Perfect Parallel Repetition Theorem for Quantum XOR Proof Systems

Cleve, Richard; Slofstra, William; Unger, Falk; Upadhyay, Sarvagya

doi:10.1109/ccc.2007.24

Cited by 59 publications

(85 citation statements)

References 20 publications

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“…In computer science, such correlations are studied in the context of XOR nonlocal games [CHTW04,CSUU08]. We say that a set of joint correlations, We prove the following theorem.…”

Section: Quantum Nonlocalitymentioning

confidence: 99%

A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement

Briët

Buhrman

Toner

2011

Commun. Math. Phys.

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Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if the local Hilbert-space dimension is at least d. This establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing a lower bound on a new generalization of Grothendieck's constant.

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“…In computer science, such correlations are studied in the context of XOR nonlocal games [CHTW04,CSUU08]. We say that a set of joint correlations, We prove the following theorem.…”

Section: Quantum Nonlocalitymentioning

confidence: 99%

A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement

Briët

Buhrman

Toner

2011

Commun. Math. Phys.

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“…We write G k for the k-fold parallel repetition of a game G. It is immediately clear that uðG k ÞR uðGÞ k ; ð5:4Þ because the players can just play each instance of the game independently, using an optimal strategy for each instance. Specializing now to the odd-cycle game, for the quantum value, the inequality in equation (5.4) is actually tight: Cleve et al (2007) have proved that…”

Section: The Odd-cycle Gamementioning

confidence: 99%

Monogamy of non-local quantum correlations

2008

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We describe a new technique for obtaining Tsirelson bounds, which are upper bounds on the quantum value of a Bell inequality. Since quantum correlations do not allow signalling, we obtain a Tsirelson bound by maximizing over all no-signalling probability distributions. This maximization can be cast as a linear programme. In a setting where three parties, A, B and C, share an entangled quantum state of arbitrary dimension, we (i) bound the trade-off between AB's and AC's violation of the Clauser-Horne-ShimonyHolt inequality and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities, relevant for interactive proof systems and cryptography.

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“…Such theorems are known for a variety of models: Boolean circuits [Yao82,GNW95], 2-prover games [Raz98], decision trees [NRS94], communication complexity [PRW97], polynomials [VW08], puzzles [BIN97], and quantum XOR games [CSUU07], just to mention a few. However, there are also examples where a direct product statement is false (see, e.g., [BIN97,PW07,Sha03]).…”

Section: Applications To Direct Product Theoremsmentioning

confidence: 99%

Constructive Proofs of Concentration Bounds

Impagliazzo

Kabanets

2010

Lecture Notes in Computer Science

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We give a simple combinatorial proof of the Chernoff-Hoeffding concentration bound [Che52,Hoe63], which says that the sum of independent {0, 1}-valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, but rather uses a simple and intuitive counting argument. In addition, our proof is constructive in the following sense: if the sum of the given random variables is not concentrated around the expectation, then we can efficiently find (with high probability) a subset of the random variables that are statistically dependent. As simple corollaries, we also get the concentration bounds for [0, 1]-valued random variables and Azuma's inequality for martingales [Azu67].We interpret the Chernoff-Hoeffding bound as a statement about Direct Product Theorems. Informally, a Direct Product Theorem says that the complexity of solving all k instances of a hard problem increases exponentially with k; a Threshold Direct Product Theorem says that it is exponentially hard in k to solve even a significant fraction of the given k instances of a hard problem. We show the equivalence between optimal Direct Product Theorems and optimal Threshold Direct Product Theorems. As an application of this connection, we get the Chernoff bound for expander walks [Gil98] from the (simpler to prove) hitting property [AKS87], as well as an optimal (in a certain range of parameters) Threshold Direct Product Theorem for weakly verifiable puzzles from the optimal Direct Product Theorem [CHS05]. We also get a simple constructive proof of Unger's result [Ung09] saying that XOR Lemmas imply Threshold Direct Product Theorems.

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Perfect Parallel Repetition Theorem for Quantum XOR Proof Systems

Cited by 59 publications

References 20 publications

A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement

A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement

Monogamy of non-local quantum correlations

Constructive Proofs of Concentration Bounds

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