) sum_(j=1)^d |j><j|",
    "parameters": {
      "delta": 0.01
    },
    "exact_expressions": {
      "trace_distance_half_norm": "T(rho,sigma_d)=delta",
      "entropy": "S(sigma_d)=h_2(delta)+delta ln d",
      "mean_number": "Tr(sigma_d a^dagger a)=delta(d+1)/2"
    },
    "rows": [
      {
        "excited_levels_d": 10,
        "trace_distance_half_norm": 0.01,
        "sigma_entropy_nats": 0.07902738528478781,
        "sigma_mean_number": 0.055
      },
      {
        "excited_levels_d": 1000,
        "trace_distance_half_norm": 0.01,
        "sigma_entropy_nats": 0.12507908714466873,
        "sigma_mean_number": 5.005
      },
      {
        "excited_levels_d": 1000000,
        "trace_distance_half_norm": 0.01,
        "sigma_entropy_nats": 0.1941566399344901,
        "sigma_mean_number": 5000.005
      }
    ],
    "lesson": "Trace-norm closeness alone gives no dimension-free entropy continuity on an oscillator; the family escapes every fixed energy set."
  },
  "sources": [
    {
      "citation": "Ke Li, Second-order asymptotics for quantum hypothesis testing, Annals of Statistics 42 (2014) 171-189",
      "doi": "10.1214/13-AOS1185",
      "url": "https://arxiv.org/abs/1208.1400",
      "locator": "Eq. (1), Theorem 2, and finite-sample expansion in Section 1"
    },
    {
      "citation": "K. M. R. Audenaert et al., Discriminating States: The Quantum Chernoff Bound, Physical Review Letters 98 (2007) 160501",
      "doi": "10.1103/PhysRevLett.98.160501",
      "url": "https://arxiv.org/abs/quant-ph/0610027",
      "locator": "Eqs. (1)-(4) for Helstrom error and the quantum Chernoff exponent; Eq. (6) for overlap-distance bounds"
    },
    {
      "citation": "Leonardo Banchi, Samuel L. Braunstein, and Stefano Pirandola, Quantum fidelity for arbitrary Gaussian states, Physical Review Letters 115 (2015) 260501",
      "doi": "10.1103/PhysRevLett.115.260501",
      "url": "https://arxiv.org/abs/1507.01941",
      "locator": "Eqs. (8)-(9) for the unsquared fidelity and displacement dependence; Eq. (29) for discrimination bounds"
    },
    {
      "citation": "Andreas Winter, Tight uniform continuity bounds for quantum entropies, Communications in Mathematical Physics 347 (2016) 291-313",
      "doi": "10.1007/s00220-016-2609-8",
      "url": "https://arxiv.org/abs/1507.07775",
      "locator": "Gibbs Hypothesis and Lemma 15 in Section IV"
    },
    {
      "citation": "M. E. Shirokov, Energy-constrained diamond norms and their use in quantum information theory, Problems of Information Transmission 54 (2018) 20-33",
      "doi": "10.1134/S0032946018010027",
      "url": "https://arxiv.org/abs/1706.00361",
      "locator": "Definition of the E-norm in Section 3 and Proposition 3"
    }
  ],
  "self_check": {
    "tolerance": 4e-13,
    "count": 41,
    "all_passed": true,
    "checks": [
      {
        "label": "normal quantile probability lies strictly between zero and one",
        "passed": true
      },
      {
        "label": "inverse-normal benchmark at epsilon=0.1",
        "passed": true
      },
      {
        "label": "homodyne relative entropy",
        "passed": true
      },
      {
        "label": "homodyne information variance",
        "passed": true
      },
      {
        "label": "full displaced-thermal relative entropy",
        "passed": true
      },
      {
        "label": "full displaced-thermal relative-entropy variance",
        "passed": true
      },
      {
        "label": "homodyne restriction loses distinguishability",
        "passed": true
      },
      {
        "label": "homodyne threshold saturates the type-I constraint at n=1",
        "passed": true
      },
      {
        "label": "homodyne type-II error is a probability at n=1",
        "passed": true
      },
      {
        "label": "homodyne threshold saturates the type-I constraint at n=4",
        "passed": true
      },
      {
        "label": "homodyne type-II error is a probability at n=4",
        "passed": true
      },
      {
        "label": "homodyne threshold saturates the type-I constraint at n=16",
        "passed": true
      },
      {
        "label": "homodyne type-II error is a probability at n=16",
        "passed": true
      },
      {
        "label": "homodyne threshold saturates the type-I constraint at n=64",
        "passed": true
      },
      {
        "label": "homodyne type-II error is a probability at n=64",
        "passed": true
      },
      {
        "label": "n=1 homodyne beta",
        "passed": true
      },
      {
        "label": "n=4 homodyne beta",
        "passed": true
      },
      {
        "label": "n=16 homodyne D_H",
        "passed": true
      },
      {
        "label": "n=64 homodyne D_H",
        "passed": true
      },
      {
        "label": "second order improves the largest-copy homodyne benchmark",
        "passed": true
      },
      {
        "label": "detector-blind one-shot divergence",
        "passed": true
      },
      {
        "label": "thermal occupation is nonnegative",
        "passed": true
      },
      {
        "label": "displacement gap squared is nonnegative",
        "passed": true
      },
      {
        "label": "Chernoff thermal occupation is nonnegative",
        "passed": true
      },
      {
        "label": "pure-state Chernoff overlap equals squared unsquared-fidelity",
        "passed": true
      },
      {
        "label": "pure-state trace distance at n=1",
        "passed": true
      },
      {
        "label": "pure-state trace distance at n=2",
        "passed": true
      },
      {
        "label": "pure-state trace distance at n=5",
        "passed": true
      },
      {
        "label": "pure-state trace distance at n=10",
        "passed": true
      },
      {
        "label": "coherent-state Helstrom error decreases with copy number",
        "passed": true
      },
      {
        "label": "thermal occupation is nonnegative",
        "passed": true
      },
      {
        "label": "displacement gap squared is nonnegative",
        "passed": true
      },
      {
        "label": "Chernoff thermal occupation is nonnegative",
        "passed": true
      },
      {
        "label": "mixed Gaussian Chernoff overlap does not exceed unsquared fidelity",
        "passed": true
      },
      {
        "label": "equal thermal noise makes the displaced pair less distinguishable",
        "passed": true
      },
      {
        "label": "coherent-probe channel lower bound increases with energy",
        "passed": true
      },
      {
        "label": "high-energy coherent probe approaches perfect channel discrimination",
        "passed": true
      },
      {
        "label": "attenuation trace-distance formula at E=1",
        "passed": true
      },
      {
        "label": "entropy counterexample grows with excited-sector dimension",
        "passed": true
      },
      {
        "label": "entropy counterexample pays increasing energy",
        "passed": true
      },
      {
        "label": "entropy counterexample keeps trace distance fixed",
        "passed": true
      }
    ]
  }
}