Quantum Hedging in Two-Round Prover-Verifier Interactions

We consider the problem of quantum correlations that arise in 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 [25] 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 [25]. Motivated by implementation concerns, 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, and show that then there is no possible hedging. 1 Overview and motivation The interactions we study consist of parallel repetitions of a game played between players Alice and Bob, also referred to as the verifier and prover respectively. The setting of the game is: 1. Alice prepares a question, and sends this question to Bob. 2. Bob generates an answer, and sends it back to Alice. 3. Alice evaluates this answer and decides if Bob wins or loses. It is assumed that Bob has complete knowledge of Alice’s specification, including both the method used to determine Alice’s question and the criteria that she uses to determine if Bob has won or lost the game. Molina and Watrous [25] consider a specific instance of this setting where Alice sends half of a 2-qubit EPR state 1 √ 2 |00〉 + 1 √ 2 |11〉 to Bob. Bob replies with a qubit and ∗This work was partially supported by Canada’s NSERC, the US ARO, the ERC Consolidator Grant QPROGRESS, the Mike and Ophelia Lazaridis Fellowship program, and the David R. Chariton Graduate Scholarship program. 1 ar X iv :1 31 0. 79 54 v4 [ qu an tph ] 2 6 Se p 20 16 Alice evaluates Bob’s answer by measuring his qubit and the second half of the EPR state against the state cos(π/8) |00〉 + sin(π/8) |11〉. A victory for Bob corresponds to obtaining cos(π/8) |00〉 + sin(π/8) |11〉 as the outcome of the measurement. When Alice and Bob play two repetitions of this game in parallel, Molina and Watrous [25] show that there exists a strategy for Bob that guarantees he wins at least one of the two repetitions with probability 1. However, when the game is played once, the probability that Bob wins is at most cos(π/8) ≈ 0.8536. Playing two repetitions in parallel leads then to a hedging phenomenon, where if Bob wants to decrease his chance of losing both repetitions, he can do so by not playing each game independently and optimally. This hedging is also perfect, in the sense that Bob can completely offset the risk of losing both games. This is a completely quantum phenomenon, with no classical counterpart. Indeed, when classical information is considered, and for any game that fits the setting we study, it is immediate to show that when Bob wants to win at least k out of n parallel repetitions, it is optimal for him to play independently (however, this is not the case when considering multiple provers [13, 12, 19]). This establishes the non-triviality of the set of outcome distributions that are possible to obtain from parallel repetition of the games that we study, when compared to the classical case. In particular, it immediately illustrates that the technique of parallel repetition cannot be used to trivially achieve strong error reduction for the complexity class QIP(2), a class studied for example in [18, 28, 33, 20]. The quantum hedging phenomenon is also an example where the quantum version of a game produces outcomes unachievable by its classical counterpart. Most famously considered by Bell [4], this type of violation has been observed in a number of game-like frameworks [9, 24, 26, 10, 5, 29, 11]. It is natural then to ask how general is the hedging phenomenon, both qualitatively and quantitatively. A complete understanding of this question would allow us to characterize the outcome distributions that can arise from Alice and Bob playing n parallel repetitions of a prover-verifier game in our setting. Consequently, it could lead to a protocol for achieving error reduction via parallel repetition for QIP(2) simpler than the one currently known [20]. The techniques used to achieve such an understanding could conceivably also extend to the analysis of prover-verifier games involving further rounds of communication, and more generally to other kinds of multi-party quantum interactions. This would lead to results for the corresponding complexity classes (and likely also for their classical parallels) about error reduction by parallel repetition. Taking a step towards such a complete understanding, we consider in Section 3 a 2-parameter generalization of the game in [25], and characterize when Bob can guarantee that he wins at least 1 out of n parallel repetitions, for every n. We give optimal strategies for Bob to win at least 1 out of n parallel repetitions, both when perfect hedging is possible and not possible. It is natural as well to consider the possibility of implementing a game that exhibits quantum hedging using existing quantum information processing devices. One possible choice would be to use optical quantum devices, but the immediate concern arises of how to account for the fact that photon losses will often occur, leading to a communication error between Alice and Bob. Even if one chose another implementation method where communication is more reliable, one would still need to consider the fact that communication errors can occur. To do so, we consider a formalism in Section 4 that takes into account communication errors, and prove that under our formalism, quantum hedging is not possible. To model communication errors, we assume that Alice cannot distinguish a communication error from Bob choosing not to return an answer. Therefore, our formalism simply allows for the possibility that Bob chooses not to return an answer, in which case the game is repeated. Bob choosing at random in our formalism whether to return an answer or not would correspond to genuine communication errors, while Bob strategizing about when to return an answer would correspond to Bob using communication errors as an excuse to avoid a losing outcome. This choice of framework can also be seen as adding postselection to our