A Lower Bound on Entanglement-Assisted Quantum Communication Complexity

  • Ashley Montanaro
  • Andreas Winter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed inputs can be turned into a quantum communication protocol. As an application of this bound, we give a very simple proof of the statement that almost all Boolean functions on n + n bits have communication complexity linear in n, even in the presence of unlimited entanglement.


  1. 1.
    Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing 1, 47–79 (2005) quant-ph/0303041MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambainis, A., Schulman, L.J., Ta-Shma, A., Vazirani, U., Wigderson, A.: The quantum communication complexity of sampling. SIAM J. Comput. 32, 1570–1585 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Spencer, J.: The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)MATHGoogle Scholar
  4. 4.
    Buhrman, H., deWolf, R.: Communication complexity lower bounds by polynomials. In: Proc. CCC 2001, pp. 120–130 (2001) cs.CC/9910010Google Scholar
  5. 5.
    Cleve, R., van Dam, W., Nielsen, M., Tapp, A.: Quantum entanglement and the communication complexity of the inner product function. In: Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications, pp.61–74, February 17-20 (1998) quant-ph/9708019Google Scholar
  6. 6.
    Fannes, M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31, 291–294 (1973)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    van Dam, W., Hayden, P.: Renyi-entropic bounds on quantum communication (2002) quant-ph/0204093Google Scholar
  8. 8.
    Gavinsky, D., Kempe, J., de Wolf, R.: Strengths and weaknesses of quantum fingerprinting. In: Proc. CCC 2006, pp. 288–298 (2006) quant-ph/0603173Google Scholar
  9. 9.
    Hausladen, P., Jozsa, R., Schumacher, B., Westmoreland, M., Wootters, W.: Classical information capacity of a quantum channel. Phys. Rev. A 54(3), 1869–1876 (1996)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Holevo, A.S.: Bounds for the quantity of information transmittable by a quantum communications channel. Problemy Peredachi Informatsii 9(3), 3–11 (1973) English translation Problems of Information Transmission 9, 177–183 (1973)Google Scholar
  11. 11.
    Horn, R.A., Johnson, C.: Matrix analysis. Cambridge University Press, Cambridge (1996)Google Scholar
  12. 12.
    Jozsa, R., Schlienz, J.: Distinguishability of states and von Neumann entropy. Phys. Rev. A 62 012301 (2000) quant-ph/9911009Google Scholar
  13. 13.
    Klauck, H.: Lower bounds for quantum communication complexity. In: Proc. FOCS 2001, pp. 288–297 (2001) quant-ph/0106160Google Scholar
  14. 14.
    Kremer, I.: Quantum communication. Master’s thesis, Hebrew University (1995)Google Scholar
  15. 15.
    Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  16. 16.
    Linial, N., Shraibman, A.: Learning complexity vs. communication complexity. Manuscript (2006), http://www.cs.huji.ac.il/~nati/PAPERS/lcc.pdf
  17. 17.
    Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. In: Proc. STOC 2007 (to appear, 2007), http://www.cs.huji.ac.il/~nati/PAPERS/quantcc.pdf
  18. 18.
    Nayak, A., Salzman, J.: On communication over an entanglement-assisted quantum channel. In: Proc. STOC 2002, pp. 698–704 (2002) quant-ph/0206122Google Scholar
  19. 19.
    Nielsen, M.A.: Quantum information theory. PhD thesis, University of New Mexico, Albuquerque (1998) quant-ph/0011036Google Scholar
  20. 20.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  21. 21.
    Raz, R.: Exponential separation of quantum and classical communication complexity, In: Proc. STOC 1999, pp. 358–367 (1999)Google Scholar
  22. 22.
    Razborov, A.A.: Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Science 67, 159–176 (2003) quant-ph/ 0204025MathSciNetGoogle Scholar
  23. 23.
    Rényi, A.: Probability theory. North-Holland, Amsterdam (1970)Google Scholar
  24. 24.
    de Wolf, R.: Quantum communication and complexity. Theoretical Computer Science 287(1), 337–353 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Yao, A.: Some complexity questions related to distributive computing. In: Proc. STOC 1979, pp. 209–213 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ashley Montanaro
    • 1
  • Andreas Winter
    • 2
  1. 1.Department of Computer Science, University of Bristol, Bristol, BS8 1UB, U.K 
  2. 2.Department of Mathematics, University of Bristol, Bristol, BS8 1TW, U.K 

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