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arXiv:1310.2689 · Physical Review A

Two-setting multisite Bell inequalities for loophole-free tests with up to 50 % loss

M. D. Reid

Published Jun 18, 2013• 2 citations
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Abstract

We consider Bell experiments with $N$ spatially separated qubits where loss is present and restrict ourselves to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid, and Drummond [E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure $W={\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}$ of overall detection efficiency, where ${\ensuremath{\eta}}_{k}$ is the efficiency at site $k$. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided ${\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}>{2}^{(2\ensuremath{-}N)}$, which implies a symmetric threshold efficiency of $\ensuremath{\eta}\ensuremath{\rightarrow}50%$, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below $0.5$, provided $N>3$.

DOI
10.1103/physreva.87.062108

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We consider Bell experiments with NNN spatially separated qubits where loss is present and restrict ourselves to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid, and Drummond [E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure W=\ensuremath∏k=1N\ensuremathηkW={\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}W=\ensuremath∏k=1N​\ensuremathηk​ of overall detection efficiency, where \ensuremathηk{\ensuremath{\eta}}_{k}\ensuremathηk​ is the efficiency at site kkk. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided \ensuremath∏k=1N\ensuremathηk>2(2\ensuremath−N){\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}>{2}^{(2\ensuremath{-}N)}\ensuremath∏k=1N​\ensuremathηk​>2(2\ensuremath−N), which implies a symmetric threshold efficiency of \ensuremathη\ensuremath→50\ensuremath{\eta}\ensuremath{\rightarrow}50%\ensuremathη\ensuremath→50, as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below 0.50.50.5, provided N>3N>3N>3.
We prove that the threshold detection efficiency for a loophole-free Bell experiment using an n-qubit Greenberger-Horne-Zeilinger state and the correlations appearing in the n-partite Mermin inequality is n/(2n-2). If the detection efficiency is equal to or lower than this value, there are local hidden variable models that can simulate all the quantum predictions. If the detection efficiency is above this value, there is no local hidden variable model that can simulate all the quantum predictions.
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We examine the NNN-partite quaternion and octonion dichotomic Bell inequalities derived by Vogel and Shchukin utilizing the square identities of Euler and Degen, which apply to experiments with MMM settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for $N\ensuremath{\ge}3$ and $2\ensuremath{\le}M\ensuremath{\le}8$. Violations are robust with respect to loss, the threshold detection efficiency being $\ensuremath{\eta}>{2}^{\frac{2}{N}\ensuremath{-}1}$ for all MMM, implying violations for efficiencies as low as $\ensuremath{\eta}\ensuremath{\sim}50%$ at each site, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$.
Bell's test, initially devised to distinguish quantum theory from local hidden variable models through {violations of local bounds}, is also a common tool for detecting entanglement. For this purpose, one can assume the quantum description of devices and use available information to strengthen the bound for separable states, which may go beyond the local bound, enabling more efficient entanglement detection. Here we present a systematic approach for strengthening Bell inequalities for qubit systems, using Mermin-Klyshko-Bell inequalities as examples, by considering measurement devices that are coarsely calibrated only by their ability to generate nonlocal correlations without requiring precise quantum characterization. In the case of bipartite and tripartite systems, we derive trade-offs between upper bounds for separable states and general states in terms of structure functions to optimize the entanglement detection. We then strengthen nnn-partite Bell inequalities for the detection of states exhibiting a diversity of entanglement structures such as genuine multipartite entanglement. For general Bell scenarios with some measurements characterized, we demonstrate that entanglement can also be detected with some local correlations by exploiting the Navascués-Pironio-Acín hierarchy of tests.
Bell inequality is a vital tool to detect the nonlocal correlations, but the construction of it for multipartite systems is still a complicated problem. In this work, inspired via a decomposition of (n+1)(n+1)(n+1)-partite Bell inequalities into nnn-partite ones, we present a generalized iterative formula to construct nontrivial (n+1)(n+1)(n+1)-partite ones from the nnn-partite ones. Our iterative formulas recover the well-known Mermin-Ardehali-Belinsk\ifmmode \breve{\imath}\else \u{\i}\fi{}-Klyshko (MABK) and other families in the literature as special cases. Moreover, a family of ``dual-use'' Bell inequalities is proposed, in the sense that for the generalized Greenberger-Horne-Zeilinger states these inequalities lead to the same quantum violation as the MABK family and, at the same time, the inequalities are able to detect the nonlocality in the entire entangled region. Furthermore, we present a generalization of the I3322 inequality to any nnn-partite case which is still tight, and of the 46 \ifmmode \acute{S}\else \'{S}\fi{}liwa's inequalities to the four-partite tight ones, by applying our iteration method to each inequality and its equivalence class.
Self-testing is a device-independent technique that allows for verification of unknown quantum systems only when observed correlations achieve some quantum supremum. In this paper, we present self-testing schemes for a class of multipartite Bell inequalities with a specific structure, which we refer to as the unit. Our research is based on the graph-theoretic approach, where correlations are characterized via Lov\'asz theta body in the framework of exclusivity graphs. The derived statements are subsequently applied to prove the self-testability of NNN-partite Mermin-Ardehali-Belinski\ifmmode \check{i}\else \v{i}\fi{}-Klyshko (MABK) family of Bell inequalities. In particular, we take special cases to demonstrate that the maximal violation of the multipartite MABK inequality is a self-test for the corresponding multipartite Greenberger-Horne-Zeilinger (GHZ) state. Moreover, we propose a device-independent verification algorithm for distributed blind quantum computation in the noisy intermediate-scale quantum era combined with multipartite GHZ state self-testing, where multiple quantum servers are involved and work in collaboration to reduce the burden on a single server.