Mathesis

Dhruv GuptaZetetic-Dhruvperson2026-09-24T00:00:00Z

The two-clique quantum junction tree theorem (Lauritzen–Zwiernik, arXiv:2605.19453, Theorem 3.1): over the acyclic two-clique base, the base's compatibility datum decides base-side reconstruction. For strictly positive consistent marginals, a quantum Markov completion exists iff Tr T(R) = 1; the completion is then unique and equals the normalized logarithmic candidate σ(R) = T(R)/Tr T(R). Together with trC_le_one (the trace bound) this is the full theorem.

DeclHiddenChannelCapacity.lz_theorem_3_1

∀ {a : Type u_1} {c : Type u_2} {b : Type u_3} [inst : Fintype a] [inst_1 : DecidableEq a] [inst_2 : Nonempty a]
  [inst_3 : Fintype c] [inst_4 : DecidableEq c] [inst_5 : Nonempty c] [inst_6 : Fintype b] [inst_7 : DecidableEq b]
  [inst_8 : Nonempty b] {ρAC : MState (a × c)} {ρCB : MState (c × b)},
  ρAC.m.PosDef →
    ρCB.m.PosDef →
      ρAC.traceLeft = ρCB.traceRight →
        (HiddenChannelCapacity.trC ρAC ρCB = 1 ↔ ∃ ω, HiddenChannelCapacity.IsMarkovCompletion ρAC ρCB ω) ∧
          ∀ (ω : MState (a × c × b)),
            HiddenChannelCapacity.IsMarkovCompletion ρAC ρCB ω → ω = HiddenChannelCapacity.sigT ρAC ρCB

TopicGraphical models

Arguments

DOI	Author	Date
MTH.R-2026-6021	Dhruv Gupta	2026-09-24T00:00:00Z

DOIMTH.C-2026-6021

Cite

Verification

Library: ZPM.Measurements.HC.QJTTwoClique
Statement digest: 0064e98c1438
First verified: 2026-09-24T00:00:00Z