The quantum switch as a commutation-syndrome filter for qubit Pauli channels: deterministic advantage over definite-order strategies and an exact geometric law

Abstract The quantum switch is known to remove noise probabilistically, but whether this constitutes an advantage over causally ordered processing has remained contentious: postselection, weak classical benchmarks, and task-dependent comparisons have all drawn criticism. Here we analyze the question exactly. For any two qubit Pauli channels, measuring the switch's control qubit implements a commutation syndrome—it sorts each noise realization by whether the two Kraus errors commuted, syndrome information that an unencoded qubit under Pauli noise does not otherwise possess—and the optimal measure-and-feed-forward protocol has the closed form F_e = Σ_{b=±} max_P c_P^(b). For the exactly solvable pair, an isotropic depolarizing channel and a flip channel along an arbitrary axis n, we prove a ceiling theorem: every definite-order protocol using each channel once, with arbitrary memoryless intermediate operations—unitary or heralded—and arbitrary receiver-side (deterministic or heralded) post-processing, has entanglement fidelity at most max_P c_P. Auxiliary side systems are covered whenever they are non-signaling about the message; sender-side encoding and pre-shared entanglement constitute stronger resource classes, which we quantify separately. The switch exceeds the ceiling deterministically, by exactly pq/3 for p ≤ 3/4 and by at least (1−p)q beyond, and the separation is generic: a survey of 5000 random Pauli pairs finds strict gain in 99.8% of cases (median 0.14); in a 200-pair stress test the gain is never overturned even when definite order is granted an optimized midpoint echo. The state-resolved gain obeys an exact geometric law, ΔF = (2pq/3)|r·n|²: alignment is precisely the condition under which the degenerate two-error syndrome becomes perfectly correctable. The separation tolerates control dephasing down to a threshold visibility that falls from 0.875 at p = q = 0.3 to 0.04 at p = q = 0.9—and at strong noise it outperforms coherent control of the two orderings even when the latter is granted twice the channel uses.