cft-cmi

(UH) proved: optimised Uhlmann bound = g_Q of the defect, for every isometric quasi-free channel

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: theorem · Knowledge-base id: uhlmann-upper-half-proved

Statement

For an isometric quasi-free channel with one-particle isometry W_I, every unitary W~ of L^2(R) with W~|_{L^2(I)} = W_I and both off-diagonal blocks P^perp W~ P and P W~ P^perp Hilbert-Schmidt (Def. 1.1 as corrected 2026-09-24, Correction C1; until then the document required only P^perp W~ P) gives -log F <= -1/2 log det(1 - V^*V) <= -1/2 log(1 - ||V||_2^2), V = P^perp W~ P (Prop. 1.3), for index-zero W~, i.e. ind(P W~ P|_{PH}) = 0: for W~ of non-zero index with ||V|| < 1 step <1>3 is false, since Gamma(W~) changes the charge and <Omega, Gamma(W~) Omega> = 0 (exact_fidelity_upper_half.tex l. 1201-1203, Correction C3). (UH) is unaffected: the product family U_s has index zero for small s, which is all Thm 6.2 / 7.1 use (l. 1192-1200). The admissible extensions form a right torsor under {1_I (+) W'}: the whole column Z E_I is prescribed by the channel and the free block is E_{I^c} Z E_{I^c}, at first order / as an exact right translation (Lemma 4.1 -- NOT a block-diagonal Z_I (+) Z_e, and NOT the truncation E_I D_chi E_I). Reduction (Thm 4.3): inf over exterior Z_e of (1/2)||P^perp(Z_0+Z_e)P||^2 = (1/2) dist(P^perp Z_0 P, Ext)^2 = (1/2)||Pi P^perp Z_0 P||^2 + (1/2) dist((1-Pi)P^perp Z_0 P, Ext)^2 = g_Q(E_I[P,Z_0]E_I), the last step because closure(Ext) = N^perp always. Hence (UH) in its PROVED, g_Q-free form (Thm 7.1, eq. (18)): for the product family U_s = W~_s e^{-lambda s G} e^{s Z_e}, limsup_{s->0} s^{-2}(-log F(omega, omega^{beta_s})) <= (1/2) dist(P^perp(D_chi + lambda G)P, Ext)^2 = (1/2)||Pi P^perp (D_chi + lambda G) P||^2. Ingredients: the exact one-parameter-group estimate ||P^perp e^{tZ}P||_2 <= |t| ||P^perp Z P||_2 (Lemma 6.1; like rate_of_remainder.tex Lemma 6.1 it needs the two blocks of e^{tZ} to have the same singular values, which the documents settle while ||P - e^{-tZ} P e^{tZ}||_2 < 1, e.g. for |t| ||[P, Z]||_2 < 1, enough for the limits s -> 0 used here, and which for larger |t| rests on the conjectural card group-index-zero-large-s) and an S_2-continuity argument for the three-factor product family (Thm 6.2); the limsup is taken at fixed Z_e and only then the infimum over Z_e (no uniformity in Z_e is claimed or needed). CAVEAT (referee R2): re-expressing the right-hand side as g_Q(delta(s))(1+o(1)) is NOT a consequence of continuity of g_Q (false in the continuum: spec Q accumulates at 0 and 1); it holds whenever S10 Lemma 6.1(2)-(3) applies, with its weight cap (Remark 7.2). Corollary 4.4: the optimised bound depends on the channel only through its first-order defect.

How to verify

rigor/exact_fidelity_upper_half.tex Secs. 1, 4, 6, 7 and Sec. 10 (referee repairs R1-R10), and Sec. 11 (Corrections of 2026-09-24, l. 1091-1205: C1-C2 both blocks, C3 the known index gap and why (UH) stands); numerically E2 card e2-optimised-uhlmann (UB_c/(1/2 s^2|u|^2) = 1.0000000 at L=128..384).

Evidence

Related claims

Review

Review record: passed 2026-09-24 by REF-CARD-ROR.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

Documentation map · All claims by area · What is open · Symbols · Definitions