Communication complexity is a task for which quantum information can beat classical informa-tion. Such tasks are rare, and finding more potential applications of quantum information is very important.
Unfortunately most quantum communication complexity problems are either extremely sensitive to noise, highly contrived, or do not offer exponential gains over the best classical protocols (in which case the advantages of quantum communication will probably be more than offset by the lower cost and higher speed of classical communication). The most interesting proposal so far is maybe the SMP model without shared randomness (a somewhat contrived model) where equality (a very natural problem) can be solved exponentially more efficiently using quantum communication.
Thus there is the tantalizing possibility that some time in the future, quantum communication complexity could be used in practical applications.
Independently of whether quantum communication complexity ever finds some real-world ap-plications, the results obtained so far have important conceptual implications. First of all they offer new insights into the power of quantum information, and in particular of quantum computing.
Indeed the basic aim of computer science, taken in a wide sense, is to accomplish a task by using the minimum amount of resources. In the usual formulation, the resource that we want to minimize is the running time of the computer. This is the most important application of quantum computing as Shor’s algorithm suggests that a quantum computer would allow exponential speedups. But in this context it is very difficult—if not impossible—to prove that quantum computers are more powerful than classical computers. The advantage of quantum computation can however be proven in simpler contexts such as the black-box model of quantum computing, where the resource that is quantified is the number of calls to an oracle; or communication complexity where the resource that is quantified is the amount of communication. The existence of these models where it can be
rigorously shown that quantum information offers important advantages over classical information reinforces our confidence that quantum computers are much more powerful than classical computers for certain tasks.
Second, the study of quantum communication complexity has led to the proposal of new tests of quantum mechanics. Indeed from Bell onwards it was known that if one wants to replace quantum mechanics by a classical model, this classical model would have to use faster than light signalling.
The discovery of fast quantum algorithms suggested that such a classical model would use an exponentially large number of resources. Quantum communication complexity has now advanced to the point where it may be possible to propose experiments in which one can prove that a classical simulation would require exponentially more resources than are used quantum mechanically.
In summary, quantum communication complexity is now a mature field that has led to some fundamental insights into the nature of computation and the foundations of physics.
References
[1] S. Aaronson and A. Ambainis. Quantum search of spatial regions. In Proceedings of 44th IEEE FOCS, pages 200–209, 2003. quant-ph/0303041.
[2] A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani. Device-independent security of quantum cryptography against collective attacks. Physical Review Letters, 98:230501, 2007.
[3] A. Ac´ın, N. Gisin, and B. Toner. Grothendieck’s constant and local models for noisy entangled quantum states. Physical Review A, 73:062105, 2006.
[4] A. Ambainis. Communication complexity in a 3-computer model. Algorithmica, 16(3):298–
301, 1996.
[5] P. K. Aravind. A simple demonstration of Bell’s theorem involving two observers and no probabilities or inequalities. quant-ph/0206070, 2002.
[6] A. Aspect, J. Dalibard, and G. Roger. Experimental test of Bell’s inequalities using time-varying analyzers. Physical Review Letters, 49:1804, 1982.
[7] A. Aspect, Ph. Grangier, and G. Roger. Experimental tests of realistic local theories via Bell’s theorem. Physical Review Letters, 47:460, 1981.
[8] A. Aspect, Ph. Grangier, and G. Roger. Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell’s inequalities. Physical Review Letters, 49:91, 1982.
[9] Z. Bar-Yossef, T. S. Jayram, and I. Kerenidis. Exponential separation of quantum and classical one-way communication complexity. In Proceedings of 36th ACM STOC, pages 128–
137, 2004.
[10] J. Barrett. Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality. Physical Review A, 65(4):042302, Mar 2002.
[11] J. Barrett, L. Hardy, and A. Kent. No signaling and quantum key distribution. Physical Review Letters, 95(1):010503, Jun 2005.
[12] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts. Nonlocal correlations as an information-theoretic resource. Physical Review A, 71:022101, 2005.
[13] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1965.
[14] C. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70:1895–1899, 1993.
[15] C. Bennett and S. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Physical Review Letters, 69:2881–2884, 1992.
[16] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher. Concentrating partial entanglement by local operations. Physical Review A, 53(4):2046–2052, Apr 1996.
[17] C. H. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, pages 175–179, 1984.
[18] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters.
Purification of noisy entanglement and faithful teleportation via noisy channels. Physical Review Letters, 76(5):722–725, Jan 1996.
