Quadratic Repetition for Commuting Projection Games and Noisy Quantum Communication

How reliably does repeating a quantum verification test amplify its soundness when the players can coordinate all their answers? This paper proves a quadratic-gap parallel-repetition bound for finite two-player games whose winning answer graphs are unions of disjoint complete bipartite blocks. The class includes noninjective and partial projection games, with arbitrary joint question distributions within each independent round and arbitrary commuting-operator strategies. If the single-game commuting value is 1 − ε, the probability of winning all n repetitions is at most exp(−2−75 ε²n). Known binary line games show that the quadratic gap power is optimal within the full block class, up to constants. A separate finite-dimensional tensor theorem uses its own single-game gap. The proof constructs one exact conditioned target with square-root-weighted rectangle labels and two independently normalized one-sided comparison states. Entropy cancellation and finite-tracial local reconstruction give a charged extension, heterogeneous repetition bounds and weighted-threshold concentration. Scalar finite-table closure then transfers the estimates to arbitrary commuting strategies. A further application bounds interactive finite-dimensional quantum protocols communicating through prescribed memoryless channels. Exponential zero-communication soundness survives a strict aggregate entanglement-assisted capacity budget. An explicit erasure-channel coding example illustrates the bound with polynomial encoding and decoding costs. The results do not establish quadratic repetition for arbitrary predicates, uniform finite-dimensional resources for arbitrary commuting strategies, or an efficient locality-preserving Hamiltonian or cryptographic compiler. The paper develops the reconstruction premise left conditional in Entropy Budgets and Weighted Concentration in Parallel Games.