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arXiv:1010.3762 · Physical Review A

Detecting full N -particle entanglement in arbitrarily-high-dimensional systems with Bell-type inequalities

Jing‐Ling Chen, Dong-Ling Deng, Hong-Yi Su, Chunfeng Wu, C. H. Oh

Published Feb 18, 2011• 39 citations
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Abstract

We derive a set of Bell-type inequalities for arbitrarily-high-dimensional systems based on the assumption of partial separability in the hybrid local-nonlocal hidden-variable model. Partially entangled states would not violate the inequalities, and thus, upon violation, these Bell-type inequalities are sufficient conditions to detect the full $N$-particle entanglement and invalidity of the hybrid local-nonlocal hidden-variable description.

DOI
10.1103/physreva.83.022316

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We derive a set of Bell-type inequalities for arbitrarily-high-dimensional systems based on the assumption of partial separability in the hybrid local-nonlocal hidden-variable model. Partially entangled states would not violate the inequalities, and thus, upon violation, these Bell-type inequalities are sufficient conditions to detect the full NNN-particle entanglement and invalidity of the hybrid local-nonlocal hidden-variable description.
Based on the assumption of partial separability, we derive Bell-type inequalities for NNN ddd-dimensional (qudit) systems, which can detect experimentally genuine multipartite entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. We also introduce the concept ``resolution'' to measure the strength of these inequalities to distinguish full entanglement from partial entanglement when white noise is added to the system.
We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e., that the N-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experi-mentally accessible sufficient conditions for full N-particle entanglement, i.e., for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne-Zeilinger states.
We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-locality has been presented.
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.
Loophole-free violations of Bell inequalities imply that at least one of the assumptions behind local hidden-variable theories must fail. Here, we show that, if only one fails, then it has to fail completely, therefore excluding models that partially constrain freedom of choice or allow for partial retrocausal influences, or allow partial instantaneous actions at a distance. Specifically, we show that (i) any hidden-variable theory with outcome independence (OI) and arbitrary joint relaxation of measurement independence (MI) and parameter independence (PI) can be experimentally excluded in a Bell-like experiment with many settings on high-dimensional entangled states, and (ii) any hidden-variable theory with MI, PI and arbitrary relaxation of OI can be excluded in a Bell-like experiment with many settings on qubit-qubit entangled states. Loophole-free violations of Bell inequalities imply that at least one of the assumptions formalising local hidden-variable theories must fail. Here, the authors show that, if only one is false, then it has to fail completely, therefore excluding models that allow partial instantaneous actions at a distance, partially constrain freedom of choice or allow for partial retrocausal influences.