[
  {
    "id": "ARR-2026-5HG8P9BAQ78C5AB2-v1",
    "type": "article",
    "title": "Petz Recoverability in AQFT via Conditional Expectations: A Framework and a Conditional Exponential Recovery Bound",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          1,
          13
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-5HG8P9BAQ78C5AB2",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-5HG8P9BAQ78C5AB2/versions/v1/",
    "note": "Archived manuscript; ARR-2026-5HG8P9BAQ78C5AB2 v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "We formulate an operational notion of recoverability in algebraic quantum field theory for type III local von Neumann algebras. Fixing a faithful normal KMS reference state and assuming a state-preserving conditional expectation, we define the recovery channel and, working in a fixed split implementation for each separation r, we assume (i) exponential decay of split-implemented conditional mutual information and (ii) a CMI-to-recovery inequality. Under these explicit bridge assumptions we obtain a conditional exponential recoverability bound E_rec(r) <= g(C1 e^(-m r)). v2 corrects the duality underlying the construction: v1 defined the Petz-type channel as the \"Accardi-Cecchini adjoint\" via the pairing omega(Z R(X)) = omega(eps(Z) X) and \"proved\" finite-dimensional consistency with the standard Petz map; that identity is false in general (the printed proof contains an invalid cyclicity step; numerically the identity fails at order 1e-1 on random faithful states, holding only in commuting/product situations). The correct statement, proved and machine-verified here, is that the standard Petz map is the trace-predual of the generalized (Accardi-Cecchini) conditional expectation; accordingly, v2 defines the recovery channel in Schrodinger picture as precomposition with the conditional expectation. This also repairs a type/direction error in v1's recovered-state definition, whose corrected form (omega restricted to AB, composed with the normal extension of id_A tensor eps) reproduces the standard Petz reconstruction exactly in finite dimensions. We further relabel v1's finite-dimensional map as the generalized conditional expectation (its Takesaki module property fails generically -- verified), record that for a true state-preserving conditional expectation the recovery is the CE-pullback, and add series positioning: this note is the AQFT capstone announced by the companions, its split-implemented CMI is one of three compatible regularizations in the series, and the numerical program deferred by v1 has since been executed. The main theorem is unchanged: a conditional framework statement isolating the missing bridge assumptions."
  }
]