. A reasonable conjecture for the answer would be \sum_{m \geq k} {n \choose m} p^m (1-p)^{n-m}, as that is the answer when it is optimal for the prover to act independently. This is known to be the correct answer when k=n, and also in the classical case. It is also correct in some generalizations of the classical case that we will discuss later. We will show how this cannot be extended to all cases, presenting an example of an interaction with k=1,n=2 in which p\approx 0.85, but it is possible to always win at least once. We will then give some upper bounds on the optimal probability for the prover of winning k times out of n parallel repetitions. These bounds are expressed as a function of p. Finally, we will connect our results to the study of error reduction for quantum interactive proofs using parallel repetition.

DOI: 10.48550/arxiv.1203.4885

Related papers

Parallel Repetition of Prover-Verifier Quantum Interactions

Abel Molina

In this thesis, we answer several questions about the behaviour of prover-verifier interactions under parallel repetition when quantum information is allowed, and the verifier acts independently in them. \n \nWe first consider the case in which a value is associated with each of the possible outcomes of an interaction. We prove that it is not possible for the prover to improve on the optimum average value per repetition by repeating the protocol multiple times in parallel. \n \nWe look then at games in which the outcomes are classified into two types, winning outcomes and losing outcomes. We ask what is the optimal probability for the prover of winning at least k times out of n parallel repetitions, given that the optimal probability of winning when only one repetition is considered is p. A reasonable conjecture for the answer would be the answer when it is optimal for the prover to act independently. This is known to be the correct answer when k=n. We will show how this cannot be extended to the general case, presenting an example of an interaction with k=1,n=2 in which p is approximately 0.85, but it is possible to always win at least once. We will then give some upper bounds on the optimal probability for the prover of winning k times out of n parallel repetitions. These bounds are expressed as a function of p. \n \nFinally, we connect our results to the study of error reduction for quantum interactive proofs using parallel repetition.In this thesis, we answer several questions about the behaviour of prover-verifier interactions under parallel repetition when quantum information is allowed, and the verifier acts independently in them. \n \nWe first consider the case in which a value is associated with each of the possible outcomes of an interaction. We prove that it is not possible for the prover to improve on the optimum average value per repetition by repeating the protocol multiple times in parallel. \n \nWe look then at games in which the outcomes are classified into two types, winning outcomes and losing outcomes. We ask what is the optimal probability for the prover of winning at least k times out of n parallel repetitions, given that the optimal probability of winning when only one repetition is considered is p. A reasonable conjecture for the answer would be the answer when it is optimal for the prover to act independently. This is known to be the correct answer when k=n. We will show how this cannot be extended to the general case, presenting an example of an interaction with k=1,n=2 in which p is approximately 0.85, but it is possible to always win at least once. We will then give some upper bounds on the optimal probability for the prover of winning k times out of n parallel repetitions. These bounds are expressed as a function of p. \n \nFinally, we connect our results to the study of error reduction for quantum interactive proofs using parallel repetition.

Jan 1, 20110 citations

Quantum interaction can superactivate cheating under parallel repetition

Archishna Bhattacharyya, Laura Mančinska, Yuming Zhao

We study interactive multiprover games and many-round protocols in which the communication between the verifier and the provers is quantum. Parallel repetition is known to suppress soundness error of classical two-prover games arbitrarily close to zero, as shown by Raz (STOC '95), even with quantum strategies as shown by Yuen (ICALP '16), and Bavarian, Vidick and Yuen (STOC '17). Yet, we show that quantum communication can have the opposite, unexpected effect. Specifically, we exhibit a one-round quantum game and two-round $\mathsf{QMIP}$ protocols whose local (unentangled) value is strictly less than one for a single instance, yet equals one under $n$ -fold parallel repetition for every $n \geq 2$ . The key is that parallel repetition not only imposes additional winning conditions but also supplies additional quantum resources as entanglement in the exchanged states can be exploited jointly across copies. Our multiround examples build on superactivation of zero-error capacities of quantum channels. To establish this connection, we introduce quantum games and interactive protocols which capture the one-shot zero-error classical and quantum capacities, both with and without entanglement assistance. Our constructions use quantum-state verification and a teleportation-based reduction from two rounds to one. In contrast, when shared entanglement is allowed, we show that parallel repetition cannot increase the entangled value when at most $3$ messages are exchanged, consistent with classical results of Bellare, Impagliazzo and Naor (FOCS '97), and that of single-prover systems by Kitaev and Watrous (STOC '00). Combining this monotonicity with our quantum game-channel correspondence, we show that entanglement-assisted zero-error classical and quantum capacities cannot be superactivated.We study interactive multiprover games and many-round protocols in which the communication between the verifier and the provers is quantum. Parallel repetition is known to suppress soundness error of classical two-prover games arbitrarily close to zero, as shown by Raz (STOC '95), even with quantum strategies as shown by Yuen (ICALP '16), and Bavarian, Vidick and Yuen (STOC '17). Yet, we show that quantum communication can have the opposite, unexpected effect. Specifically, we exhibit a one-round quantum game and two-round $\mathsf{QMIP}$ protocols whose local (unentangled) value is strictly less than one for a single instance, yet equals one under $n$-fold parallel repetition for every $n \geq 2$. The key is that parallel repetition not only imposes additional winning conditions but also supplies additional quantum resources as entanglement in the exchanged states can be exploited jointly across copies. Our multiround examples build on superactivation of zero-error capacities of quantum channels. To establish this connection, we introduce quantum games and interactive protocols which capture the one-shot zero-error classical and quantum capacities, both with and without entanglement assistance. Our constructions use quantum-state verification and a teleportation-based reduction from two rounds to one. In contrast, when shared entanglement is allowed, we show that parallel repetition cannot increase the entangled value when at most $3$ messages are exchanged, consistent with classical results of Bellare, Impagliazzo and Naor (FOCS '97), and that of single-prover systems by Kitaev and Watrous (STOC '00). Combining this monotonicity with our quantum game-channel correspondence, we show that entanglement-assisted zero-error classical and quantum capacities cannot be superactivated.