[19] R. Bhatia. Matrix Analysis. Number 169 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
[20] G. Brassard. Quantum communication complexity. Foundations of Physics, 33(11):1593–
1616, 2003. quant-ph/0101005.
[21] G. Brassard, A. Broadbent, and A. Tapp. Quantum pseudo-telepathy. Foundations of Physics, 35(11):1877–1907, 2005.
[22] G. Brassard, H. Buhrman, N. Linden, A. A. M´ethot, A. Tapp, and F. Unger. Limit on nonlocality in any world in which communication complexity is not trivial. Physical Review Letters, 96:250401, 2006.
[23] G. Brassard, R. Cleve, and A. Tapp. Cost of exactly simulating quantum entanglement with classical communication. Physical Review Letters, 83(9):1874–1877, Aug 1999. arXiv:quant-ph/9901035.
[24] G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. In Quantum Computation and Quantum Information: A Millennium Volume, volume 305 of AMS Contemporary Mathematics Series, pages 53–74. 2002. quant-ph/0005055.
[25] J. Bri¨et, H. Buhrman, and B. Toner. A generalized Grothendieck inequality and entanglement in XOR games. Arxiv preprint arXiv:0901.2009, 2009.
[26] ˇC. Brukner, M. ˙Zukowski, J.-W. Pan, and A. Zeilinger. Bell’s inequalities and quantum communication complexity. Physical Review Letters, 92(12):127901, Mar 2004.
[27] N. Brunner, N. Gisin, S. Popescu, and V. Scarani. Simulation of partial entanglement with no-signaling resources. arXiv:0803.2359, 2008.
[28] N. Brunner, N. Gisin, and V. Scarani. Entanglement and non-locality are different resources.
New Journal of Physics, 7:88, 2005.
[29] N. Brunner, N. Gisin, V. Scarani, and C. Simon. Detection loophole in asymmetric Bell experiments. Physical Review Letters, 98:220403, 2007.
[30] N. Brunner, S. Pironio, A. Acin, N. Gisin, A.A. M´ethot, and V. Scarani. Testing the dimension of Hilbert spaces. Physical Review Letters, 100(21):210503–210503, 2008.
[31] H. Buhrman, R. Cleve, and W. van Dam. Quantum entanglement and communication com-plexity. SIAM Journal on Computing, 30(8):1829–1841, 2001. quant-ph/9705033.
[32] H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf. Quantum fingerprinting. Physical Review Letters, 87(16), September 26, 2001. quant-ph/0102001.
[33] H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and com-putation. In Proceedings of 30th ACM STOC, pages 63–68, 1998. quant-ph/9802040.
[34] H. Buhrman, W. van Dam, P. Høyer, and A. Tapp. Multiparty quantum communication complexity. Physical Review A, 60(4):2737–2741, 1999. quant-ph/9710054.
[35] H. Buhrman, P. Høyer, S. Massar, and H. R¨ohrig. Combinatorics and quantum nonlocality.
Physical Review Letters, 91(047903), 2003. quant-ph/0209052.
[36] H. Buhrman, P. Høyer, S. Massar, and H. R¨ohrig. Multipartite nonlocal quantum correlations resistant to imperfections. Physical Review A, 73(012321), 2006. quant-ph/0410139.
[37] A. Cabello. All versus Nothing inseparability for two observers. Physical Review Letters, 87(1):010403, Jun 2001.
[38] A. Cabello. Bell’s theorem without inequalities and without probabilities for two observers.
Physical Review Letters, 86(10):1911–1914, Mar 2001.
[39] A. Cabello. Stronger two-observer all-versus-nothing violation of local realism. Physical Review Letters, 95(21):210401, 2005.
[40] A. Cabello and J.-A. Larsson. Minimum detection efficiency for a loophole-free atom-photon Bell experiment. Physical Review Letters, 98:220402, 2007.
[41] A. Cabello and A. J. L´opez-Tarrida. Proposed experiment for the quantum guess my number protocol. Physical Review A, 71:020301(R), 2005.
[42] N. J. Cerf, N. Gisin, S. Massar, and S. Popescu. Simulating maximal quantum entanglement without communication. Physical Review Letters, 94:220403, 2005.
[43] B. S. Tsirelson (Cirel’son). Quantum generalizations of Bell’s inequality. Letters in Mathe-matical Physics, 4(2):93–100, 1980.
[44] J. F. Clauser, M. A. Horne, A. Shimoney, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23(15):880–884, 1969.
