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 give an exact analysis of the question for a qubit under Pauli noise. 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 does not otherwise possess, and the optimal measure-and-feed-forward protocol has a closed form. The analytical results have two levels of generality. For the exactly solvable pair, an isotropic depolarizing channel and a flip channel along an arbitrary axis, we prove a complete ceiling theorem: every definite-order protocol using each channel once, with arbitrary memoryless intermediate operations and receiver-side post-processing, deterministic or heralded, has entanglement fidelity at most maxPcP, and the switch exceeds this ceiling deterministically, by pq/3 over most of parameter space. For arbitrary qubit Pauli pairs, the ceiling is proven against arbitrary receiver-side post-processing, and the gain is strictly positive if and only if the two syndrome branches call for different Pauli corrections, confining the exceptions to error-dominated noise. Side systems are covered whenever they are non-signaling about the message. Beyond these theorems the evidence is numerical: strict gain in 99.8% of 5000 random Pauli pairs, and no case, within a 200-pair sample, in which an optimized midpoint echo overturns the gain. The state-resolved gain obeys an exact geometric law, ΔF=2pq/3|r^⋅n^|2: alignment is the condition under which the degenerate two-error syndrome becomes perfectly correctable. The separation survives control dephasing down to a threshold visibility that falls with noise strength, and at strong noise it outperforms coherent control of the two orderings even with twice the channel uses.