Tensor norms and the classical communication complexity of nonlocal quantum measurement

SIAM Journal on Computing

Volumn 38, Issue 3, 2008, Pages 753-766

  Shi, Yaoyun ^a   Zhu, Yufan ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Bell Inequality; Classical Simulation; Communication Complexity; Quantum entanglement; Tensor Norms

Indexed keywords

BELL INEQUALITY; CLASSICAL COMMUNICATIONS; CLASSICAL SIMULATION; COMMUNICATION COMPLEXITY; ENTANGLED QUANTUM STATES; GENERAL UPPER BOUNDS; LOCAL MEASUREMENTS; NON LOCALITIES; OUTPUT DISTRIBUTIONS; QUANTUM MEASUREMENTS; SHARED ENTANGLEMENTS; TENSOR NORMS;

DISTRIBUTION FUNCTIONS; TENSORS;

COMMUNICATION;

EID: 55249085625     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/050644768     Document Type: Article

References (44)

1. D. AHARONOV, A. KITAEV, AND N. NISAN, Quantum circuits with mixed states, in Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 1998, pp. 20-30.
2. D. BACON AND B. F. TONER, Bell inequalities with auxiliary communication, Phys. Rev. Lett., 90 (2003), 157904.
3. J. S. BELL, On the Einstein-Podolsky-Rosen paradox, Physics, 1 (1964), pp. 195-200.
4. C. H. BENNETT, G. BRASSARD, C. CRÉPEAU, R. JOZSA, A. PERES, AND W. K. WOOTTERS, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett., 70 (1993), pp. 1895-1899.
5. C. H. BENNETT AND G. BRASSARD, Quantum cryptography: Public key distribution and cointossing, in Proceedings of the IEEE international Conference on Computers, Systems and Signal Processing (Bangalore, India), IEEE Press, New York, 1984, pp. 175-179.
6. C. H. BENNETT, A. W. HARROW, D. W. LEUNG, AND J. A. SMOLIN, On the capacities of bipartite Hamiltonians and unitary gates, IEEE Trans. Inform. Theory, 49 (2003), pp. 1895-1911.
7. D. BOHM, The Paradox of Einstein, Rosen, and Podolsky, Quantum Theory and Measurement, Prentice-Hall, Englewood Cliffs, NJ, 1951, pp. 611-623.
8. G. BRASSARD, R. CLEVE, AND A. TAPP, Cost of exactly simulating quantum entanglement with classical communication, Phys. Rev. Lett., 83 (1999), pp. 1874-1877.
9. H. BUHRMAN, R. CLEVE, AND W. VAN DAM, Quantum entanglement and communication complexity, SIAM J. Compute 30 (2001), pp. 1829-1841.
10. H. BUHRMAN, R. CLEVE, J. WATROUS, AND R. DE WOLF, Quantum fingerprinting, Phys. Rev. Lett., 87 (2001), 167902.
11. A. M. CHILDS, H. L. HASELGROVE, AND M. A. NIELSEN, Lower bounds on the complexity of simulating quantum gates, Phys. Rev. A, 68 (2003), pp. 52311-52316.
12. A. M. CHILDS, D. W. LEUNG, F. VERSTRAETE, AND G. VIDAL, Asymptotic entanglement capacity of the Ising and anisotropic Heisenberg interactions, Quantum Inf. Compute., 3 (2003), pp. 97-105.
13. R. CLEVE AND H. BUHRMAN, Substituting quantum entanglement for communication, Phys. Rev. A, 56 (1997), pp. 1201-1204.
14. R. CLEVE, W. VAN DAM, M. NIELSEN, AND A. TAPP, Quantum entanglement and the communication complexity of the inner product function, in Quantum Computing and Quantum Communications, Lecture Notes in Comput. Sci. 1509, Springer, Berlin, 1999, pp. 61-74.
15. J. A. CSIRIK, Cost of exactly simulating a bell pair using classical communication, Phys. Rev. A, 66 (2002), 014302.
16. A. DEFANT AND K. FLORET, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland., Amsterdam, 1993.
17. A. EINSTEIN, B. PODOLSKY, AND N. ROSEN, Can quantum-mechanical description of reality be considered complete?, Phys. Rev., 47 (1935), pp. 777-780.
18. A. K. EKERT, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett., 67 (1991), pp. 661-663.
19. D. GAVINSKY, On the role of shared entanglement, Quantum Inform. Comput., to appear. Available online at http://www.arxiv.org/ebs/quant- ph/0604052.
20. M. X. GOEMANS AND D. P. WILLIAMSON, .879-approximation algorithms for MAX CUT and MAX 2SAT, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 1994, pp. 422-431.
