Quantitative preservation of the XOR-game quantum–classical gap

Determine whether there exists a universal constant c>0 such that every XOR game G satisfies val_+(G) ≤ c val(G) + (1−c) val*(G), thereby quantitatively preserving a fixed fraction of the single-copy quantum–classical gap between the asymptotic classical and quantum winning rates.

Background

The paper proves that for every XOR game G, the Dinur–Steurer relaxed value satisfies val_+(G)≤val*(G), with equality if and only if the classical and entangled single-copy values coincide. Combined with the multiplicative upper bound on repeated-game values, this establishes that any strict quantum–classical gap remains strict asymptotically.

The unresolved issue is whether this qualitative preservation can be made quantitative uniformly over all XOR games. Specifically, the proposed affine inequality would imply that a universal fraction c of the gap between val(G) and val*(G) persists when comparing the asymptotic classical rate, controlled by val_+(G), with the entangled value. The paper gives c=1/2 as an example whose validity would have this interpretation, but does not establish any such universal constant.