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arXiv:quant-ph/0606138 · Physical Review A

Grothendieck’s constant and local models for noisy entangled quantum states

Antonio Acín, Nicolas Gisin, Ben Toner

Published Jun 9, 2006• 200 citations
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Abstract

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states ${\ensuremath{\rho}}_{p}^{W}=p\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{-}}⟩⟨{\ensuremath{\psi}}^{\ensuremath{-}}\ensuremath{\mid}+(1\ensuremath{-}p)\mathbb{1}∕4$, we show that there is a local model for projective measurements if and only if $p\ensuremath{\leqslant}1∕{K}_{G}(3)$, where ${K}_{G}(3)$ is Grothendieck's constant of order 3. Known bounds on ${K}_{G}(3)$ prove the existence of this model at least for $p\ensuremath{\lesssim}0.66$, quite close to the current region of Bell violation, $p\ensuremath{\sim}0.71$. We generalize this result to arbitrary quantum states.

DOI
10.1103/physreva.73.062105

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We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states \ensuremathρpW=p\ensuremath∣\ensuremathψ\ensuremath−⟩⟨\ensuremathψ\ensuremath−\ensuremath∣+(1\ensuremath−p)1∕4{\ensuremath{\rho}}_{p}^{W}=p\ensuremath{\mid}{\ensuremath{\psi}}^{\ensuremath{-}}⟩⟨{\ensuremath{\psi}}^{\ensuremath{-}}\ensuremath{\mid}+(1\ensuremath{-}p)\mathbb{1}∕4\ensuremathρpW​=p\ensuremath∣\ensuremathψ\ensuremath−⟩⟨\ensuremathψ\ensuremath−\ensuremath∣+(1\ensuremath−p)1∕4, we show that there is a local model for projective measurements if and only if p\ensuremath⩽1∕KG(3)p\ensuremath{\leqslant}1∕{K}_{G}(3)p\ensuremath⩽1∕KG​(3), where KG(3){K}_{G}(3)KG​(3) is Grothendieck's constant of order 3. Known bounds on KG(3){K}_{G}(3)KG​(3) prove the existence of this model at least for p\ensuremath≲0.66p\ensuremath{\lesssim}0.66p\ensuremath≲0.66, quite close to the current region of Bell violation, p\ensuremath∼0.71p\ensuremath{\sim}0.71p\ensuremath∼0.71. We generalize this result to arbitrary quantum states.
We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state ρ=v|ψ−⟩⟨ψ−|+(1−v)14 via a local hidden variable (LHV) model, where |ψ−⟩ denotes the singlet state. We show analytically that these correlations are local for v=999×689×10−6 cos2⁡(π/50)≃0.6829 . In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant KG(3)≤1/v≃1.4644 . We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for v≃0.4553 . The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.
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.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
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 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 a lower bound on a new generalization of Grothendieck’s constant.
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.