All-channel second-order optimum = quasi-free SDP value

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: theorem · Knowledge-base id: all-channel-equals-quasi-free

Statement

Dual value sup_t Delta_z(t)_+^2/(8 V(t)) equals min_F g_Q (Sion, one compact set: F is bounded and WOT-closed because its constraints are affine positivity conditions); every channel's error is >= the dual value up to the remainder control. Numerically the exact all-channel optimum on tiny chains (one SDP over Choi matrices) is a Gaussian state to 1e-6; the relaxation is tight.

How to verify

rigor/optimality_all_channels.tex Sec. 4-5; numerics/optimality_all/all_channels_sdp.py, analyze_opt.py, relaxation_test.py.

Evidence

• rigor/optimality_all_channels.tex — document, at thm:main
• refs/Sion1958_minimax.pdf — source, at Cor3.3; held privately, not redistributed
• numerics/optimality_all/all_channels_sdp.py — numerics

Related claims

• Every recovery channel's recovered symbol is quasi-free feasible — proved
• Exact single-observable fidelity bound (Alberti-Uhlmann) and its reduction to the recovered symbol — proved

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A1.

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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