Exact single-observable fidelity bound (Alberti-Uhlmann) and its reduction to the recovered symbol

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: theorem · Knowledge-base id: single-observable-bound

Statement

Lemma lem:single (rigor/optimality_all_channels.tex, Lemma 2.2): let phi, psi be normal states on a von Neumann algebra M, T = T* in M, m := phi(T), Delta := phi(T) - psi(T), N := ||T - m||. For every 0 < eps < 1/N, F(phi,psi) <= 1 - (eps/2) Delta + (eps^2/2) psi((T-m)^2)/(1 - eps N); hence, for every V' >= psi((T-m)^2) with 0 < Delta N < 2V', -log F(phi,psi) >= 1 - F(phi,psi) >= Delta^2/(8V') (1 - Delta N/V'). Prop. prop:red (Prop. 2.5): for a recovery channel beta with -log F(omega, omega^beta) <= E and a finite-rank self-adjoint t on h_I, T = dGamma(t), delta^beta = Q - Q^beta, V(t) = Tr(tQt(1-Q)): if Delta := Tr t delta^beta > 0 and V' := V(t) + 2 sqrt(2E) ||t||_1^2 > Delta ||t||_1/2, then -log F(omega, omega^beta) >= Delta^2/(8V') (1 - Delta ||t||_1/V') >= (Tr t delta^beta)^2/(8(V(t) + 2 sqrt(2E) ||t||_1^2)) (1 - |Tr t delta^beta| ||t||_1/V(t)), a bound that depends on beta only through the recovered symbol. These are lower bounds; they say nothing about the upper half. Not claimed here: 'for EVERY recovery channel -log F >= g_Q(Q - Q^beta)(1 - o(1)) in Legendre form' is Remark rem:use (2.6, :146-157), not a proof: it needs a near-maximising test operator with V(t) and ||t||_1 of order one, while the Legendre maximiser (the SLD) need not be finite rank, or even Hilbert-Schmidt in the continuum. The every-channel bound that is proved is Thm. thm:main(2) (card all-channel-equals-quasi-free, which uses this card): E_rec >= D_z (1 - o(1)), D_z = sup_t Delta_z(t)_+^2/(8V(t)), along any family of finite-rank t_z with ||t_z||_1^2/V(t_z) bounded and Delta_z(t_z) ||t_z||_1/V(t_z) -> 0, in particular liminf z^-2 E_rec >= sup over such families of liminf z^-2 Delta_z(t_z)^2/(8V(t_z)) (eq:liminf); by thm:main(3) the remainder control is automatic for finite-dimensional h_I.

How to verify

rigor/optimality_all_channels.tex Lemma 2.2 (lem:single, :61-89) with Lemma 2.1 (lem:alberti = Alberti-Uhlmann Thm. 2), Lemma 2.4 (lem:variance) and Prop. 2.5 (prop:red, :114-144); Remark 2.6 (rem:use, :146-157) is the heuristic reading; Thm. 4.1(2)-(3) (thm:main, :325-340) states the every-channel form.

Evidence

• rigor/optimality_all_channels.tex — document, at lem:single
• rigor/optimality_all_channels.tex — document, at prop:red
• refs/AlbertiUhlmann02_math-ph-0202038.pdf — source, at Thm2; held privately, not redistributed

Related claims

• All-channel second-order optimum = quasi-free SDP value — proved

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A3.

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.

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