21. D. R. JOCIĆ, The Cauchy-Schwarz norm inequality for elementary operators in Schatten ideals, J. London Math. Soc., 60 (1999), pp. 925-934.
22. A. KITAEV, AND J. WATROUS, Parallelization, amplification, and exponential time simulation of quantum interactive proof systems, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 2000, pp. 608-617.
23. A. Y. KITAEV, A. H. SHEN, AND M. N. VYALYI, Classical and Quantum Computation, Grad. Stud. Math, 47, AMS, Providence, RI, 2002. Translated by Lester J. Senechal from the 1999 Russian original.
24. H. KLAUCK, Lower bounds for quantum communication complexity, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), B. Werner, ed., IEEE Press, Piscataway, NJ, 2001, pp. 288-297.
25. I. KREMER, Quantum Communication, Master's thesis, Computer Science Department, Hebrew University, Jerusalem, Israel, 1995.
26. I. KREMER, N. NISAN, AND D. RON, On randomized one-round communication complexity, in Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 1995, pp. 596-605.
27. E. KUSHILEVITZ AND N. NISAN, Communication Complexity, Cambridge University Press, Cambridge, UK, 1997.
28. H.-K. LO AND H. F. CHAU, Unconditional security of quantum key distribution over arbitrarily long distances, Science, 283 (1999), pp. 2050-2056.
29. T. MAUDLIN, Bell's inequality, information transmission and prism Models, in the Proceedings of PSA, Vol, 1 D. Hull, M. Forbes, and K. Okrnhlik, eds., Philosophy of Science Association, Bloomsburg, PA, 1992, pp. 404-417.
30. D. MAYERS, Unconditional security in quantum cryptography, J. ACM, 48 (2001), pp. 351-406.
31. A. NAYAK AND J. SALZMAN, On communication over an entanglement-assisted quantum channel, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 2002, pp. 698-704.
32. I. NEWMAN, Private vs. common random bits in communication complexity, Inform. Process. Lett., 39 (1991), pp. 67-71.
33. M. A. NIELSEN AND I. L. CHUANG, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000.
34. J. PRESKILL, Quantum Information and Computation, Lecture Notes for course Physics 229, Caltech, Pasadena, CA, 2007; available online at http://www.theory.altech.edu/̃preskill/ph229.
35. R. RAZ, Exponential separation of quantum and classical communication complexity, in Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 1999, pp. 358-367.
36. A. A. RAZBOROV, Quantum communication complexity of symmetric predicates, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), pp. 145-159 (in Russian). English translation available at http://genesis.mi.ras.ru/ ̃razborov/qcc_eng.ps.
37. O. RUDOLPH, A separability criterion for density operators, J. Phys. A., 33 (2000). pp. 3951-3955.
38. O. RUDOLP, Further results on the cross norm criterion for separability, Quantum Inform. Process., 4 (2005), pp. 219-239.
39. M. STEINER, Towards quantifying non-local information transfer: Finite-bit non-locality, Phys. Lett. A, 270 (2000), pp, 239-244.
40. W. TITTEL, J. BRENDEL, H. ZBINDEN, AND N. GISIN, Violation of Bell inequalities by photons more than 10 km apart, Phys. Rev. Lett., 81 (1998), pp. 3563-3566.
41. B. TONER AND D. BACON, Communication cost of simulating Bell correlations, Phys Rev Lett., 91 (2003), 187904.
42. A. C.-C. YAO, Some complexity questions related to distributive computing, in Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 1979, pp. 209-213.
43. A. C.-C. YAO, Quantum circuit complexity, in Proceedings of the 34th Annual Symposium on Foundations of Computer Science (FOCS), IEEE Press, Piscataway, NJ, 1993, pp. 352-361.
44. A. C.-C. YAO, On the power of quantum fingerprinting, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 2003, pp. 77-81.