Second-order problem over quasi-free channels is a convex SDP

Status: proved — a complete structured proof.

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

Statement

Gauge-invariant quasi-free channels have data (X,Y), CP iff Y >= 0 and XX* + Y <= 1; in the variables (X, Z = X Q_BB X* + Y) the feasible set F = {X Q_BB X* <= Z <= 1 - X(1-Q_BB)X*} is convex and semidefinite representable, the recovered symbol is affine, and the SLD form g_Q is a convex quadratic; every local minimum is global. Caveat: g_Q is the second-order term only for defects small relative to e^{-kappa_max} (order of limits); cap weights, never drop modes.

How to verify

rigor/optimality_second_order.tex Sec. 3.

Evidence

• rigor/optimality_second_order.tex — document, at thm:convex-set
• numerics/optimality/run_exact.py — numerics

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A2.

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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