Second-order (SLD / quantum Fisher) expansion of the quasi-free root fidelity in finite dimensions (Note 5 Prop. 2.3)

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: quasi-free-second-order-fidelity-expansion

Statement

For gauge-invariant quasi-free states on a finite-dimensional one-particle space with symbols Q and Q + eps D (0 < Q < 1, D = D^*), -log F(omega_Q, omega_{Q+eps D}) = (eps^2/4) sum_{i,k} |D_ik|^2 / (q_i(1-q_k) + q_k(1-q_i)) + O(eps^3) in an eigenbasis of Q, with weight 1 + cosh((kappa_i+kappa_k)/2)/cosh((kappa_i-kappa_k)/2) in the modular variables q = (1+e^kappa)^{-1}, and the analogous Kubo-Mori expansion of the relative entropy.

How to verify

No verification recipe recorded on this card.

Evidence

• fidelity_recovered_quasifree.tex — document, at prop:second-order

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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