{
  "title": "Objects, hypotheses, licensed conclusions, and boundary tests",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "noncommutative-information-measurement-aqec-hypothesis-conclusion-table",
  "caption": "Objects, hypotheses, licensed conclusions, and boundary tests",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Essential hypotheses",
      "html": "Essential hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or extension",
      "html": "Excluded converse or extension"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Heisenberg map $\\Phi:\\mathcal M\\to\\mathcal N$",
        "html": "Heisenberg map $\\Phi:\\mathcal M\\to\\mathcal N$"
      },
      "Essential hypotheses": {
        "text": "Normal, unital, and completely positive; a representation and predual are fixed.",
        "html": "Normal, unital, and completely positive; a representation and predual are fixed."
      },
      "Licensed conclusion": {
        "text": "Normal states pull back to normal states, normalization is preserved, and finite ancillary amplifications remain positive.",
        "html": "Normal states pull back to normal states, normalization is preserved, and finite ancillary amplifications remain positive."
      },
      "Excluded converse or extension": {
        "text": "Positivity alone is not complete positivity; complete positivity alone is not normality or spacetime localization.",
        "html": "Positivity alone is not complete positivity; complete positivity alone is not normality or spacetime localization."
      },
      "Adversarial check": {
        "text": "Transpose one half of an entangled qubit pair, or use a singular state map on $B(\\ell^2)$.",
        "html": "Transpose one half of an entangled qubit pair, or use a singular state map on $B(\\ell^2)$."
      }
    },
    {
      "Object and domain": {
        "text": "Relative modular operator for normal $\\rho,\\sigma$",
        "html": "Relative modular operator for normal $\\rho,\\sigma$"
      },
      "Essential hypotheses": {
        "text": "Support conventions are explicit; the channel is normal UCP; an $L^p$ parameter lies in its proved range.",
        "html": "Support conventions are explicit; the channel is normal UCP; an $L^p$ parameter lies in its proved range."
      },
      "Licensed conclusion": {
        "text": "Araki data processing, and sandwiched-Rényi data processing for the stated parameter range.",
        "html": "Araki data processing, and sandwiched-Rényi data processing for the stated parameter range."
      },
      "Excluded converse or extension": {
        "text": "A finite value is not automatic, and one parameter theorem does not extend to every divergence.",
        "html": "A finite value is not automatic, and one parameter theorem does not extend to every divergence."
      },
      "Adversarial check": {
        "text": "Choose $s(\\rho)\\nleq s(\\sigma)$ or move $\\alpha$ outside the proved interpolation range.",
        "html": "Choose $s(\\rho)\\nleq s(\\sigma)$ or move $\\alpha$ outside the proved interpolation range."
      }
    },
    {
      "Object and domain": {
        "text": "Equality in data processing or inclusion $\\mathcal N\\subset\\mathcal M$",
        "html": "Equality in data processing or inclusion $\\mathcal N\\subset\\mathcal M$"
      },
      "Essential hypotheses": {
        "text": "A faithful reference or support reduction, a declared state family, and modular invariance for a preserving expectation.",
        "html": "A faithful reference or support reduction, a declared state family, and modular invariance for a preserving expectation."
      },
      "Licensed conclusion": {
        "text": "A Petz-type recovery for that family, or a unique reference-preserving normal conditional expectation.",
        "html": "A Petz-type recovery for that family, or a unique reference-preserving normal conditional expectation."
      },
      "Excluded converse or extension": {
        "text": "Equality for one pair does not recover every state; an arbitrary algebraic projection need not be positive or state preserving.",
        "html": "Equality for one pair does not recover every state; an arbitrary algebraic projection need not be positive or state preserving."
      },
      "Adversarial check": {
        "text": "Perturb the reference so its modular flow moves the candidate sufficient subalgebra.",
        "html": "Perturb the reference so its modular flow moves the candidate sufficient subalgebra."
      }
    },
    {
      "Object and domain": {
        "text": "Local instrument or system–probe scattering morphism",
        "html": "Local instrument or system–probe scattering morphism"
      },
      "Essential hypotheses": {
        "text": "Normal CP outcome maps summing to a channel, compact coupling support, time-slice identifications, and causal factorization.",
        "html": "Normal CP outcome maps summing to a channel, compact coupling support, time-slice identifications, and causal factorization."
      },
      "Licensed conclusion": {
        "text": "No-signalling nonselective action, induced observables, state updates, and ordered composition.",
        "html": "No-signalling nonselective action, induced observables, state updates, and ordered composition."
      },
      "Excluded converse or extension": {
        "text": "A selected branch need not leave remote conditional expectations fixed; overlapping schemes need not commute.",
        "html": "A selected branch need not leave remote conditional expectations fixed; overlapping schemes need not commute."
      },
      "Adversarial check": {
        "text": "Give the smearing a spacelike tail or swap two causally unordered coupling regions.",
        "html": "Give the smearing a spacelike tail or swap two causally unordered coupling regions."
      }
    },
    {
      "Object and domain": {
        "text": "Correctable von Neumann algebra $\\mathfrak A$",
        "html": "Correctable von Neumann algebra $\\mathfrak A$"
      },
      "Essential hypotheses": {
        "text": "A normal noise channel, a declared code representation, and commutation of $\\mathfrak A$ with all compressed error products.",
        "html": "A normal noise channel, a declared code representation, and commutation of $\\mathfrak A$ with all compressed error products."
      },
      "Licensed conclusion": {
        "text": "A normal recovery fixes every observable in $\\mathfrak A$.",
        "html": "A normal recovery fixes every observable in $\\mathfrak A$."
      },
      "Excluded converse or extension": {
        "text": "Protected logical observables do not imply recovery of the gauge factor, complementary correlations, or all code operators.",
        "html": "Protected logical observables do not imply recovery of the gauge factor, complementary correlations, or all code operators."
      },
      "Adversarial check": {
        "text": "Add a logical generator acting on an erased factor; two distinguishable inputs then have identical outputs.",
        "html": "Add a logical generator acting on an erased factor; two distinguishable inputs then have identical outputs."
      }
    },
    {
      "Object and domain": {
        "text": "Split pair, finite-index inclusion, or sharp type-III factor",
        "html": "Split pair, finite-index inclusion, or sharp type-III factor"
      },
      "Essential hypotheses": {
        "text": "A positive split collar, or a finite Pimsner–Popa index; otherwise use normal modular quantities and explicit energy constraints.",
        "html": "A positive split collar, or a finite Pimsner–Popa index; otherwise use normal modular quantities and explicit energy constraints."
      },
      "Licensed conclusion": {
        "text": "Normal product states and type-I approximants, finite sector/entropy bounds, or intrinsic relative entropy.",
        "html": "Normal product states and type-I approximants, finite sector/entropy bounds, or intrinsic relative entropy."
      },
      "Excluded converse or extension": {
        "text": "Zero collar, infinite index, and a sharp type-III algebra do not supply a canonical reduced density matrix.",
        "html": "Zero collar, infinite index, and a sharp type-III algebra do not supply a canonical reduced density matrix."
      },
      "Adversarial check": {
        "text": "Collapse the collar, replace a finite group by $U(1)$, or demand trace-one local density operators.",
        "html": "Collapse the collar, replace a finite group by $U(1)$, or demand trace-one local density operators."
      }
    }
  ]
}