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International Colloquium on Automata, Languages, and Programming

ICALP 2012: Automata, Languages, and Programming pp 617–628Cite as

Classical and Quantum Partition Bound and Detector Inefficiency

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7391))

Abstract

We study randomized and quantum efficiency lower bounds in communication complexity. These arise from the study of zero-communication protocols in which players are allowed to abort. Our scenario is inspired by the physics setup of Bell experiments, where two players share a predefined entangled state but are not allowed to communicate. Each is given a measurement as input, which they perform on their share of the system. The outcomes of the measurements should follow a distribution predicted by quantum mechanics; however, in practice, the detectors may fail to produce an output in some of the runs. The efficiency of the experiment is the probability that neither of the detectors fails.

When the players share a quantum state, this leads to a new bound on quantum communication complexity (eff*) that subsumes the factorization norm. When players share randomness instead of a quantum state, the efficiency bound (eff), coincides with the partition bound of Jain and Klauck. This is one of the strongest lower bounds known for randomized communication complexity, which subsumes all the known combinatorial and algebraic methods including the rectangle (corruption) bound, the factorization norm, and discrepancy. The lower bound is formulated as a convex optimization problem. In practice, the dual form is more feasible to use, and we show that it amounts to constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for eff*). For one-way communication, we show that the quantum one-way partition bound is tight for classical communication with shared entanglement up to arbitrarily small error.

Full version available as arXiv:1203.4155 and ECCC TR12-023.

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Author information

Authors and Affiliations

  1. LRI, Université Paris-Sud 11, France

    Sophie Laplante & Virginie Lerays

  2. ULB, QuIC, Ecole Polytechnique de Bruxelles, Belgium

    Jérémie Roland

Authors
  1. Sophie Laplante
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  2. Virginie Lerays
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  3. Jérémie Roland
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Centre for Discrete Mathematics and its Applications, University of Warwick, Warwick, UK

    Artur Czumaj

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Kurt Mehlhorn

  3. Computer Laboratory, University of Cambridge, UK

    Andrew Pitts

  4. ETH Zurich, Switzerland

    Roger Wattenhofer

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© 2012 Springer-Verlag Berlin Heidelberg

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Laplante, S., Lerays, V., Roland, J. (2012). Classical and Quantum Partition Bound and Detector Inefficiency. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_52

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  • DOI: https://doi.org/10.1007/978-3-642-31594-7_52

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31593-0

  • Online ISBN: 978-3-642-31594-7

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