In this work, we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value ω*. We show that the value ω* can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of ω* under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication ω qow . In particular, we show that ω qow can be computed efficiently and it satisfies a perfect parallel repetition theorem.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Arora S., Lund C., Motwani R., Sudan M., Szegedy M. (1998) Proof verification and the hardness of approximation problems. Journal of the ACM 45(3): 501–555
Arora S., Safra S. (1998) Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM 45(1): 70–122
M. Ben-Or, M. Hassidim & H. Pilpel (2008). Quantum Multiprover Interactive Proofs with Communicating Provers. In Proceedings of 49th IEEE Symposium on Foundations of Computer Science, 467–476.
H. Buhrman, N. Chandran, S. Fehr, R. Gelles, V. Goyal, R. Ostrovsky & C. Schaffner (2011). Position-Based Quantum Cryptography: Impossibility and Constructions. In Advances in Cryptology - CRYPTO 2011: 31st Annual Cryptology Conference, Proceedings, number 6841 in Lecture Notes in Computer Science, 429–446.
Chan J. T. (1994) Facial Structure of the Trace Class. Arch. Math.64(3): 185–187
R. Cleve, W. Slofstra, F. Unger & S. Upadhyay (2007). Strong parallel repetition theorem for quantum XOR proof systems. In Proceedings of 22nd IEEE Conference on Computational Complexity, 501–555.
Devetak I., Junge M., King C., Ruskai M.B. (2006) Multiplicativity of completely bounded p-norms implies a new additivity result. Comm. Math. Phys.266(1): 37–63
E. Effros & Z.-J. Ruan (2000). Operator Spaces, volume 23 of London Mathematical Society Monographs New Series. Oxford University Press, Oxford.
U. Feige (1991). On the success probability of two provers in one-round proof systems. In Proceedings of 6th IEEE Structure in Complexity Theory, 116–123.
M. Grötschel, L. Lovasz & A. Schrijver (1993). Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin.
G. Gutoski (2010). Quantum Strategies and Local Operations. Technical report arXiv:1003.0038.
Haagerup U., Musat M. (2007) On the best constants in noncommutative Khintchine-type inequalities. J. Funct. Anal.250(2): 588–624
Haagerup U., Musat M. (2008) The Effros-Ruan conjecture for bilinear forms on C*-algebras. Invent. Math.174: 139–163
T. Ito, H. Kobayashi & K. Matsumoto (2009). Oracularization and two-prover one-round interactive proofs against nonlocal strategies. In Proceedings of 24th Annual IEEE Conference on Computational Complexity, 217–228.
N. Johnston, D. W. Kribs & V. I. Paulsen (2009). Computing Stabilized Norms for Quantum Operations via the Theory of Completely Bounded Maps. Quantum Inf. Comput. 9 (1-2), 16–35.
Johnston N., Kribs D.W., Paulsen V.I., Pereira R. (2011) Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory. J. Funct. Anal.260: 2407–2423
Junge M. (2005) Embedding of the operator space OH and the logarithmic “little Grothendieck inequality”. Invent. Math. 161 (2): 225–286
M. Junge, M. Navascues, C. Palazuelos, D. Pérez-García, V. B. Scholz & R. F. Werner (2011). Connes’ embedding problem and Tsirelson’s problem. J. Math. Phy. 52, 012 102.
Junge M., Palazuelos C. (2011) Large violation of Bell inequalities with low entanglement. Comm. Math. Phys. 306(3): 695–746
M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva & M.M. Wolf (2010). Operator Space theory: a natural framework for Bell inequalities. Phys. Rev. Lett. 104, 170 405.
Junge M., Xu Q. (2010) Representation of certain homogeneous Hilbertian operator spaces and applications. Invent. Math. 179(1): 75–118
J. Kempe, H. Kobayashi, K. Matsumoto, B. Toner & T. Vidick (2008a). Entangled Games are Hard to approximate. In Proceedings of 49th IEEE Symposium on Foundations of Computer Science, 447–456.
J. Kempe & O. Regev (2010). No Strong Parallel Repetition with Entangled and Non-signaling Provers. In Proceedings of 25th Annual IEEE Conference on Computational Complexity, 7–15.
J. Kempe, O. Regev & B. Toner (2008b). Unique Games with Entangled Provers are Easy. In Proceedings of 49th IEEE Symposium on Foundations of Computer Science, 457–466.
J. Kempe & T. Vidick (2011). Parallel Repetition of Entangled Games. In 43th Annual ACM Symposium on Theory of Computing, 353–362.
Kitaev A. (1997) Quantum computations: Algorithms and error correction. Russian Math. Surveys 52(6): 1191–1249
A. Kitaev & J. Watrous (2000). Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In 32nd Annual ACM Symposium on Theory of Computing, 608–617.
Kobayashi H., Matsumoto K. (2003) Quantum multi-prover interactive proof systems with limited prior entanglement. J. Comput. Syst. Sci. 66(3): 429–450
D. Leung, B. Toner & J. Watrous (2008). Coherent state exchange in multi-prover quantum interactive proof systems. Technical report arXiv:0804.4118.
Oikhberg T. (2010) Completely bounded and ideal norms of multiplication operators and Schur multipliers. Integr. Equ. Oper. Theory 66(3): 425–440
V. Paulsen (2002). Completely bounded maps and operator algebras, volume 78 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.
Pérez-García D., Wolf M.M., Palazuelos C., Villanueva I., Junge M. (2008) Unbounded violation of tripartite Bell inequalities. Comm. Math. Phys. 279(2): 455–486
G. Pisier (1998). Non-Commutative Vector Valued L p -Spaces and Completely p-Summing Maps. Asterisque 247.
G. Pisier (2003). Introduction to operator space theory, volume 294 of London Mathematical Society Lecture Note Series. Cambridge University Press.
Pisier G. (2012) Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. (N.S.) 49(2): 237–323
Pisier G., Shlyakhtenko D. (2002) Grothendieck’s theorem for operator spaces. Invent. Math.150(1): 185–217
A. Rapaport,Ta-Shma (2007). On the power of quantum, one round, two prover interactive proof systems. Quantum Inf. Process. 6(6), 445–459.
Raz R. (1998) A parallel repetition theorem. SIAM Journal on Computing 27: 763–803
O. Regev & T. Vidick (2012a). Elementary Proofs of Grothendieck Theorems for Completely Bounded Norms. J. Operator Theory.
O. Regev & T. Vidick (2012b). Quantum XOR games. Technical report arXiv:1207.4939
B. Simon (1981). Pointwise domination of matrices and comparison of S p norms. Pacific J. Math. 97, 471Ð475.
Smith R.R. (1983) Completely bounded maps between C*-algebras. J. London Math. Soc.27: 157–166
T. Vidick (2013). Three-player entangled XOR games are NP-hard to approximate. Technical report arXiv:1302.1242.
Watrous J. (2003) PSPACE has constant-round quantum interactive proof systems. Theoretical Computer Science 292(3): 575–588
Watrous J. (2009) Semidefinite programs for completely bounded norms. Theory of Computing 5: 217–238
About this article
Cite this article
Cooney, T., Junge, M., Palazuelos, C. et al. Rank-one quantum games. comput. complex. 24, 133–196 (2015). https://doi.org/10.1007/s00037-014-0096-x