[45] R. Cleve and H. Buhrman. Substituting quantum entanglement for communication. Physical Review A, 56(2):1201–1204, 1997. quant-ph/9704026.
[46] R. Cleve, W. van Dam, M. Nielsen, and A. Tapp. Quantum entanglement and the com-munication complexity of the inner product function. In Proceedings of 1st NASA QCQC conference, volume 1509 of Lecture Notes in Computer Science, pages 61–74. Springer, 1998.
quant-ph/9708019.
[47] R. Cleve, P. Høyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies.
In 19th IEEE Annual Conference on Computational Complexity, 2004. Proceedings, pages 236–249, 2004.
[48] D. Collins and N. Gisin. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. Journal of Physics A-Mathematical and General, 37(5):1775–1788, 2004.
[49] D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu. Bell inequalities for arbitrarily high-dimensional systems. Physical Review Letters, 88(4):40404–40404, 2002.
[50] W. van Dam. Nonlocality & Communication Complexity. PhD thesis, University of Oxford, Department of Physics, 2000.
[51] W. van Dam. Implausible consequences of superstrong nonlocality, 2005. arXiv:quant-ph/0501159v1.
[52] W. van Dam, R. D. Gill, and P. D. Gr¨unwald. The statistical strength of nonlocality proofs.
IEEE Transactions on Information Theory, 51(8):2812–2835, 2005.
[53] J. Niel de Beaudrap. One-qubit fingerprinting schemes. Physical Review A, 69(2):022307, Feb 2004.
[54] J. Degorre, S. Laplante, and J. Roland. Simulating quantum correlations as a distributed sampling problem. Physical Review A, 72:062314, 2005.
[55] J. Degorre, S. Laplante, and J. Roland. Classical simulation of traceless binary observables on any bipartite quantum state. Physical Review A, 75:012309, 2007.
[56] D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. In Proceedings of the Royal Society of London, volume A439, pages 553–558, 1992.
[57] A.C. Doherty, Y.C. Liang, B. Toner, and S. Wehner. The quantum moment problem and bounds on entangled multi-prover games. In Proceedings of 23rd IEEE Conference on Com-putational Complexity, pages 199–210, 2008.
[58] J. Du, P. Zou, X. Peng, D. K. Oi, L. C. Kwek, and A. Ekert. Experimental quantum multimeter and one-qubit fingerprinting. Physical Review A, 74:042319, 2006.
[59] H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41–44, 1964.
[60] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47:777–780, 1935.
[61] M. Forster, S. Winkler, and S. Wolf. Distilling Non-Locality. Physical review letters, 102:120401, 2009.
[62] P. Frankl and V. R¨odl. Forbidden intersections. Transactions of the American Mathematical Society, 300(1):259–286, 1987.
[63] S. J. Freedman and J. F. Clauser. Experimental test of local hidden-variable theories. Physical Review Letters, 28(14):938–941, Apr 1972.
[64] E. F. Galv˜ao. Feasible quantum communication complexity protocol. Physical Review A, 65(1):012318, Dec 2001.
[65] D. Gavinsky. Classical interaction cannot replace a quantum message. In Proceedings of 40th ACM STOC, pages 95–102, 2008. quant-ph/0703215.
[66] D. Gavinsky. Classical interaction cannot replace quantum nonlocality. arXiv:0901.0956, 2008.
[67] D. Gavinsky, J. Kempe, I. Kerenidis, R. Raz, and R. de Wolf. Exponential separations for one-way quantum communication complexity, with applications to cryptography. In Proceedings of 39th ACM STOC, pages 516–525, 2007. quant-ph/0611209.
[68] D. Gavinsky, J. Kempe, O. Regev, and R. de Wolf. Bounded-error quantum state identifica-tion and exponential separaidentifica-tions in communicaidentifica-tion complexity. In Proceedings of 38th ACM STOC, pages 594–603, 2006. quant-ph/0511013.
[69] D. Gavinsky, J. Kempe, and R. de Wolf. Strengths and weaknesses of quantum fingerprinting.
In Proceedings of 21st IEEE Conference on Computational Complexity, pages 288–295, 2006.
quant-ph/0603173.
[70] D. Gavinsky, O. Regev, and R. de Wolf. Simultaneous communication protocols with quantum and classical messages. Chicago Journal of Theoretical Computer Science, 7, 2008. quant-ph/0807.2758.
