Fidelity limits and purity requirements for entanglement swapping with one auxiliary photon

Internal-state mismatch limits heralded entanglement swapping. We determine the limits of using one independent auxiliary photon (a helper) with passive optics acting identically on every internal state, complete photon counting and deterministic local recovery. For pure data states of overlap $0<c<1$, put $t=1-c$. Optimizing over the helper and all allowed finite networks reduces the minimum Bell error from (1-c²)/2~ t without a helper to t²/(2-c+2c²)~ t²/3. Attaining this minimum requires a pure midpoint helper and three equally spaced logical Bloch directions (a trine); the largest single-record probability at equality is (2-c+2c²)/72. For a mixed helper supported on the data-state span, let its eigenvalues be (1± r)/2 and η=1-r. Under a complete-record probability floor p≥ p₀, with fixed 0<p₀<1/64, the optimal error has order min²+η,t√t+η\ as t,η→0. Quadratic suppression is possible precisely when η=O(t²), requiring an ideal Hong--Ou--Mandel helper self-visibility deficit of quadratic or smaller order in the data-visibility deficit. When η is proportional to t, exchange-invariant helpers leave linear error, whereas a tilted helper and a real eight-mode network attain the optimal order t3/2. The converse follows from compatibility of the optical amplitudes shared by all photon assignments. The same order holds for local unitaries, deterministic local operations and classical communication, and deterministic separable recovery, and for the average over a bounded number of accepted records at nonvanishing total acceptance probability.