Results on threshold repetition, purified effects and conditioned strategies

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Universal threshold repetition — Theorem 2.1. Assume (S) (Song's conditioning estimate; not proved in the programme). For a finite game with entangled value v < 1, Pr[W_k >= ceil((v+delta)k)] <= exp(-kappa_0 delta^5 k/(1 + log d)) for every finite-dimensional strategy, with a universal kappa_0.
• Threshold repetition for a fixed question distribution — Theorem 2.2. For a finite game with entangled value v < 1, Pr[W_k >= ceil((v+delta)k)] <= exp(-delta^3 k/(16384 (1 + kappa_mu)(1 + log d))) for every finite-dimensional strategy; kappa_mu depends on the support components of the question distribution.
• One coordinate at a time — Lemma 3.1. In every finite-dimensional strategy for the k-fold repetition, each coordinate is won with probability at most the entangled value.
• Sampling the upper tail — Lemma 1.1. For arbitrary jointly distributed win indicators with Pr(W >= qk) = p_H > 0, some set C of at most ell coordinates is won with probability at least p_H q^ell/2 and, given that, W < q_0 k with probability at most epsilon.
• From conditioning to threshold bounds — Proposition 2.1. If conditioning cost at most T forces the average conditional win rate outside the tested set below v + delta/4, then the threshold probability is at most exp(-delta T k/(16(1 + log d))) for every k.
• Two-sided error amplification — Corollary 4.1. Assume (S). Accepting when at least ceil(tk) of k repetitions are won, with t the midpoint of the completeness and soundness levels, makes both errors exponentially small in k.
• Resolvent purification — Lemma 1.1. The maps F_T h(s) = T(T+s)^(-1) h satisfy F_T^* F_T = T, and averaging effects moves them by at most the operator entropy gap of t log t.
• Transporting measurements — Lemma 2.1. If F^* F = T and positive G_a sum to T, there is a POVM with F^* G~_a F = G_a for every answer a.
• Exact conditional laws — Lemma 2.1. Conditional vectors and transported local measurements reproduce exactly the law of the answers at an untested coordinate given that all tested coordinates are won.
• Mixed exposure — Lemma 3.1. Replacing one player's averaged effect by its average over the other player's question costs, on average over the untested coordinates, at most p tau in squared distance.
• Drift under conditioning — Lemma 4.1. For independent identically distributed coordinates and an event of probability p, the average total-variation distance of the conditional marginals from the original law is at most sqrt(log(1/p)/(2m)).
• Conditioning through support components — Proposition 2.1. The average conditional win rate outside a tested set is at most v + sqrt(log(1/p)/(2m)) + 4 sqrt(kappa_mu tau) + 2 kappa_mu tau.

Reading order

Five lessons, to be read in order. The cubic threshold bound for a fixed question distribution is proved in full. The universal fifth-power bound and the soundness half of two-sided amplification use the named hypothesis (S), a conditioning estimate of Song that is not proved in the programme.