[71] N. Gisin and B. Gisin. A local hidden variable model of quantum correlation exploiting the detection loophole. Physics Letters A, 260:323–327, 1999.
[72] D. M. Greenberger, M. Horne, and A. Zeilinger. Going beyond Bell’s theorem. In M. Kafatos, editor, Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, pages 69–72.
Kluwer Academic, 1989.
[73] D. Gross, S. T. Flammia, and J. Eisert. Most quantum states are too entangled to be useful as computational resources. Physical Review Letters, 102:190501, 2009.
[74] L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of 28th ACM STOC, pages 212–219, 1996. quant-ph/9605043.
[75] A. S. Holevo. Bounds for the quantity of information transmitted by a quantum commu-nication channel. Problemy Peredachi Informatsii, 9(3):3–11, 1973. English translation in Problems of Information Transmission, 9:177–183, 1973.
[76] R. T. Horn, S. A. Babichev, K.-P. Marzlin, A. I. Lvovsky, and B. C. Sanders. Single-qubit optical quantum fingerprinting. Physical Review Letters, 95:150502, 2005.
[77] Juraj Hromkoviˇc. Communication complexity and parallel computing. Springer-Verlag, New York, 1997.
[78] B. Kalyanasundaram and G. Schnitger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545–557, 1992. Earlier version in Structures’87.
[79] L. A. Khalfi and B. S. Tsirelson. Quantum and quasi-classical analogs of Bell inequalities.
In P. Lahti and P. Mittelstaedt, editors, Symposium on the Foundations of Modern Physics, pages 441–460. World Scientific, Singapore, 1985.
[80] H. Klauck, R. ˇSpalek, and R. de Wolf. Quantum and classical strong direct product theo-rems and optimal time-space tradeoffs. SIAM Journal on Computing, 36(5):1472–1493, 2007.
Earlier version in FOCS’03. quant-ph/0402123.
[81] D. E. Knuth. Combinatorial matrices. In Selected Papers on Discrete Mathematics, volume 106 of CSLI Lecture Notes. Stanford University, 2003.
[82] I. Kremer. Quantum communication. Master’s thesis, Hebrew University, Computer Science Department, 1995.
[83] E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997.
[84] N. Linial and A. Shraibman. Lower bounds in communication complexity based on factor-ization norms. In Proceedings of 39th ACM STOC, pages 699–708, 2007.
[85] I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legr´e, and N. Gisin. Distribution of time-bin entangled qubits over 50 km of optical fiber. Physical Review Letters, 93:180502, 2004.
[86] Ll. Masanes, A. Acin, and N. Gisin. General properties of nonsignaling theories. Physical Review A, 73:012112, 2006.
[87] S. Massar. Nonlocality, closing the detection loophole, and communication complexity. Phys-ical Review A, 65:032121, 2002.
[88] S. Massar. Quantum fingerprinting with a single particle. Physical Review A, 71:012310, 2005.
[89] S. Massar, D. Bacon, N. J. Cerf, and R. Cleve. Classical simulation of quantum entanglement without local hidden variables. Physical Review A, 63(5):052305, Apr 2001.
[90] S. Massar, S. Pironio, J. Roland, and B. Gisin. Bell inequalities resistant to detector ineffi-ciency. Phys. Rev. A, 66(5):052112, Nov 2002.
[91] D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe. Bell inequality violation with two remote atomic qubits. Physical Review Letters, 100:150404, 2008.
[92] T. Maudlin. Bell’s inequality, information transmission, and prism models. In D. Hull, M. Forbes, and K. Okruhlik, editors, PSA: Proceedings of the Biennial Meeting of the Phi-losophy of Science Association, volume 1, pages 404–417, 1992.
[93] N. D. Mermin. Simple unified form for the major no-hidden-variables theorems. Physical Review Letters, 65:3373–3376, 1990.
[94] N. D. Mermin. What’s wrong with these elements of reality? Physics Today, 43:9–11, 1990.
[95] N. D. Mermin. Hidden variables and the two theorems of John Bell. Reviews of Modern Physics, 65(3):803–815, 1993.
[96] D. L. Moehring, M. J. Madsen, B. B. Blinov, and C.Monroe. Experimental Bell inequality violation with an atom and a photon. Physical Review Letters, 93:090410, 2004.
[97] M. Navascues, S. Pironio, and A. Acin. Bounding the set of quantum correlations. Physical Review Letters, 98(1):10401, 2007.
