Common mode dephasing in a boson fermion pair: recoverability beyond the one-pair DFS

A boson–fermion mode pair under common-mode dephasing contains an exactdecoherence-free subspace in its single-excitation manifold and a distinctive surroundingoff-code dephasing spectrum. This paper asks whether, under noncommuting transferout of the exact code, the surrounding structure has recoverability value. For weak noncommutingdrift that populates only adjacent sectors, we introduce a first-shell recoverabilitybenchmark based on the code-to-shell map. By that benchmark, in the one-pairsetting, the adjacent off-code band is of limited recovery value. The first setting in whichexact protection no longer fixes the code uniquely is the two-pair boson–fermion system:a two-dimensional logical code may now be chosen inside a larger exact protectedmanifold. The corresponding first-shell problem is formulated in terms of compressedshell metrics and simplified by pair-exchange symmetry. In the matched case, this yieldsan antisymmetric code for which the lower and upper shell responses are scalar on thelogical qubit. A truncated n-pair appendix then shows that larger pair number broadensthe near-ΔS = 0 band structure; recoverability metrics follow a different scaling.The boson–fermion encoding is thus one more protected-subspace option whose valuedepends on the perturbation model and physical setting.