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Conditional mutual information of adjacent intervals as a quantum-Markov defect. These pages are generated from the project's knowledge base by tools/build_docs.py, so what they say about a result is what the knowledge base says about it, with the same status word.

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Foundations: CMI as a Markov defect (Notes 1-3)

• proved Three faces of the Bures form (type III)
• proved CMI as the data-processing defect of tracing out C (Note 2)
• proved Longo-Xu CMI of adjacent intervals for the free fermion net A_r, its finite-index subnets and extensions
• proved Type-III second-variation lemma (fidelity half), under the vector hypothesis (V)
• conjectural Longo-Xu Remark 4.3: adjacent-interval CMI (c/6) log(1+z) for every rational conformal net (conjecture)

Theorem A: exact recovery error of the free fermion

• refereed Theorem A: lim Phi/zeta^2 = c/(12 pi^2)
• proved The C^{1,1} extension of the zero-collar map is implementable on Fock space (uhlmann_upper_bound.tex Cor. 2.7, Prop. 3.1)
• proved f2 = c/(12 pi^2)
• proved Free-fermion one-particle metrics of the compression direction: g_B = 2r/(3 pi^2), g_KM = r/6 (c = r)
• proved Zero-collar Petz map of the free fermion is a geometric compression
• proved Phi(zeta) and D(zeta) are nondecreasing in zeta (check_note5_theory.tex Prop. 8.1)
• proved Exact root fidelity of two quasi-free CAR states from their symbols (Note 5 Thm. 2.1)
• proved Remainder of Theorem A: global upper bound Phi <= -(1/2) log(1 - 2 f2 zeta^2); c3 <= 0 as a limsup; no positive log^-2 term
• proved Remainder bound in the settled range: Phi(zeta) <= -1/2 log(1 - 2 f2 zeta^2) for 0 <= zeta < (4 f2)^{-1/2} = pi sqrt(3)
• proved Ultraviolet tail bound for the windowed fidelity (shape proved; explicit constants 0.173, 0.88 rest on numerical inputs)
• proved Two-chirality law for the massless Dirac fermion: -log F^(lambda) = Phi(z(1+e^{-pi lambda})) + Phi(z(1+e^{pi lambda})), even in lambda
• proved Exact x-space Mellin transform W-hat(k) of the <TT> kernel and its KMS relation
• numerical Certified enclosures of the compressed recovery error
• numerical Continuum Phi at large zeta and the drifting exponent
• numerical Weak link: Phi(zeta) <= 0.0085 zeta^2 on (0, 1] (rigorous on (0, 0.8868], numerical on [0.8868, 1])
• numerical Weak link: numerical values of the tail constants A_1 <= 2.0 and A_3 <= 0.0374
• conjectural Index zero for e^{sK} beyond ||G_s||_2 < 1: for a unitary group e^{sK} with [P_-, K] Hilbert-Schmidt the two off-diagonal blocks have the same singular values also where |s| ||[P_-, K]||_2 >= 1 (open; no document contains an argument)
• conjectural Two-chirality law for a general 2D CFT (rests on an unproved chiral factorisation)
• conjectural Two-chirality minimum: for the massless Dirac fermion -log F^(lambda) is minimal at lambda = 0 for every z (unproved)

Theorem B: universality for diffeomorphism-covariant nets

• refereed (H1)-(H3) hold for all diffeomorphism-covariant nets
• refereed Theorem B: f2 = c/(12 pi^2) for every diffeomorphism-covariant net
• proved g_B = 2c/(3 pi^2) for the compression direction of every diffeomorphism-covariant net (corollary of Theorem B)
• proved Stress-tensor normalisation chain
• numerical Bosonic check: U(1) current net agrees

Kubo-Mori (relative entropy) coefficient s2

• proved Exact relative-entropy identity on unitary orbits (bounded generators; affiliated generators with Omega in D(A))
• conjectural s2 = c/12 for general nets
• conjectural g_KM = c/6 for the compression direction of a general conformal net (conjecture; only the analytically continued value exists)

Optimality of recovery channels (O8, Phases 3 and 3b)