[98] M. Navascues, S. Pironio, and A. Ac´ın. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10(073013):073013, 2008.
[99] A. Nayak. Optimal lower bounds for quantum automata and random access codes. In Proceedings of 40th IEEE FOCS, pages 369–376, 1999. quant-ph/9904093.
[100] A. Nayak and J. Salzman. On communication over an entanglement-assisted quantum chan-nel. In Proceedings of 34th ACM STOC, pages 698–704, 2002.
[101] I. Newman. Private vs. common random bits in communication complexity. Information Processing Letters, 39(2):67–71, 1991.
[102] I. Newman and M. Szegedy. Public vs. private coin flips in one round communication games.
In Proceedings of 28th ACM STOC, pages 561–570, 1996.
[103] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cam-bridge University Press, 2000.
[104] J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature, 403:515, 2000.
[105] R. Paturi. On the degree of polynomials that approximate symmetric Boolean functions. In Proceedings of 24th ACM STOC, pages 468–474, 1992.
[106] P. M. Pearle. Hidden-variable example based upon data rejection. Physical Review D, 2:1418, 1970.
[107] D. Perez-Garcia, M.M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge. Unbounded viola-tion of tripartite Bell inequalities. Communicaviola-tions in Mathematical Physics, 279:455, 2008.
arXiv:quant-ph/0702189.
[108] S. Popescu and D. Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24:379, 1994.
[109] A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche. Step-by-step engineered multiparticle entanglement. Science, 288:5473, 2000.
[110] R. Raz. Exponential separation of quantum and classical communication complexity. In Proceedings of 31st ACM STOC, pages 358–367, 1999.
[111] A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385–390, 1992.
[112] A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Sciences, mathematics, 67(1):159–176, 2003. quant-ph/0204025.
[113] O. Regev and B. Toner. Simulating quantum correlations with finite communication. In Pro-ceedings of 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07), pages 384–394, 2007.
[114] T. J. Rivlin and E. W. Cheney. A comparison of uniform approximations on an interval and a finite subset thereof. SIAM Journal on Numerical Analysis, 3(2):311–320, 1966.
[115] M. A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W. M. Itano, C. Monroe, and D. J.
Wineland. Experimental violation of a Bell’s inequality with efficient detection. Nature, 409:791, 2001.
[116] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A.
Turchette, W. M. Itano, D. J. Wineland, and C. Monroe. Experimental entanglement of four particles. Nature, 404:256, 2000.
[117] V. B. Scholz and R. F. Werner. Tsirelson’s Problem. Arxiv preprint arXiv:0812.4305, 2008.
[118] E. Schr¨odinger. Discussion of probability relations between separated systems. Proceedings of the Cambridge Philosophical Society, 31:555–563, 1935.
[119] E. Schr¨odinger. Probability relations between separated systems. Proceedings of the Cam-bridge Philosophical Society, 32:446–451, 1936.
[120] B. Schumacher. Quantum coding. Physical Review A, 51(4):2738–2747, Apr 1995.
[121] C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379–423, 623–656, 1948.
[122] A. Sherstov. The pattern matrix method for lower bounds on quantum communication. In Proceedings of 40th ACM STOC, pages 85–94, 2008.
[123] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, 1997. Earlier version in FOCS’94. quant-ph/9508027.
[124] M. Steiner. Towards quantifying non-local information transfer: finite-bit non-locality.
Physics Letters A, 270:239–244, 2000. arXiv:quant-ph/9902014.
[125] R. T. Thew, A. Ac´ın, H. Zbinden, and N. Gisin. Bell-type test of energy-time entangled qutrits. Physical Review Letters, 93(1):010503, Jul 2004.
[126] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. Violation of Bell inequalities by photons more than 10 km apart. Physical Review Letters, 81:3563, 1998.
[127] B. F. Toner and D. Bacon. Communication cost of simulating Bell correlations. Physical Review Letters, 91(18):187904, Oct 2003.
[128] P. Trojek, C. Schmid, M. Bourennane, C. Brukner, M. ˙Zukowski, and H. Weinfurter. Exper-imental quantum communication complexity. Physical Review A, 75:050305(R), 2005.
[129] B. S. Tsirelson. Quantum analogues of the Bell inequalities. the case of two spatially separated domains. Journal of Soviet Mathematics, 36:557–570, 1987.
[130] A. Vaziri, G. Weihs, and A. Zeilinger. Experimental two-photon, three-dimensional entangle-ment for quantum communication. Physical Review Letters, 89(24):240401, Nov 2002.
