Abstract
Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if the local Hilbert-space dimension is at least d. This establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing a lower bound on a new generalization of Grothendieck’s constant.
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Acknowledgments
We thank Ronald de Wolf for discussions about the Hidden Matching problem and we thank Vid Stojevic for pointing out a problem with an earlier version of Lemma 1.
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Communicated by M.B. Ruskai
Supported by Vici grant 639-023-302 from the Netherlands Organization for Scientific Research (NWO), by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848, and EU QCS grant.
Part of this work was completed at CWI. Supported by NWO Vici grant 639-023-302, by the European Commission under the Integrated Project QAP funded by the IST directorate as Contract Number 015848.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Briët, J., Buhrman, H. & Toner, B. A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement. Commun. Math. Phys. 305, 827–843 (2011). https://doi.org/10.1007/s00220-011-1280-3
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DOI: https://doi.org/10.1007/s00220-011-1280-3