• proved A-C restriction bound: (Tcond) implies E_AC = 0 exactly; applied on the hopping chain (4,8,2), (4,12,2), (4,16,2)
• proved All-channel second-order optimum = quasi-free SDP value
• proved Every recovery channel's recovered symbol is quasi-free feasible
• proved The first-order compression defect is a commutator with the vacuum symbol: ddelta = [Q, E_I D_chi E_I] (F1 Lemma 6.1)
• proved Exterior directions exhaust the invisible space: closure(Ext) = N^perp (question (Q) settled, YES)
• proved Exact KKT certificate at the compression; Moebius rigidity
• proved KKT sign criterion: when the isometric orbit carries the all-channel second-order optimum
• proved Moebius reduction: the fidelity and relative entropy of the geometric recovery depend on (a, L, s) only through zeta = a s/(a+L), including the half-line chart a = infinity (zeta = s) (Note 5 Lemma 1.1; check_note5_theory prop:closed)
• proved The Petz map is not second-order optimal (compression 4x better at leading order)
• proved Second-order problem over quasi-free channels is a convex SDP
• proved Second-order (SLD / quantum Fisher) expansion of the quasi-free root fidelity in finite dimensions (Note 5 Prop. 2.3)
• proved Exact single-observable fidelity bound (Alberti-Uhlmann) and its reduction to the recovered symbol
• proved Exact (N,Y,V) chart of F and the tangent problem at the compression
• proved The theta-problem in the modular frame of the half-line is a parameter-free quadratic Wiener-Hopf problem (F1 Prop. 6.2)
• proved (UH) proved: optimised Uhlmann bound = g_Q of the defect, for every isometric quasi-free channel
• numerical Hopping chain: T = Q_AB^+ Q_AC satisfies the A-C intertwining conditions (Tcond) for (L_A,L_B,L_C) = (4,8,2), (4,12,2), (4,16,2)
• numerical The all-channel second-order optimum equals the isometric-orbit value: kappa_opt = 1 - theta = 0.616 +- 0.003 (numerical; conditional on H1, H2, H3)
• numerical S10's circle model: the ultraviolet taper suppresses theta by ~20%, and there is no taper-free limit
• numerical E_rec <= (1-theta) f2 z^2 (1+o(1)): the zero-collar compression is NOT optimal in exact fidelity
• numerical The compression is not the minimiser of the second-order functional
• numerical Exact -1/2 log det(1-V^*V) with the optimised extension confirms (UH) beyond first order
• numerical NS-circle model: Ext = N^perp exactly, with the dimension identity (L-|I|)^2
• numerical mpmath at 60-80 digits: dist((1-Pi)M, Ext) = 0 exactly
• numerical Optimised exterior extension: UB_c = (1/2)s^2|u|^2 exactly and UB_rot/UB_c = 1-theta
• numerical Is the compression beaten in EXACT fidelity?
• numerical Finite chains: exterior directions span exactly the invisible space (Ext = N^perp)
• numerical Full primal SDP over F in the modular Galerkin frame: min_F g_Q = 8.41 and 7.13 times g_Q(delta_c) at h = 0.4, 0.3 -- the S11 representability obstruction, not a statement about non-isometric channels
• numerical KKT criterion numerics: every finite run FAILS (a); the violation extrapolates to 0^- as h -> 0, Ymax -> inf; under H3, (e) follows from the proved E(zeta_inf) >= 0 alone, and the finite-h junction values are positive
• numerical Structure of the numerical optimum
• numerical Tangent SDP over (G, N1, Y1) in the circle model: orbit-only optimum matches (1-theta) to 1.1e-7, full optimum not resolvable (measured Gram cond 2.4e22)
• numerical The first-order gain constant theta = 0.384 +- 0.003 (S10's 0.300 is a taper-suppressed lower bound)
• conjectural Hypothesis H1 of the tangent problem: g_Q-differentiability along the scaled family
• conjectural Hypothesis H2 of the tangent problem: localisation of the minimisers of the second-order programme
• conjectural Hypothesis H3 of the tangent problem (cone hypothesis): every admissible isometry generator with a feasible first-order datum lies in the model cone modulo the linear class

Lattice, thermal, massive and gapped geometries (O7, O9, O10)

• proved Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions
• proved Separation constraint: all-weak L->R Petz chains have e^{Delta_k} >= 2/eps (sharp)
• proved Amplification: the corner strength in the chain frame is zeta_step (1 + z')
• proved Corner data of all one-sided protocols for n = 4 and n = 5: the holonomy is the starting block
• proved Angle form of the sequential recovery bound, conditional on Hypothesis (U) (vacuous: (U) is refuted)
• proved Type-III sequential recovery bound: Bures distances add along a chain of von Neumann inclusions (chordal and second-order forms, no Hypothesis (U))
• proved Vacuum rigidity: the pulled-back vacuum is the vacuum iff the map is Moebius; no protocol recovers exactly
• numerical Separated intervals: CMI with a gap and linearity off criticality
• numerical Markov c-function along the massive flow
• numerical Thermal states: two-branch law at the cylinder cross ratio, exponential shielding
• numerical Two-branch law confirmed on the lattice
• conjectural Second-order network law: Phi_tot = sum zeta_j zeta_k G(y_j - y_k), G = c Ghat universal
• conjectural VWZ four-interval protocols: Phi(1):Phi(2):Phi(3) -> 1:2:4; middle-out is the worst order

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