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arXiv:quant-ph/0410139 · Physical Review A

Multipartite nonlocal quantum correlations resistant to imperfections

Harry Buhrman, Peter Høyer, Hein Röhrig, Serge Massar

Published Jan 18, 2006• 7 citations
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Abstract

We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of superluminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in Einstein-Podolsky-Rosen (EPR) experiments in the presence of noise. We apply our method to an example involving $n$ parties sharing a Greenberger-Horne-Zeilinger-type state on which they carry out local measurements. For this example, we show that for local hidden-variable theories to reproduce the quantum correlations, the amount of superluminal classical communication $c$ and the detector efficiency $\ensuremath{\eta}$ are constrained by $\ensuremath{\eta}{2}^{\ensuremath{-}c∕n}\ensuremath{\leqslant}O({n}^{\ensuremath{-}1∕6})$. This result holds even if the classical models are allowed to make an error with constant probability.

DOI
10.1103/physreva.73.012321

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We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of superluminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in Einstein-Podolsky-Rosen (EPR) experiments in the presence of noise. We apply our method to an example involving nnn parties sharing a Greenberger-Horne-Zeilinger-type state on which they carry out local measurements. For this example, we show that for local hidden-variable theories to reproduce the quantum correlations, the amount of superluminal classical communication ccc and the detector efficiency \ensuremathη\ensuremath{\eta}\ensuremathη are constrained by \ensuremathη2\ensuremath−c∕n\ensuremath⩽O(n\ensuremath−1∕6)\ensuremath{\eta}{2}^{\ensuremath{-}c∕n}\ensuremath{\leqslant}O({n}^{\ensuremath{-}1∕6})\ensuremathη2\ensuremath−c∕n\ensuremath⩽O(n\ensuremath−1∕6). This result holds even if the classical models are allowed to make an error with constant probability.
We use techniques for lower bounds on communication to derive necessary conditions (in terms of detector efficiency or amount of superluminal communication) for being able to reproduce the quantum correlations occurring in Einstein-Podolsky-Rosen-type experiments with classical local hidden-variable theories. As an application, we consider n parties sharing a Greenberger-Horne-Zeilinger-type state and show that the amount of superluminal classical communication required to reproduce the correlations is at least n(log((2)n-3) bits and the maximum detector efficiency eta(*) for which the resulting correlations can still be reproduced by a local hidden-variable theory is upper bounded by eta(*)</=8/n and thus decreases with n.
To reproduce in a local hidden variables theory correlations that violate Bell inequalities, communication must occur between the parties. We show that the amount of violation of a Bell inequality imposes a lower bound on the average communication needed to produce these correlations. Moreover, for every probability distribution there exists an optimal inequality for which the degree of violation gives the minimal average communication. As an example, to produce using classical resources the correlations that maximally violate the Clauser-Horne-Shimony-Holt inequality, $\sqrt{2}\ensuremath{-}1\ensuremath{\simeq}0.4142$ bits of communication are necessary and sufficient. For Bell tests performed on two entangled states of dimension d> 3d>~3d> 3 where each party has the choice between two measurements, our results suggest that more communication is needed to simulate outcomes obtained from certain nonmaximally entangled states than maximally entangled ones.
In this paper, we investigate the communication cost of reproducing Einstein-Podolsky-Rosen (EPR) steering correlations arising from bipartite quantum systems. We characterize the set of bipartite quantum states which admits a local hidden state model augmented with c bits of classical communication from an untrusted party (Alice) to a trusted party (Bob). In case of one bit of information (c = 1), we show that this set has a nontrivial intersection with the sets admitting a local hidden state and a local hidden variables model for projective measurements. On the other hand, we find that an infinite amount of classical communication is required from an untrusted Alice to a trusted Bob to simulate the EPR steering correlations produced by a two-qubit maximally entangled state. It is conjectured that a state-of-the-art quantum experiment would be able to falsify two bits of communication this way.
The nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs. The maximal number m of groups and the minimal number k of broadcasting parties that allow for the reproduction of a given set of correlations quantify their multipartite nonlocal content. We show how upper bounds on m and lower bounds on k can be computed from the violation of the Mermin-Svetlichny inequalities. While n-partite Greenberger-Horne-Zeilinger states violate these inequalities maximally, we find that W states violate them only by a very small amount.
The Einstein-Podolsky-Rosen (EPR) nonlocality puzzle has been recognized as one of the most important unresolved issues in the foundational aspects of quantum mechanics. We show that the problem is more or less entirely resolved, if the quantum correlations are calculated directly from local quantities, which preserve the phase information in the quantum system. We assume strict locality for the probability amplitudes instead of local realism for the outcomes and calculate an amplitude correlation function. Then the experimentally observed correlation of outcomes is calculated from the square of the amplitude correlation function. Locality of amplitudes implies that measurement on one particle does not collapse the companion particle to a definite state. Apart from resolving the EPR puzzle, this approach shows that the physical interpretation of apparently “nonlocal” effects, such as quantum teleportation and entanglement swapping, are different from what is usually assumed. Bell-type measurements do not change distant states. Yet the correlations are correctly reproduced, when measured, if complex probability amplitudes are treated as the basic local quantities. As examples, we derive the quantum correlations of two-particle maximally entangled states and the three-particle Greenberger-Horne-Zeilinger entangled state.
It is well known that entangled quantum states are nonlocal: the corrrelations between local measurements carried out on these states cannot be reproduced by local hidden variable models. Svetlichny, followed by others, showed that multipartite quantum states are more nonlocal than bipartite ones in the sense that even some nonlocal classical models with (super-luminal) communication between some of the parties cannot reproduce the quantum correlations. Here we study in detail the kinds of nonlocality present in quantum states. More precisely, we enquire what kinds of classical communication patterns cannot reproduce quantum correlations. By studying the extremal points of the space of all multiparty probability distributions, in which all parties can make one of a pair of measurements each with two possible outcomes, we find a necessary condition for classical nonlocal models to reproduce the statistics of all quantum states. This condition extends and generalizes work of Svetlichny and others in which it was showed that a particular class of classical nonlocal models, the ``separable'' models, cannot reproduce the statistics of all multiparticle quantum states. Our condition shows that the nonlocality present in some entangled multiparticle quantum states is much stronger than previously thought. We also study the sufficiency of our condition.