[131] S. Wehner, M. Christandl, and A.C. Doherty. Lower bound on the dimension of a quantum system given measured data. Physical Review A, 78(6), 2008.
[132] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger. Violation of Bell’s inequality under strict Einstein locality conditions. Physical Review Letters, 81:5039, 1998.
[133] R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Physical Review A, 40(8):4277–4281, Oct 1989.
[134] R. F. Werner and M. M. Wolf. All-multipartite Bell-correlation inequalities for two dichotomic observables per site. Physical Review A, 64:032112, 2001.
[135] R. F. Werner and M. M. Wolf. Bell inequalities and entanglement. Quantum Information and Computation, 1(3):1–25, 2001. Arxiv preprint quant-ph/0107093.
[136] A. C-C. Yao. Some complexity questions related to distributive computing. In Proceedings of 11th ACM STOC, pages 209–213, 1979.
[137] A. C-C. Yao. Quantum circuit complexity. In Proceedings of 34th IEEE FOCS, pages 352–360, 1993.
[138] A. C-C. Yao. On the power of quantum fingerprinting. In Proceedings of 35th ACM STOC, pages 77–81, 2003.
[139] J. Zhang, X.-H. Bao, T.-Y. Chen, T. Yang, A. Cabello, and J.-W. Pan. Experimental quantum guess my number protocol using multiphoton entanglement. Physical Review A, 75:022302, 2007.
[140] Z. Zhao, T. Yang, Y.-A. Chen, A.-N. Zhang, M. ˙Zukowski, and J.-W. Pan. Experimental violation of local realism by four-photon Greenberger-Horne-Zeilinger entanglement. Physical Review Letters, 91(18):180401, Oct 2003.
[141] M. ˙Zukowski and ˇC. Brukner. Bell’s theorem for general n-qubit states. Physical Review Letters, 88(21):210401, May 2002.
A Nayak’s Proof of a Consequence of Holevo’s Bound
Here we prove that if we are encoding n bits in d-dimensional quantum states, then the average recovery probability is at most d/2n. Therefore, an exact procedure requires d≥ 2n, and thus at least n qubits.
Let ρ0, . . . , ρ2n−1be the d-dimensional states that encode the elements of{0, 1}n(which we iden-tify with {0, 1, . . . , 2n− 1} in the obvious way). Let E0, . . . , E2n−1 be the measurement operators applied for decoding (they sum to the d-dimensional identity). The probability of successfully re-covering x∈ {0, 1}nfrom its encoding is Tr(Exρx). Therefore, we can bound the success probability for a uniformly random x∈ {0, 1}n by
1 2n
2n−1
X
x=0
Tr(Exρx) ≤ 1 2n
2n−1
X
x=0
Tr(Ex)
= 1
2nTr
2n−1
X
x=0
Ex
!
= 1
2nTr(I)
= d
2n. (30)
The first inequality follows because the density operator ρxis positive semi-definite and has trace 1, therefore it can be unitarily diagonalized: U∗ρxU = D, where D is diagonal with diagonal entries that are non-negative and sum to 1. Because the trace is invariant under cyclic permutations of the matrices, we now have Tr(Exρx) = Tr(U∗ExU U∗ρxU ) = Tr(U∗ExU D)≤ Tr(U∗ExU I) = Tr(Ex).
B Rectangles and the Lower Bound for Distributed Deutsch-Jozsa
Separations between quantum and classical communication complexity always require two things:
an efficient quantum protocol for some problem, and a lower bound on the communication of all classical protocols solving that same problem. In this appendix we will give some tools for lower bounding classical communication complexity, leading eventually to the lower bound on classical protocols for the Distributed Deutsch-Jozsa problem that we mentioned in Section 3.4.
B.1 Rectangles
Consider some communication complexity problem f : X × Y → {0, 1}, where Alice starts with an input x ∈ X and Bob starts with an input y ∈ Y . We start by introducing the crucial combinatorial notion for classical lower bounds. A rectangle is a set R ⊆ X × Y that is of the
form R = A× B with A ⊆ X and B ⊆ Y . For example, if n = 2 and A = {00, 01}, B = {01, 10}
then R = A× B = {(00, 01), (00, 10), (01, 01), (01, 10))} is a rectangle. The following result is a
then R = A× B = {(00, 01), (00, 10), (01, 01), (01, 10))} is a rectangle. The following result is a