[Paper Review] On the Power of Entangled Provers: Immunizing games against entanglement

Julia Kempe, Hirotada Kobayashi+3 more|arXiv (Cornell University)|Apr 23, 2007

Cryptography and Data Security39 references11 citations

TL;DR

This paper introduces two methods to make multi-prover classical games resistant to entangled provers: using quantum communication with a quantum verifier, or adding an extra prover. It establishes that NEXP ⊆ QMIP1,s(2,1) and NEXP ⊆ MIP1,s(3,1) with soundness 1−2−poly(n), and PSPACE ⊆ MIP1,s(2,1) with soundness 1−1/poly(n), providing the first non-trivial bounds in this setting and showing that entangled prover game values cannot be computed via polynomial-sized semidefinite programs unless P=NP.

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.

Motivation & Objective

• To address the challenge of entangled provers undermining the soundness of multi-prover interactive proof systems.
• To develop generic techniques that make classical games resistant to quantum entanglement.
• To establish new complexity-theoretic bounds for proof systems with entangled provers.
• To demonstrate limitations of semidefinite programming in computing the value of entangled prover games.

Proposed method

• Using quantum communication and a quantum verifier to prevent entangled provers from coordinating effectively.
• Introducing an additional prover to disrupt entanglement-based strategies in multi-prover games.
• Leveraging quantum information techniques to design games where entanglement provides no advantage.
• Applying complexity-theoretic reductions to show that NEXP and PSPACE are contained in specific classes of interactive proof systems with entangled provers.
• Using soundness analysis to bound the advantage of entangled provers in the constructed games.
• Establishing that polynomial-sized semidefinite programs cannot compute the value of these games unless P=NP.

Experimental results

Research questions

• RQ1 Can multi-prover classical games be made robust against entangled provers using quantum verification or additional provers?
• RQ2 What are the complexity-theoretic implications of making games resistant to entanglement?
• RQ3 Can the value of entangled prover games be computed efficiently using semidefinite programming?
• RQ4 What is the precise relationship between NEXP, PSPACE, and proof systems with entangled provers?
• RQ5 Are there inherent limitations to semidefinite programming in solving entangled prover games?

Key findings

• NEXP is contained in QMIP1,s(2,1) with soundness 1−2−poly(n), demonstrating that entangled provers can verify NEXP languages with high soundness.
• NEXP is also contained in MIP1,s(3,1) with soundness 1−2−poly(n), showing that adding a third prover can immunize games against entanglement.
• PSPACE is contained in MIP1,s(2,1) with soundness 1−1/poly(n), providing the first non-trivial bound for PSPACE in this setting.
• The results imply that unless P=NP, the value of entangled prover games cannot be computed by polynomial-sized semidefinite programs.
• The paper establishes that quantum communication and additional provers are effective tools for immunizing games against entanglement.
• These findings represent the first non-trivial complexity-theoretic bounds for games with entangled provers in the MIP* and QMIP frameworks.