A generalized Grothendieck inequality and entanglement in XOR games

Abstract

Suppose Alice and Bob make local two-outcome measurements on a shared entangled state. For any d, we show that there are correlations that can only be reproduced if the local dimension is at least d. This resolves a conjecture of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and 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 the first lower bounds on a new generalization of Grothendieck's constant.

arXiv commentVersion submitted to QIP on 10-20-08. See also arxiv:0812.1572 for related results, obtained independently

References in corpus (5)

• Randomizing quantum states: Constructions and applications
• Testing the Hilbert space dimension
• Grothendieck's constant and local models for noisy entangled quantum states
• A lower bound on the dimension of a quantum system given measured data
• Simulating Quantum Correlations with Finite Communication

Cited by in corpus (9)

• Non-locality and Communication Complexity
• Device-independent quantum key distribution secure against collective attacks
• Large violation of Bell inequalities with low entanglement
• Lower bounds on the entanglement needed to play XOR non-local games
• Grothendieck inequalities for semidefinite programs with rank constraint
• The positive semidefinite Grothendieck problem with rank constraint
• Optimisation of Bell inequalities with invariant Tsirelson bound
• Better Non-Local Games from Hidden Matching
• A Lower Bound on the Value of Entangled Binary Games