1 · The essentials
Consider a game in which two players each receive a question and give an answer, with only finitely many possible questions and answers. The players may share a quantum resource that helps coordinate their answers. Let be their best possible chance of winning under the allowed rules. Now give them many independently drawn copies of the game, allowing them to use one combined quantum strategy across all copies. The main result says that the chance of winning a fraction greater than by any fixed allowed margin decreases exponentially as the number of copies grows. This remains true when the two players’ questions within each copy are related rather than independently chosen. The result covers quantum strategies of every finite dimension, and the main theorem has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine two quiz partners receiving separate question sheets for many copies of the same quiz and trying to beat their usual best success rate by a fixed margin. The result says that even a combined plan across the copies, using the allowed shared quantum resource, cannot prevent the chance of exceeding that target from falling exponentially; ordinary quiz teamwork is only an image, not a model of that resource.
Why it matters
The result controls unusually high success rates, not just the chance of winning every copy. Its scope includes joint quantum strategies across repetitions and correlations between the players’ questions within a repetition. It also gives quantitative decay rates, including a universal rate expressed in terms of the target margin and the numbers of possible answers.
2 · Going further
For a finite two-player, one-round game, let denote its entangled value. For independently sampled repetitions and , the problem is to bound the probability of winning at least a fraction of the copies. The bound must hold uniformly over all joint finite-dimensional entangled strategies, rather than only strategies that treat each copy separately.
The main manuscript gives exponential decay in . The universal rate is proportional to , where and are the players’ answer alphabets. For each fixed question distribution, the abstract also reports an explicit rate cubic in . Correlation between the two players’ questions is allowed, while the repeated question pairs are sampled independently.
The supplied abstract states the result and its quantitative rates but does not describe the argument, so no proof mechanism can be extracted from this source. Only the main manuscript is supplied; no companion manuscripts or earlier results are described. The main result has a Lean formalization, but the source does not separately specify formalization coverage for each rate refinement.
Useful notions
- Probability of winning a game
- Independent repetitions
- Shared quantum resources
- Exponential decay
- Finite sets of questions and answers
What was known before / what is new
Before
The source frames the problem using a game’s entangled value and independently sampled repetitions. It does not identify earlier repetition bounds or give a historical account, so it does not support a more specific comparison with prior work.
New
The main manuscript establishes threshold repetition for every finite two-player, one-round game whose entangled value is below one, uniformly over joint finite-dimensional entangled strategies. It allows arbitrary correlated question distributions and gives a universal rate proportional to the fifth power of the margin, divided by a logarithmic answer-size factor. It also reports an explicit cubic rate for each fixed question distribution.
Limits and assumptions
The hypotheses require a finite two-player, one-round game, entangled value , an integer , and a margin . Repetitions are independent, even though the players’ questions within a copy may be correlated and their strategy may act jointly across copies. The stated scope is all finite-dimensional entangled strategies; the source makes no claim here about infinite-dimensional strategies. The main result has a Lean formalization, but the supplied text does not detail coverage of every quantitative refinement. It names no remaining open problems.
Mini glossary
- Entangled value —
- The best winning probability available to the players using the allowed shared quantum strategies.
- Threshold repetition —
- Repeating a game and asking for the probability of winning at least a specified fraction of its copies.
- Joint finite-dimensional entangled strategy —
- A strategy using a shared quantum resource of finite dimension that can coordinate answers across the repeated games.
- Correlated question distribution —
- A rule for drawing questions under which the two players’ questions need not be independent of each other.
- Answer alphabet —
- The finite set of answers a player is allowed to give.
- Exponential decay —
- A decrease bounded by an expression with a negative exponent proportional to the number of repetitions.
Keywords : Entangled games · Threshold repetition · Exponential decay
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction of k independent repetitions decays exponentially in k, for . Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Threshold parallel repetition for finite-dimensional entangled games
25/09/2026For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every and k ≥ 1. Its universal rate is proportional to , where are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.