Exact root fidelity of two quasi-free CAR states from their symbols (Note 5 Thm. 2.1)

Status: proved — a complete structured proof.

Area: Theorem A: exact recovery error of the free fermion · Card type: theorem · Knowledge-base id: quasi-free-fidelity-formula

Statement

For two gauge-invariant quasi-free states on a FINITE-DIMENSIONAL one-particle space h with symbols Q1, Q2 (0 < Q < 1), G = Q/(1-Q): -log F = (1/2) sum log(1+t) + (1/2) sum log(1+s) - sum log(1 + sqrt(tau)), t = eig G1, s = eig G2, tau = eig(G1^{1/2} G2 G1^{1/2}) (Note 5 Thm. 2.1, thm:finite-fidelity); equivalently F = det(1-Q1)^{1/2} det(1-Q2)^{1/2} prod_i (1 + sigma_i(C)), C = (1-Q1)^{-1/2} Q1^{1/2} Q2^{1/2} (1-Q2)^{-1/2}, the form that is numerically stable when eigenvalues sit near 0 and 1. For infinite-dimensional h the formula is obtained as the monotone limit over finite-dimensional compressions (Note 5, Thm det-rep) when the two states are quasi-equivalent. Implemented in mpmath at dps 60-80 and validated against the brute-force many-body fidelity on three random modes to 12 digits.

How to verify

Note 5 Thm. 2.1; numerics/optimality/petz_ref.py (mpmath, 60-90 digits); selftest() in numerics/optimality_all/exact_fidelity_rotated.py.

Evidence

• fidelity_recovered_quasifree.tex — document, at thm:finite-fidelity
• numerics/optimality/petz_ref.py — numerics
• numerics/optimality_all/exact_fidelity_rotated.py — numerics

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B2.

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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