A Parallel Repetition Theorem for Entangled Two-Player One-Round Games under Product Distributions

Jain, Rahul; Pereszlényi, Attila; Yao, Penghui

doi:10.1109/ccc.2014.29

Cited by 28 publications

(41 citation statements)

References 29 publications

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“…Very recently Jain et al [36] extended the result to product distributions on the questions, while also removing the dependence on the number of questions: they obtained the bound…”

Section: Additional Related Workmentioning

confidence: 95%

A Parallel Repetition Theorem for Entangled Projection Games

Dinur

Steurer

Vidick

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1 − ε < 1, the value of the k-fold repetition of G goes to zero as O((1 − ε c) k), for some universal constant c ≥ 1. If furthermore the constraint graph of G is expanding we obtain the optimal c = 1. Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call "vector quantum strategy" to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, the algorithm relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our "quantum correlated sampling lemma" generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two non-communicating parties, given classical descriptions of bipartite states |ψ, |ϕ respectively such that |ψ ≈ |ϕ, are able to locally generate a joint entangled state |Ψ ≈ |ψ ≈ |ϕ using an initial entangled state that is independent of their inputs.

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“…Very recently Jain et al [36] extended the result to product distributions on the questions, while also removing the dependence on the number of questions: they obtained the bound…”

Section: Additional Related Workmentioning

confidence: 95%

A Parallel Repetition Theorem for Entangled Projection Games

Dinur

Steurer

Vidick

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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“…Section 5 gives the proof of the central security claims. The appendix proves supporting propositions, including the parallel repetition result derived from [8,16,9,4].…”

Section: Security Analysis and Proof Techniquesmentioning

confidence: 60%

Parallel Device-Independent Quantum Key Distribution

Jain

Miller

Shi

2020

IEEE Trans. Inform. Theory

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A prominent application of quantum cryptography is the distribution of cryptographic keys that are provably secure. Recently, such security proofs were extended by Vazirani and Vidick (Physical Review Letters, 113, 140501, 2014) to the device-independent (DI) scenario, where the users do not need to trust the integrity of the underlying quantum devices. The protocols analyzed by them and by subsequent authors all require a sequential execution of N multiplayer games, where N is the security parameter. In this work, we prove unconditional security of a protocol where all games are executed in parallel. Besides decreasing the number of timesteps necessary for key generation, this result reduces the security requirements for DI-QKD by allowing arbitrary information leakage of each user's inputs within his or her lab. To the best of our knowledge, this is the first parallel security proof for a fully device-independent QKD protocol. Our protocol tolerates a constant level of device imprecision and achieves a linear key rate.

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“…This enables improvements in the state of the art projection PCP theorem, while bypassing some of the difficulty with general parallel repetition. There also has been a lot of work around parallel repetition for games with entanglement [7,15,11,14,6]. …”

Section: Previous Workmentioning

confidence: 99%

Small Value Parallel Repetition for General Games

Braverman

Garg

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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We prove a parallel repetition theorem for general games with value tending to 0. Previously Dinur and Steurer proved such a theorem for the special case of projection games. We use information theoretic techniques in our proof. Our proofs also extend to the high value regime (value close to 1) and provide alternate proofs for the parallel repetition theorems of Holenstein and Rao for general and projection games respectively. We also extend the example of Feige and Verbitsky to show that the small-value parallel repetition bound we obtain is tight. Our techniques are elementary in that we only need to employ basic information theory and discrete probability in the small-value parallel repetition proof.

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A Parallel Repetition Theorem for Entangled Two-Player One-Round Games under Product Distributions

Cited by 28 publications

References 29 publications

A Parallel Repetition Theorem for Entangled Projection Games

A Parallel Repetition Theorem for Entangled Projection Games

Parallel Device-Independent Quantum Key Distribution

Small Value Parallel Repetition for General Games

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