quant-pharXiv:2609.31223Sep 28, 20260 citations

Concentration Bounds from Parallel Repetition Theorems

Taylor Hornby

This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions: \n \n- A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances). \n \n- Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies. \n \n- A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case.This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions: \n \n- A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances). \n \n- Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies. \n \n- A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case.

Aug 22, 20181 citation

Quantum Information and Variants of Interactive Proof Systems

Sarvagya Upadhyay

For nearly three decades, the model of interactive proof systems and its variants have been central to many important and exciting developments in computational complexity theory such as exact characterization of some well known complexity classes, development of probabilistically checkable proof systems and theory of hardness of approximation, and formalization of fundamental cryptographic primitives. \n \nOn the other hand, the theory of quantum information, which is primarily concerned with harnessing quantum mechanical features for algorithmic, cryptographic, and information processing tasks has found many applications. 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Aug 24, 20110 citations

Quantum Hedging in Two-Round Prover-Verifier Interactions

Srinivasan Arunachalam, Abel Molina, Russo, Vincent

We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely offset the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in arXiv:1104.1140. Exploring in a different direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely offset the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in arXiv:1104.1140. Exploring in a different direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.

arXiv:1310.7954Leibniz international proceedings in informaticsOct 29, 20134 citations

No Strong Parallel Repetition with Entangled and Non-signaling Provers

Julia Kempe, Oded Regev

We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: If the game is played H times in parallel, does the maximum winning probability decay exponentially in ℓ? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form (1 - ⊖ (ε))ℓwhere 1 - ε is the value of the game and H is the number of repetitions. This question is known as the strong parallel repetition question and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does not hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact perfect) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form (1 - ⊖ (ε2))ℓ. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: If the game is played H times in parallel, does the maximum winning probability decay exponentially in ℓ? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form (1 - ⊖ (ε))ℓwhere 1 - ε is the value of the game and H is the number of repetitions. This question is known as the strong parallel repetition question and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does not hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact perfect) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form (1 - ⊖ (ε2))ℓ. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.

arXiv:0911.0201IEEE Conference on Computational Complexity (CCC)Jun 1, 201033 citations

p

\mathsf{QMIP}

protocols whose local (unentangled) value is strictly less than one for a single instance, yet equals one under

n

-fold parallel repetition for every

n \geq 2

. The key is that parallel repetition not only imposes additional winning conditions but also supplies additional quantum resources as entanglement in the exchanged states can be exploited jointly across copies. Our multiround examples build on superactivation of zero-error capacities of quantum channels. To establish this connection, we introduce quantum games and interactive protocols which capture the one-shot zero-error classical and quantum capacities, both with and without entanglement assistance. Our constructions use quantum-state verification and a teleportation-based reduction from two rounds to one. In contrast, when shared entanglement is allowed, we show that parallel repetition cannot increase the entangled value when at most

3

messages are exchanged, consistent with classical results of Bellare, Impagliazzo and Naor (FOCS '97), and that of single-prover systems by Kitaev and Watrous (STOC '00). Combining this monotonicity with our quantum game-channel correspondence, we show that entanglement-assisted zero-error classical and quantum capacities cannot be superactivated.