Exact -1/2 log det(1-V^*V) with the optimised extension confirms (UH) beyond first order

Status: numerical — measured, with an error bar and a convergence order; not a theorem.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: e2-exact-fidelity-optimised-extension

Statement

With the Tikhonov-optimised exterior generator Z_e (mu = 1e-8; the mu = 1e-4 rows overshoot by 3e-3 and are not used) and W~ = exp(Z_pre + Z_e), Z_pre = -s D_chi resp. -s(D_chi + eps_* G_*), the EXACT Uhlmann exponent UH = -(1/2) log det(1 - V^*V), V = P^perp W~ P (singular values, log1p), satisfies at L = 128..384 and s = 0.001, 0.002, 0.005: UH_compr / ((1/2) s^2 |u|^2) between 1.00000 (s = 0.001) and 0.99981 (s = 0.005, L = 384), i.e. equal to the first-order UB to within 2e-4, the deficit a clean O(s^2); UH_rot / ((1/2) s^2 |u|^2) = 0.70885 (L=128), 0.71125 (192), 0.71057 (256), 0.70764 (384) at s = 0.001, against 1 - theta = 0.70890, 0.71120, 0.71051, 0.70767: agreement to about 5e-5, i.e. four significant digits; decreasing slightly with s (0.70714 at s = 0.005, L = 384). The two second-order-inequivalent constructions exp(Z_pre) exp(Z_e) and W_s exp(-s eps_* G_*) exp(Z_e) give values below the sum version (0.70596 and 0.70810 at L = 128, s = 0.001). Hence (UH) holds in this model beyond first order, with room to spare.

How to verify

run numerics/optimality_all/e2_exact.py; compare the columns 'UB', 'exact sum', 'prod' against the printed (1-theta)*(1/2 s^2|u|^2)

Evidence

• numerics/optimality_all/e2_exact.py — numerics
• rigor/exterior_extension_numerics.tex — document, at sec:exact

Related claims

• Optimised exterior extension: UB_c = (1/2)s^2|u|^2 exactly and UB_rot/UB_c = 1-theta — numerical
• Is the compression beaten in EXACT fidelity? — numerical
• phase4-plan — done, no page of its own

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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