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On the Power of Entangled Provers: Immunizing games against entanglement

Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto, Ben Toner, Thomas Vidick

Published Apr 23, 2007• 10 citations
OpenAlex

Abstract

We describe two generic ways to make multi-prover classical games resistant against entangled provers. The first uses quantum communication and a quantum verifier, the second adds an additional prover. This leads to several new results on the power of proof systems with entangled provers. We show that NEXP ⊆ QMIP1,s(2, 1) and NEXP ⊆ MIP1,s(3, 1) with soundness s = 1− 2− poly(n) and PSPACE ⊆ MIP1,s(2, 1) with soundness s = 1 − 1/ poly(n), providing the first non-trivial bounds in this setting. Moreover, our results imply that, unless P = NP, the value of entangled prover games cannot be computed by semi-definite programs that are polynomial in the size of the verifier’s system, a method that has been successful for more restricted quantum games. ∗also at CNRS & LRI, Univerite de Paris-Sud, Orsay, France †Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848 and by an Alon Fellowship of the Israeli Higher Council of Academic Research. ‡Part of this work was completed at Caltech. Supported by the National Science Foundation under Grants PHY-0456720 and CCF-0524828, by EU project QAP, and by NWO VICI project 639-023-302. Part of this research has been funded by the Dutch BSIK/BRICKS project. §Work partly done while at LRI, Univ. de Paris-Sud, Orsay.

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We describe two generic ways to make multi-prover classical games resistant against entangled provers. The first uses quantum communication and a quantum verifier, the second adds an additional prover. This leads to several new results on the power of proof systems with entangled provers. We show that NEXP ⊆ QMIP1,s(2, 1) and NEXP ⊆ MIP1,s(3, 1) with soundness s = 1− 2− poly(n) and PSPACE ⊆ MIP1,s(2, 1) with soundness s = 1 − 1/ poly(n), providing the first non-trivial bounds in this setting. Moreover, our results imply that, unless P = NP, the value of entangled prover games cannot be computed by semi-definite programs that are polynomial in the size of the verifier’s system, a method that has been successful for more restricted quantum games. ∗also at CNRS & LRI, Univerite de Paris-Sud, Orsay, France †Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848 and by an Alon Fellowship of the Israeli Higher Council of Academic Research. ‡Part of this work was completed at Caltech. Supported by the National Science Foundation under Grants PHY-0456720 and CCF-0524828, by EU project QAP, and by NWO VICI project 639-023-302. Part of this research has been funded by the Dutch BSIK/BRICKS project. §Work partly done while at LRI, Univ. de Paris-Sud, Orsay.
We show that the value of a general two-prover quantum game cannot be computed by a semi-definite program ofvpolynomial size (unless P=NP), a method that has been successful in more restricted quantum games. More precisely, we show that proof of membership in the NP-complete problem GAP-3D-Matching can be obtained by a 2-prover, 1-round quantum interactive proof system where the provers share entanglement, with perfect completeness and soundness s=1-2^(-O(n)), and such that the space of the verifier and the size of the messages are O(log n). This implies that QMIP^*_{log n,1,1-2^(-O(n))} \nsubseteq P unless P = NP and provides the first non-trivial lower bound on the power of entangled quantum provers, albeit with an exponentially small gap. The gap achievable by our proof system might in fact be larger, provided a certain conjecture on almost commuting versus nearly commuting projector matrices is true.
We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1−1/ poly1-1/\,\mathrm{poly}1−1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP\mathrm{P}=\mathrm{NP}P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.
Multi Prover Interactive Proof systems (MIPs)were first presented in a cryptographic context, but ever since they were used in various fields. Understanding the power of MIPs in the quantum context raises many open problems, as there are several interesting models to consider. For example, one can study the question when the provers share entanglement or not, and the communication between the verifier and the provers is quantum or classical. While there are several partial results on the subject, so far no one presented an efficient scheme for recognizing NEXP (or NP with logarithmic communication), except for [KM03], in the case there is no entanglement (and of course no communication between the provers). We introduce another variant of Quantum MIP, where the provers do not share entanglement, the communication between the verifier and the provers is quantum, but the provers are unlimited in the classical communication between them. At first, this model may seem very weak, as provers who exchange information seem to be equivalent in power to a simple prover. This in fact is not the case - we show that any language in NEXP can be recognized in this model efficiently, with just two provers and two rounds of communication, with a constant completeness-soundness gap.
This paper presents three results on the power of two-prover one-round interactive proof systems based on oracularization under the existence of prior entanglement between dishonest provers. It is proved that the two-prover one-round interactive proof system for PSPACE by Cai, Condon, and Lipton [JCSS 48:183-193, 1994] still achieves exponentially small soundness error in the existence of prior entanglement between dishonest provers (and more strongly, even if dishonest provers are allowed to use arbitrary no-signaling strategies). It follows that, unless the polynomial-time hierarchy collapses to the second level, two-prover systems are still advantageous to single-prover systems even when only malicious provers can use quantum information. It is also shown that a "dummy" question may be helpful when constructing an entanglement-resistant multi-prover system via oracularization. This affirmatively settles a question posed by Kempe et al. [FOCS 2008, pp. 447-456] and every language in NEXP is proved to have a two-prover one-round interactive proof system even against entangled provers, albeit with exponentially small gap between completeness and soundness. In other words, it is NP-hard to approximate within an inverse-polynomial the value of a classical two-prover one-round game against entangled provers. Finally, both for the above proof system for NEXP and for the quantum two-prover one-round proof system for NEXP proposed by Kempe et al., it is proved that exponentially small completeness-soundness gaps are best achievable unless soundness analysis uses the structure of the underlying system with unentangled provers.
We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) quantum verifier and two entangled provers or (ii) classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation to within a constant. We start our proof by describing two ways to modify classical multi-prover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be ``rounded'' to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.
We introduce a new variant of quantum multiprover interactive proofs (QMIP) where the provers and the verifier are quantum. The verifier can exchange quantum messages with the provers. The provers cannot communicate quantumly between themselves and do not share entanglement, but are unlimited in the classical communication between them, even after receiving messages from the verifier. We show that any language in nondeterministic exponential time (NEXP) can be recognized in this model efficiently, with just two provers and two rounds of communication, and with a constant completeness/soundness gap. This is in contrast to the result of [R. Jain et al., Comm. ACM, 53 (2010), pp. 102--109], which shows that QIP = PSPACE, or equivalently that the set of languages that can be recognized by a quantum verifier communicating with a single quantum prover is equal to PSPACE. To analyze the cheating power of the provers, we give them more power and allow them to perform any separable operation. We then show a unique two-phase protocol in which the provers first commit to a superposition of correct answers to all possible questions, and then in the second phase the verifier opens up the committed answer and checks for correctness and consistency.